%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: Convexity.dvi %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips Convexity %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1994.06.30:1218 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N 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Fn(\014)1749 955 y Fm(\003)0 1026 y Fp(as)g(the)h(map)f Fn(N)k Fo(3)16 b Fn(n)f Fo(7\000)-8 b(!)15 b Fn(\013)544 1033 y Ft(n)571 1026 y Fp(+)586 1058 y Fo(\001)610 1026 y Fn(\014)638 1033 y Ft(n)681 1026 y Fo(2)g Fn(R)j Fp(then)g Fn(R)938 1008 y Ft(N)993 1026 y Fp(b)q(ecomes)g(a)f(semiring.)f(If)i Fn(r)f Fo(2)f Fn(R)i Fp(and)f(if)0 1105 y(w)o(e)g(denote)h(the)f(constan)o(t)g(map)g Fn(N)j Fo(\000)-8 b(!)15 b Fn(R)j Fp(with)f(v)m(alue)h Fn(r)h Fp(b)o(y)e Fn(r)1225 1112 y Fm(\003)1266 1105 y Fp(then)g Fn(R)f Fo(3)f Fn(r)i Fo(7!)e Fn(r)1608 1112 y Fm(\003)1647 1105 y Fo(2)h Fn(R)1734 1086 y Ft(N)0 1164 y Fp(is)g(a)h(homomorphis)o(m)c(of)k(semirings.)71 1224 y(Let)g Fn(M)22 b Fp(b)q(e)17 b(a)f(semimo)q(dule)f(\(in)h(the)h (sense)f(of)g([5],)g Fo(x)p Fp(1\))h(o)o(v)o(er)e(the)i(semiring)e Fn(R)p Fp(.)h(Then)g(w)o(e)0 1284 y(can)f(again)f(form)g Fn(M)390 1266 y Ft(N)429 1284 y Fp(.)g(Let)i Fn(\013)577 1291 y Fm(\003)614 1284 y Fo(2)e Fn(R)699 1266 y Ft(N)752 1284 y Fp(and)g Fn(\026)877 1266 y Fm(0)877 1296 y(\003)900 1284 y Fn(;)8 b(\026)952 1291 y Fm(\003)989 1284 y Fo(2)14 b Fn(M)1089 1266 y Ft(N)1127 1284 y Fp(,)h(and)f(de\014ne)h Fn(\013)1426 1291 y Fm(\003)1448 1284 y Fn(\026)1478 1291 y Fm(\003)1516 1284 y Fp(and)f Fn(\026)1641 1266 y Fm(0)1641 1296 y(\003)1672 1284 y Fp(+)8 b Fn(\026)1749 1291 y Fm(\003)0 1344 y Fp(p)q(oin)o(t)o(wise)k(|)h(as)g(in)f(the)i (case)f(of)g Fn(R)h Fp(|)f(then)g Fn(M)908 1326 y Ft(N)960 1344 y Fp(b)q(ecomes)f(a)i Fn(R)1231 1326 y Ft(N)1269 1344 y Fp(-semimo)q(dule.)c(As)k(b)q(efore)0 1403 y(w)o(e)g(let)h Fn(m)185 1410 y Fm(\003)208 1403 y Fp(,)f Fn(m)f Fo(2)h Fn(M)5 b Fp(,)16 b(b)q(e)e(the)h(constan)o(t)e(map)h(with)g(v)m(alue)h Fn(m)p Fp(.)f(Then)g Fn(M)19 b Fo(3)14 b Fn(m)g Fo(7!)g Fn(m)1597 1410 y Fm(\003)1633 1403 y Fo(2)g Fn(M)1733 1385 y Ft(N)0 1463 y Fp(is)i(a)h(homomorphis)o(m)c(of)k Fn(R)p Fp(-semimo)q(dules.)71 1523 y(If)e Fn(R)f Fp(is)g(a)h(partially) f(ordered)f(semiring)f(and)i Fn(M)20 b Fp(is)14 b(a)h(partially)e (ordered)h Fn(R)p Fp(-semimo)q(dule)0 1583 y(\(in)k(the)g(sense)f(of)h ([5],)f Fo(x)p Fp(1\))h(then)g(w)o(e)f(de\014ne)g Fn(\026)885 1565 y Fm(0)885 1595 y(\003)924 1583 y Fo(\024)f Fn(\026)1009 1590 y Fm(\003)1032 1583 y Fp(,)h(where)h Fn(\026)1239 1565 y Fm(0)1239 1595 y(\003)1261 1583 y Fn(;)8 b(\026)1313 1590 y Fm(\003)1352 1583 y Fo(2)16 b Fn(M)1454 1565 y Ft(N)1493 1583 y Fp(,)i(as)f Fn(\026)1617 1565 y Fm(0)1617 1595 y Ft(n)1660 1583 y Fo(\024)f Fn(\026)1745 1590 y Ft(n)0 1643 y Fp(for)h(all)h Fn(n)d Fo(2)h Fn(N)5 b Fp(.)18 b(This)e(mak)o(es)h Fn(M)637 1624 y Ft(N)693 1643 y Fp(a)h(partially)f (ordered)f(semimo)q(dule)g(o)o(v)o(er)h(the)g(partially)0 1702 y(ordered)e(semiring)g Fn(R)418 1684 y Ft(N)456 1702 y Fp(.)i(Ob)o(viously)l(,)e Fn(R)g Fo(3)f Fn(r)i Fo(7!)e Fn(r)955 1709 y Fm(\003)993 1702 y Fo(2)h Fn(R)1079 1684 y Ft(N)1133 1702 y Fp(and)i Fn(M)j Fo(3)14 b Fn(m)g Fo(7!)g Fn(m)1512 1709 y Fm(\003)1549 1702 y Fo(2)h Fn(M)1650 1684 y Ft(N)1705 1702 y Fp(are)0 1762 y(order)j(preserving.)f Fn(\026)419 1769 y Fm(\003)460 1762 y Fo(2)h Fn(M)564 1744 y Ft(N)622 1762 y Fp(is)h(said)f(to)h(b)q(e)h(b)q(ounded)e(if)h (there)g(are)g Fn(m)1428 1744 y Fm(0)1441 1762 y Fn(;)8 b(m)1507 1744 y Fm(00)1551 1762 y Fo(2)18 b Fn(M)25 b Fp(with)0 1822 y Fn(m)44 1804 y Fm(0)44 1834 y(\003)81 1822 y Fo(\024)13 b Fn(\026)163 1829 y Fm(\003)200 1822 y Fo(\024)h Fn(m)297 1804 y Fm(00)297 1834 y(\003)322 1822 y Fp(,)j(and)f(a)h(family)f Fo(f)p Fn(\026)699 1804 y Ft(i)699 1834 y Fm(\003)736 1822 y Fp(:)e Fn(i)g Fo(2)g Fn(I)t Fo(g)j Fp(is)f(called)g(uniformly)f(b)q(ounded)h(if)h(there)f (are)0 1882 y Fn(m)44 1864 y Fm(0)58 1882 y Fn(;)8 b(m)124 1864 y Fm(00)163 1882 y Fo(2)14 b Fn(M)22 b Fp(with)17 b Fn(m)438 1864 y Fm(0)438 1894 y(\003)474 1882 y Fo(\024)d Fn(\026)557 1864 y Ft(i)557 1894 y Fm(\003)594 1882 y Fo(\024)f Fn(m)690 1864 y Fm(00)690 1894 y(\003)732 1882 y Fp(for)j(all)g Fn(i)e Fo(2)g Fn(I)t Fp(.)71 1941 y(If)j Fn(M)23 b Fp(is)17 b(an)g Fn(R)p Fp(-prenormed)e Fn(R)p Fp(-semimo)q(dule)g(o)o(v)o(er)h(the)h(prenormed)e(semiring)h Fn(R)h Fp(and)g(if)0 2001 y Fo(k)g(k)g Fp(denotes)g(the)h(prenorm)d (with)i(v)m(alue)g(cone)h Fn(C)i Fp(\(see)e([5],)e Fo(x)p Fp(2\))i(then,)f(with)g Fn(\026)1506 2008 y Fm(\003)1544 2001 y Fo(2)e Fn(M)1645 1983 y Ft(N)1684 2001 y Fp(,)i(w)o(e)0 2061 y(denote)i(b)o(y)g Fo(k)p Fn(\026)290 2068 y Fm(\003)313 2061 y Fo(k)g Fp(the)g(map)f Fn(N)23 b Fo(3)c Fn(n)f Fo(7!)f(k)p Fn(\026)843 2068 y Ft(n)870 2061 y Fo(k)h(2)g Fn(C)t Fp(;)h(hence)f Fo(k)p Fn(\026)1232 2068 y Fm(\003)1255 2061 y Fo(k)h Fp(is)g(in)f Fn(C)1452 2043 y Ft(N)1489 2061 y Fp(.)h(This)g(mak)o(es)0 2121 y Fn(M)53 2103 y Ft(N)104 2121 y Fp(a)12 b Fn(R)179 2103 y Ft(N)217 2121 y Fp(-prenormed)d Fn(R)516 2103 y Ft(N)554 2121 y Fp(-semimo)q(dule)h (o)o(v)o(er)h(the)i(prenormed)c(semiring)i Fn(R)1501 2103 y Ft(N)1551 2121 y Fp(with)h(v)m(alue)0 2181 y(cone)k Fn(C)153 2162 y Ft(N)190 2181 y Fp(.)71 2240 y(Finally)d(a)h (construction)f(that)i(will)f(b)q(e)g(used)g(later.)g(Let)h Fn(A)f Fp(b)q(e)h(some)e(set)i(and)e Fn(\037)h Fp(:)g Fn(N)19 b Fo(\000)-8 b(!)0 2300 y Fn(A)15 b Fp(a)f(set)h(map.)e(Let)i (furthermore)e Fn(M)20 b Fp(b)q(e)15 b(an)f Fn(R)p Fp(-semimo)q(dule)f (and)g Fn(\026)1301 2307 y Fm(\003)1338 2300 y Fo(2)h Fn(M)1438 2282 y Ft(N)1477 2300 y Fp(.)g(Giv)o(en)g Fn(a)g Fo(2)g Fn(A)0 2372 y Fp(w)o(e)i(denote)h(b)o(y)f Fn(\026)333 2346 y Ft(\037)358 2331 y Fk(\000)p Fj(1)404 2346 y Fs(\()p Ft(a)p Fs(\))333 2379 y Fm(\003)476 2372 y Fp(the)h(map)f Fn(N)j Fo(\000)-9 b(!)14 b Fn(M)22 b Fp(giv)o(en)16 b(b)o(y)550 2524 y Fn(N)j Fo(3)14 b Fn(n)g Fo(7!)764 2454 y Fi(\032)810 2496 y Fn(\026)840 2503 y Ft(n)1429 2496 y Fp(,)i(if)h Fn(\037)p Fp(\()p Fn(n)p Fp(\))e(=)e Fn(a)p Fp(;)810 2556 y(0)594 b(,)16 b(otherwise.)0 2700 y(In)g(other)g(w)o(ords,)f Fn(\026)379 2674 y Ft(\037)404 2659 y Fk(\000)p Fj(1)451 2674 y Fs(\()p Ft(a)p Fs(\))379 2707 y Fm(\003)522 2700 y Fp(is)h(giv)o(en)h(b)o(y)f(the)h(form)o(ulae)16 2837 y Fn(\026)46 2811 y Ft(\037)71 2796 y Fk(\000)p Fj(1)117 2811 y Fs(\()p Ft(a)p Fs(\))46 2844 y Fm(\003)172 2837 y Fo(j)p Fn(\037)217 2817 y Fm(\000)p Fs(1)271 2837 y Fp(\()p Fn(a)p Fp(\))e(=)e Fn(\026)432 2844 y Fm(\003)455 2837 y Fo(j)p Fn(\037)500 2817 y Fm(\000)p Fs(1)554 2837 y Fp(\()p Fn(a)p Fp(\))117 b(and)e Fn(\026)961 2811 y Ft(\037)986 2796 y Fk(\000)p Fj(1)1033 2811 y Fs(\()p Ft(a)p Fs(\))961 2844 y Fm(\003)1088 2837 y Fo(j)p Fn(N)16 b Fh(r)11 b Fn(\037)1239 2817 y Fm(\000)p Fs(1)1293 2837 y Fp(\()p Fn(a)p Fp(\))k(=)e(0)1449 2844 y Fm(\003)1472 2837 y Fo(j)p Fn(N)k Fh(r)10 b Fn(\037)1623 2817 y Fm(\000)p Fs(1)1677 2837 y Fp(\()p Fn(a)p Fp(\))p Fn(:)p eop %%Page: 3 3 3 2 bop 1747 42 a Fp(3)0 167 y(Note)19 b(that)g(an)o(y)f(subset)g(of)g Fn(N)24 b Fp(can)19 b(b)q(e)f(obtained)g(as)g Fn(\037)1069 149 y Fm(\000)p Fs(1)1123 167 y Fp(\()p Fn(a)p Fp(\),)h(pro)o(vided)e (that)i Fn(A)g Fp(con)o(tains)0 227 y(at)h(least)f(t)o(w)o(o)g(elemen)o (ts,)f(and)g(that)i(eac)o(h)f(partition)f(of)i Fn(N)k Fp(can)19 b(b)q(e)h(written)f(as)g Fo(f)p Fn(\037)1621 209 y Fm(\000)p Fs(1)1674 227 y Fp(\()p Fn(a)p Fp(\))h(:)0 286 y Fn(a)14 b Fo(2)g Fn(A)124 268 y Fm(0)139 286 y Fo(g)i Fp(for)g(some)g(subset)g Fn(A)571 268 y Fm(0)602 286 y Fp(of)h Fn(A)p Fp(,)f(pro)o(vided)f(that)i Fn(A)g Fp(is)f(large)g(enough.)71 346 y(In)22 b(the)h(follo)o(wing)e (de\014nition)h(w)o(e)g(refer)g(to)h(the)g(concept)g(of)g(p)q(ositiv)o (e)f(semiring.)f(Ac-)0 406 y(cording)16 b(to)h([5],)f Fo(x)q Fp(1,)g(a)h(p)q(ositiv)o(e)g(semiring)e(is)i(a)g(partially)f (ordered)f(semiring)g(with)i(0)g(as)g(its)0 466 y(smallest)f(elemen)o (t.)0 588 y Fg(\(1.1\))24 b(De\014nition.)f Fp(Let)f Fn(C)j Fp(b)q(e)d(a)f(p)q(ositiv)o(e)h(semiring.)d(By)j(a)f Ff(left)g Fn(N)5 b Fp(-)p Ff(summation)22 b Fp(for)f Fn(C)0 648 y Fp(is)f(mean)o(t)g(a)h(pair)f(\()p Fn(S)412 655 y Ft(C)445 648 y Fn(;)8 b Fp(\006)503 655 y Ft(C)537 648 y Fp(\))21 b(consisting)e(of)i(a)g Fn(C)t Fp(-subsemim)o(o)q(dule)d Fn(S)1357 655 y Ft(C)1411 648 y Fp(of)i Fn(C)1511 630 y Ft(N)1569 648 y Fp(and)g(a)h Fn(C)t Fp(-)0 708 y(homomorphism)13 b(\006)388 715 y Ft(C)435 708 y Fp(:)h Fn(S)494 715 y Ft(C)541 708 y Fo(\000)-9 b(!)14 b Fn(C)20 b Fp(suc)o(h)15 b(that)64 768 y(\(i\))26 b Fn(C)182 750 y Fs(\()p Ft(N)t Fs(\))277 768 y Fp(:=)g Fo(f)p Fn(\013)413 775 y Fm(\003)463 768 y Fo(2)h Fn(C)563 750 y Ft(N)627 768 y Fp(:)g(supp)15 b Fn(\013)819 775 y Fm(\003)866 768 y Fp(is)24 b(\014nite)p Fo(g)g Fp(is)g(con)o(tained)g(in)g Fn(S)1473 775 y Ft(C)1530 768 y Fp(and)g(for)g(all)142 828 y Fn(\013)174 835 y Fm(\003)219 828 y Fo(2)g Fn(C)316 810 y Fs(\()p Ft(N)t Fs(\))384 828 y Fp(,)e(\006)456 835 y Ft(C)489 828 y Fp(\()p Fn(\013)540 835 y Fm(\003)563 828 y Fp(\))i(=)667 790 y Fi(P)720 803 y Fm(0)734 828 y Fo(f)p Fn(\013)791 835 y Ft(n)841 828 y Fp(:)f Fn(n)f Fo(2)i Fp(supp)15 b Fn(\013)1138 835 y Fm(\003)1161 828 y Fo(g)p Fp(,)22 b(where)1371 790 y Fi(P)1424 803 y Fm(0)1460 828 y Fp(stands)f(for)g (the)142 887 y(usual)15 b(sum)h(of)g(\014nitely)h(man)o(y)e(elemen)o (ts)h(in)g Fn(C)t Fp(;)50 947 y(\(ii\))26 b(for)21 b(all)g Fn(\013)329 954 y Fm(\003)374 947 y Fo(2)h Fn(S)460 954 y Ft(C)515 947 y Fp(and)f Fn(\014)645 954 y Fm(\003)690 947 y Fo(2)h Fn(C)785 929 y Ft(N)844 947 y Fp(with)f Fn(\014)990 954 y Fm(\003)1035 947 y Fo(\024)h Fn(\013)1128 954 y Fm(\003)1151 947 y Fn(;)8 b(\014)1201 954 y Fm(\003)1246 947 y Fp(is)21 b(in)g Fn(S)1395 954 y Ft(C)1450 947 y Fp(and)f(\006)1587 954 y Ft(C)1621 947 y Fp(\()p Fn(\014)1668 954 y Fm(\003)1691 947 y Fp(\))j Fo(\024)142 1007 y Fp(\006)178 1014 y Ft(C)211 1007 y Fp(\()p Fn(\013)262 1014 y Fm(\003)285 1007 y Fp(\);)37 1082 y(\(iii\))i(for)c(ev)o(ery)g Fn(\013)391 1089 y Fm(\003)435 1082 y Fo(2)h Fn(S)521 1089 y Ft(C)575 1082 y Fp(and)f(ev)o(ery)g(map)f Fn(')h Fp(:)h Fn(N)27 b Fo(\000)-9 b(!)21 b Fn(N)s(;)8 b(\013)1283 1056 y Ft(')1309 1041 y Fk(\000)p Fj(1)1356 1056 y Fs(\()p Ft(n)p Fs(\))1283 1089 y Fm(\003)1435 1082 y Fp(is)21 b(in)g Fn(S)1584 1089 y Ft(C)1638 1082 y Fp(for)g(all)142 1157 y Fn(n)e Fo(2)g Fn(N)5 b Fp(,)20 b(and)f(the)h(map)f Fn(\013)657 1133 y Ft(')683 1118 y Fk(\000)p Fj(1)657 1164 y Fm(\003)752 1157 y Fp(giv)o(en)g(b)o(y)g Fn(N)25 b Fo(3)19 b Fn(n)g Fo(7!)g Fp(\006)1228 1164 y Ft(C)1261 1157 y Fp(\()p Fn(\013)1312 1131 y Ft(')1338 1116 y Fk(\000)p Fj(1)1385 1131 y Fs(\()p Ft(n)p Fs(\))1312 1164 y Fm(\003)1444 1157 y Fp(\))g Fo(2)g Fn(C)k Fp(is)c(in)h Fn(S)1739 1164 y Ft(C)142 1230 y Fp(and)c(satis\014es)f(\006)458 1237 y Ft(C)492 1230 y Fp(\()p Fn(\013)543 1206 y Ft(')569 1191 y Fk(\000)p Fj(1)543 1237 y Fm(\003)618 1230 y Fp(\))f(=)g(\006) 740 1237 y Ft(C)773 1230 y Fp(\()p Fn(\013)824 1237 y Fm(\003)847 1230 y Fp(\);)38 1305 y(\(iv\))26 b(if)17 b Fn(\013)220 1312 y Fm(\003)260 1305 y Fp(is)g(in)g Fn(C)410 1287 y Ft(N)464 1305 y Fp(and)g(there)g(exists)g(a)h(map)e Fn(')f Fp(:)g Fn(N)20 b Fo(\000)-8 b(!)14 b Fn(N)23 b Fp(suc)o(h)16 b(that)i Fn(\013)1531 1279 y Ft(')1557 1264 y Fk(\000)p Fj(1)1604 1279 y Fs(\()p Ft(n)p Fs(\))1531 1312 y Fm(\003)1679 1305 y Fp(is)f(in)142 1378 y Fn(S)173 1385 y Ft(C)222 1378 y Fp(for)g(all)f Fn(n)e Fo(2)g Fn(N)22 b Fp(and)16 b(that)g Fn(\013)757 1354 y Ft(')783 1339 y Fk(\000)p Fj(1)757 1385 y Fm(\003)849 1378 y Fp(is)g(in)g Fn(S)988 1385 y Ft(C)1038 1378 y Fp(then)h Fn(\013)1184 1385 y Fm(\003)1223 1378 y Fp(is)f(in)g Fn(S)1362 1385 y Ft(C)1395 1378 y Fp(.)0 1474 y Fg(\(1.2\))25 b(Lemma.)i Ff(L)m(et)c Fn(C)j Ff(b)m(e)d(a)g(p)m(ositive)i(semiring)e(with)g(left) g Fn(N)5 b Ff(-summation)23 b Fp(\()p Fn(S)1612 1481 y Ft(C)1645 1474 y Fn(;)8 b Fp(\006)1703 1481 y Ft(C)1737 1474 y Fp(\))p Ff(.)0 1534 y(Then)19 b(the)g(c)m(onditions)i(\(1.1\),)g (\(i\)-\(iii\),)h(imply)d(that)g(for)h(every)e Fn(\013)1275 1541 y Fm(\003)1314 1534 y Fo(2)e Fn(S)1394 1541 y Ft(C)1446 1534 y Ff(the)j(ine)m(qualities)0 1593 y Fn(\013)32 1600 y Ft(n)73 1593 y Fo(\024)c Fp(\006)163 1600 y Ft(C)196 1593 y Fp(\()p Fn(\013)247 1600 y Fm(\003)270 1593 y Fp(\))p Fn(;)8 b(n)15 b Fo(2)g Fn(N)5 b Ff(,)18 b(hold;)i(in)e(p)m (articular,)i(if)f Fp(sup)o Fo(f)p Fn(\013)1091 1600 y Ft(n)1132 1593 y Fp(:)14 b Fn(n)g Fo(2)h Fn(N)5 b Fo(g)19 b Ff(exists,)g Fp(sup)o Fo(f)p Fn(\013)1625 1600 y Ft(n)1666 1593 y Fp(:)14 b Fn(n)g Fo(2)0 1653 y Fn(N)5 b Fo(g)14 b(\024)g Fp(\006)173 1660 y Ft(C)206 1653 y Fp(\()p Fn(\013)257 1660 y Fm(\003)280 1653 y Fp(\))19 b Ff(is)f(satis\014e)m(d.)1182 b Fo(u)-33 b(t)71 1749 y Fp(The)19 b(next)h(de\014nition)e(uses)h(the)h (notion)e(of)i(prenormed)d(semiring.)g(Due)j(to)g([5],)e(\(2.1\),)0 1809 y(this)d(is)h(a)f(semiring)f Fn(R)i Fp(together)g(with)f(a)h(map)f Fo(k)h(k)d Fp(:)h Fn(R)g Fo(\000)-9 b(!)14 b Fn(C)t Fp(,)h(where)g Fn(C)k Fp(is)c(a)h(p)q(ositiv)o(e)f(and)0 1869 y(complete)h(\(with)h (resp)q(ect)g(to)g(the)f(partial)g(order\))g(semiring,)e(suc)o(h)i (that)53 1929 y(\(o\))26 b Fo(k)p Fp(0)p Fo(k)13 b Fp(=)h(0)66 b(and)g Fo(k)p Fp(1)p Fo(k)13 b Fp(=)h(1;)64 1988 y(\(i\))26 b Fo(k)p 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Fo(k)p Fn(\014)949 2776 y Fm(\003)972 2769 y Fo(k)e(\024)f(k)p Fn(\013)1120 2776 y Fm(\003)1143 2769 y Fo(k)p Fp(,)j Fn(\014)1226 2776 y Fm(\003)1266 2769 y Fp(is)f(in)h Fn(S)1404 2776 y Ft(R)1453 2769 y Fp(and)g Fo(k)p Fp(\006)1611 2776 y Ft(R)1643 2769 y Fn(\014)1671 2776 y Fm(\003)1694 2769 y Fo(k)e(\024)142 2828 y Fp(\006)178 2835 y Ft(C)211 2828 y Fo(k)p Fn(\013)268 2835 y Fm(\003)291 2828 y Fo(k)p Fp(;)p eop %%Page: 4 4 4 3 bop 0 42 a Fp(4)349 b Fv(Section)17 b(3:)61 b(B.)15 b(P)o(areigis,)i(D.)f(Pumpl)q(\177)-24 b(un,)15 b(and)f(H.)h(R\177)-23 b(ohrl)37 176 y Fp(\(iii\))25 b(for)18 b(ev)o(ery)g Fn(\013)385 183 y Fm(\003)425 176 y Fo(2)f Fn(S)506 183 y Ft(R)556 176 y Fp(and)h(ev)o(ery)g Fn(')f Fp(:)f Fn(N)22 b Fo(\000)-8 b(!)16 b Fn(N)s(;)8 b(\013)1124 150 y Ft(')1150 135 y Fk(\000)p Fj(1)1197 150 y Fs(\()p Ft(n)p Fs(\))1124 183 y Fm(\003)1273 176 y Fp(is)18 b(in)g Fn(S)1416 183 y Ft(R)1467 176 y Fp(for)g(all)g Fn(n)e Fo(2)h Fn(N)5 b Fp(,)142 251 y(and)12 b(the)h(map)g 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y Fp(:=)e Fo(f)p Fn(\026)420 1239 y Fm(\003)466 1232 y Fo(2)g Fn(M)576 1214 y Ft(N)638 1232 y Fp(:)g(supp)15 b Fn(\026)825 1239 y Fm(\003)871 1232 y Fp(is)22 b(\014nite)p Fo(g)g Fp(is)g(con)o(tained) g(in)g Fn(S)1468 1239 y Ft(M)1534 1232 y Fp(and)g(for)g(all)142 1292 y Fn(\026)172 1299 y Fm(\003)217 1292 y Fo(2)h Fn(M)326 1274 y Fs(\()p Ft(N)t Fs(\))396 1292 y Fn(;)8 b Fp(\006)454 1299 y Ft(M)498 1292 y Fp(\()p Fn(\026)547 1299 y Fm(\003)570 1292 y Fp(\))23 b(=)674 1255 y Fi(P)726 1267 y Fm(0)740 1292 y Fo(f)p Fn(\026)795 1299 y Ft(n)845 1292 y Fp(:)f Fn(n)h Fo(2)g Fp(supp)15 b Fn(\026)1139 1299 y Fm(\003)1162 1292 y Fo(g)p Fp(,)21 b(where)1372 1255 y Fi(P)1424 1267 y Fm(0)1460 1292 y Fp(stands)g(for)g(the)142 1352 y(usual)15 b(sum)h(of)g(\014nitely)h(man)o(y)e(elemen)o(ts)h(in)g Fn(M)5 b Fp(;)50 1412 y(\(ii\))26 b(for)13 b(all)g Fn(\026)311 1419 y Fm(\003)348 1412 y Fo(2)h Fn(S)426 1419 y Ft(M)484 1412 y Fp(and)e Fn(\027)602 1419 y Fm(\003)639 1412 y Fo(2)i Fn(M)739 1394 y Ft(N)791 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Fo(7!)g Fp(\006)1076 1628 y Ft(M)1120 1621 y Fp(\()p Fn(\026)1169 1596 y Ft(')1195 1581 y Fk(\000)p Fj(1)1242 1596 y Fs(\()p Ft(n)p Fs(\))1169 1628 y Fm(\003)1301 1621 y Fp(\))g Fo(2)g Fn(M)28 b Fp(is)22 b(in)f Fn(S)1625 1628 y Ft(M)1691 1621 y Fp(and)142 1694 y(satis\014es)15 b(\006)361 1701 y Ft(M)405 1694 y Fp(\()p Fn(\026)454 1671 y Ft(')480 1655 y Fk(\000)p Fj(1)454 1701 y Fm(\003)530 1694 y Fp(\))f(=)g(\006)652 1701 y Ft(M)696 1694 y Fp(\()p Fn(\026)745 1701 y Fm(\003)768 1694 y Fp(\).)71 1820 y(Note)f(that)f(\(1.3\))h(is)e(a)h(sp)q(ecial)g(case)g(of)g(\(1.4\))h (and)e(that)i(\(1.1\))f(can)g(b)q(e)h(view)o(ed)e(as)h(a)g(sp)q(ecial)0 1880 y(case)20 b(of)h(\(1.3\))g(b)o(y)f(setting)g Fn(R)h Fp(=)f Fn(C)q(;)8 b Fo(k)20 b(k)g Fp(=)h(id)901 1887 y Ft(C)935 1880 y Fn(;)8 b(S)988 1887 y Ft(R)1040 1880 y Fp(=)20 b Fn(S)1130 1887 y Ft(C)1163 1880 y Fp(,)h(and)f(\006)1335 1887 y Ft(R)1388 1880 y Fp(=)g(\006)1483 1887 y Ft(C)1516 1880 y Fp(.)g(Hence)h(the)0 1940 y(statemen)o(ts)16 b(follo)o(wing)f (\(1.5\))i(concerning)e(semimo)q(dules)g(apply)h(to)h(prenormed)d (semirings)0 1999 y(and)i(p)q(ositiv)o(e)g(semirings)e(as)j(w)o(ell.)0 2125 y Fg(\(1.5\))24 b(De\014nition.)f Fp(Let)e Fn(M)27 b Fp(and)21 b Fn(M)751 2107 y Fm(0)787 2125 y Fp(b)q(e)g Fn(R)p Fp(-prenormed)e Fn(R)p Fp(-semimo)q(dule)g(with)i(left)h Fn(N)5 b Fp(-)0 2185 y(summation)15 b(\()p Fn(S)308 2192 y Ft(M)352 2185 y Fn(;)8 b Fp(\006)410 2192 y Ft(M)454 2185 y Fp(\))18 b(resp.)d(\()p Fn(S)660 2192 y Ft(M)702 2182 y Fk(0)718 2185 y Fn(;)8 b Fp(\006)776 2192 y Ft(M)818 2182 y Fk(0)834 2185 y Fp(\).)17 b(Then)f(a)h Ff(homomorphism)i Fp(from)d Fn(M)23 b Fp(to)17 b Fn(M)1707 2167 y Fm(0)1738 2185 y Fp(is)0 2245 y(a)g(map)e Fn(f)20 b Fp(:)13 b Fn(M)20 b Fo(\000)-8 b(!)13 b Fn(M)438 2227 y Fm(0)469 2245 y Fp(suc)o(h)j(that)64 2304 y(\(i\))26 b Fn(f)c Fp(is)16 b(a)h(homomorphis)o(m)d(of)i(left)h Fn(R)p Fp(-semimo)q(dules)e(from)h Fn(M)22 b Fp(to)17 b Fn(M)1425 2286 y Fm(0)1456 2304 y Fp(satisfying)194 2400 y Fn(f)223 2380 y Ft(N)262 2400 y Fp(\()p Fn(S)312 2407 y Ft(M)356 2400 y Fp(\))d Fo(\022)f Fn(S)472 2407 y Ft(M)514 2398 y Fk(0)596 2400 y Fp(and)66 b(\006)779 2407 y Ft(M)821 2398 y Fk(0)836 2400 y Fp(\()p Fn(f)884 2380 y Ft(N)923 2400 y Fp(\()p Fn(\026)972 2407 y Fm(\003)996 2400 y Fp(\)\))14 b(=)g Fn(f)5 b Fp(\(\006)1185 2407 y Ft(M)1230 2400 y Fp(\()p Fn(\026)1279 2407 y Fm(\003)1302 2400 y Fp(\)\))53 b(,)17 b(for)f(all)g Fn(\026)1599 2407 y Fm(\003)1636 2400 y Fo(2)e Fn(S)1714 2407 y Ft(M)1758 2400 y Fp(;)50 2496 y(\(ii\))26 b(there)16 b(is)g(a)h Fn(c)c Fo(2)h Fn(C)20 b Fp(\(dep)q(ending)c(on)g Fn(f)5 b Fp(\))18 b(with)447 2592 y Fo(k)p Fp(\006)508 2599 y Ft(M)550 2589 y Fk(0)566 2592 y Fn(f)595 2571 y Ft(N)633 2592 y Fp(\()p Fn(\026)682 2599 y Fm(\003)706 2592 y Fp(\))p Fo(k)c(\024)f Fp(\(\006)871 2599 y Ft(C)905 2592 y Fp(\()p Fo(k)p Fn(\026)979 2599 y Fm(\003)1003 2592 y Fo(k)p Fp(\)\))p Fn(c)305 b Fp(,)17 b(for)f(all)g Fn(\026)1599 2599 y Fm(\003)1636 2592 y Fo(2)e Fn(S)1714 2599 y Ft(M)1758 2592 y Fn(:)71 2718 y Fp(It)f(is)f(clear)g(from)g(\(1.18\))h(that)g (the)g(totalit)o(y)g(of)g Fn(R)p Fp(-prenormed)d Fn(R)p Fp(-semimo)q(dules)h(with)h(left)0 2777 y Fn(N)5 b Fp(-summation)13 b(together)i(with)f(their)h(homomorphis)o(ms)c(and)j(the)h (set-theoretical)g(comp)q(o-)0 2837 y(sition)h(of)h(these)f(forms)g(a)g (category)h Fn(N)740 2844 y Ft(R)773 2837 y Fp(pnSmo)q(d)n(.)p eop %%Page: 5 5 5 4 bop 1747 42 a Fp(5)71 167 y(A)21 b(homomorphism)c(of)k Fn(R)p Fp(-prenormed)e Fn(R)p Fp(-semimo)q(dules)f(is)j(called)f Ff(c)m(ontr)m(active)i Fp(\(or)f(a)0 227 y Ff(c)m(ontr)m(action)p Fp(\))f(if)f Fn(c)e Fp(=)g(1)h(can)h(b)q(e)f(c)o(hosen)g(in)g(\(1.5\),) h(\(ii\).)f(Again)h(it)g(is)f(easy)g(to)h(see)g(that)f(the)0 286 y(totalit)o(y)d(of)f Fn(R)p Fp(-prenormed)d Fn(R)p Fp(-semimo)q(dules)h(together)i(with)g(their)g(con)o(tractiv)o(e)f (homomor-)0 346 y(phisms)i(forms)g(a)i(sub)q(category)f Fn(N)663 353 y Ft(R)695 346 y Fp(pnSmo)q(d)874 324 y Fs(1)913 346 y Fp(of)h Fn(N)1010 353 y Ft(R)1042 346 y Fp(pnSmo)q(d)o(.)71 406 y(W)l(e)g(close)f(with)g(three)h(statemen)o (ts)f(directly)g(related)g(to)h(the)g(ab)q(o)o(v)o(e)f(de\014nitions.)0 488 y Fg(\(1.6\))23 b(Lemma.)j Ff(\(1.4\),)d(\(o\),)g(implies)e(that,)h (whenever)f Fn(\026)1159 495 y Fm(\003)1201 488 y Fo(2)f Fn(S)1285 495 y Ft(M)1349 488 y Ff(and)i Fn(\027)1474 495 y Fm(\003)1516 488 y Fo(2)e Fn(M)1622 470 y Ft(N)1681 488 y Ff(with)0 548 y Fo(k)p Fn(\027)50 555 y Fm(\003)72 548 y Fo(k)14 b(\024)g(k)p Fn(\026)219 555 y Fm(\003)242 548 y Fo(k)j Ff(holds)j(then)d Fn(\027)547 555 y Fm(\003)584 548 y Fo(2)d Fn(S)662 555 y Ft(M)723 548 y Ff(is)19 b(satis\014e)m(d.)0 685 y(Pr)m(o)m(of.)h Fp(Due)c(to)g(\(1.4\),)h(\(o\),)f Fo(k)p Fn(\026)594 692 y Fm(\003)617 685 y Fo(k)g Fp(is)g(in)g Fn(S)797 692 y Ft(C)830 685 y Fp(.)g(Hence)h(\(1.1\),)f(\(ii\),)g(sho)o (ws)f(that)i Fo(k)p Fn(\027)1537 692 y Fm(\003)1559 685 y Fo(k)f Fp(is)g(in)g Fn(S)1739 692 y Ft(C)0 745 y Fp(and)g(th)o(us,)f (b)o(y)h(\(1.4\),)h(\(o\),)g Fn(\027)541 752 y Fm(\003)581 745 y Fp(is)f(in)g Fn(S)720 752 y Ft(M)763 745 y Fp(.)961 b Fo(u)-33 b(t)0 827 y Fg(\(1.7\))18 b(Lemma.)27 b Ff(\(1.4\),)19 b(\(o\))g(and)f(\(ii\),)g(imply)g(that,)g(whenever)f Fn(\026)1286 834 y Fm(\003)1323 827 y Fo(2)d Fn(S)1401 834 y Ft(M)1462 827 y Ff(and)k Fn(')13 b Fp(:)h Fn(N)19 b Fo(\000)-8 b(!)0 902 y Fn(N)23 b Ff(is)c(a)f(map,)h Fn(\026)315 876 y Ft(')341 861 y Fk(\000)p Fj(1)387 876 y Fs(\()p Ft(n)p Fs(\))315 909 y Fm(\003)463 902 y Ff(is)g(in)f Fn(S)609 909 y Ft(M)653 902 y Ff(,)g(for)g(al)s(l)g Fn(n)c Fo(2)g Fn(N)5 b Ff(,)18 b(and)h Fn(\026)1133 878 y Ft(')1159 863 y Fk(\000)p Fj(1)1133 909 y Fm(\003)1226 902 y Ff(is)f(in)g Fn(S)1371 909 y Ft(M)1415 902 y Ff(.)0 1052 y(Pr)m(o)m(of.)j Fp(Again)16 b Fo(k)p Fn(\026)350 1059 y Fm(\003)373 1052 y Fo(k)h Fp(is)g(in)g Fn(S)556 1059 y Ft(C)589 1052 y Fp(.)g(Since)f Fo(k)p Fn(\026)805 1026 y Ft(')831 1011 y Fk(\000)p Fj(1)878 1026 y Fs(\()p Ft(n)p Fs(\))805 1059 y Fm(\003)936 1052 y Fo(k)f(\024)f(k)p Fn(\026)1084 1059 y Fm(\003)1107 1052 y Fo(k)p Fp(,)j Fo(k)p Fn(\026)1218 1026 y Ft(')1244 1011 y Fk(\000)p Fj(1)1290 1026 y Fs(\()p Ft(n)p Fs(\))1218 1059 y Fm(\003)1349 1052 y Fo(k)g Fp(is)f(in)h Fn(S)1531 1059 y Ft(C)1581 1052 y Fp(and)g(th)o(us)0 1127 y Fn(\026)30 1101 y Ft(')56 1086 y Fk(\000)p Fj(1)103 1101 y Fs(\()p Ft(n)p Fs(\))30 1134 y Fm(\003)177 1127 y Fp(is)g(in)f Fn(S)317 1134 y Ft(M)360 1127 y Fp(.)h(By)g(\(1.4\),)g (\(ii\),)317 1231 y Fo(k)p Fp(\006)378 1238 y Ft(M)422 1231 y Fp(\()p Fn(\026)471 1206 y Ft(')497 1191 y Fk(\000)p Fj(1)544 1206 y Fs(\()p Ft(n)p Fs(\))471 1238 y Fm(\003)603 1231 y Fp(\))p Fo(k)d(\024)f Fp(\006)749 1238 y Ft(C)783 1231 y Fp(\()p Fo(k)p Fn(\026)857 1206 y Ft(')883 1191 y Fk(\000)p Fj(1)930 1206 y Fs(\()p Ft(n)p Fs(\))857 1238 y Fm(\003)988 1231 y Fo(k)p Fp(\))h(=)g(\006)1135 1238 y Ft(C)1168 1231 y Fp(\()p Fo(k)p Fn(\026)1242 1238 y Fm(\003)1265 1231 y Fo(k)1290 1211 y Ft(')1316 1196 y Fk(\000)p Fj(1)1363 1211 y Fs(\()p Ft(n)p Fs(\))1421 1231 y Fp(\))p Fn(:)0 1336 y Fp(Due)h(to)h(\(1.1\),)f(\(iii\),)g Fn(N)20 b Fo(3)14 b Fn(n)f Fo(7!)h Fp(\006)653 1343 y Ft(C)687 1336 y Fp(\()p Fn(\026)736 1310 y Ft(')762 1295 y Fk(\000)p Fj(1)809 1310 y Fs(\()p Ft(n)p Fs(\))736 1343 y Fm(\003)867 1336 y Fp(\))g Fo(2)g Fn(C)k Fp(is)d(in)g Fn(S)1138 1343 y Ft(C)1186 1336 y Fp(as)g Fo(k)p Fn(\026)1301 1343 y Fm(\003)1324 1336 y Fo(k)g Fp(is)g(in)f Fn(S)1500 1343 y Ft(C)1533 1336 y Fp(.)h(Therefore)0 1411 y(the)j(map)e Fn(N)21 b Fo(3)15 b Fn(n)g Fo(7!)g(k)p Fp(\006)478 1418 y Ft(M)522 1411 y Fp(\()p Fn(\026)571 1385 y Ft(')597 1370 y Fk(\000)p Fj(1)644 1385 y Fs(\()p Ft(n)p Fs(\))571 1418 y Fm(\003)702 1411 y Fp(\))p Fo(k)h(2)f Fn(C)21 b Fp(is)c(in)g Fn(S)1008 1418 y Ft(C)1058 1411 y Fp(and)g(hence)g(the)h (map)e Fn(\026)1523 1387 y Ft(')1549 1372 y Fk(\000)p Fj(1)1523 1418 y Fm(\003)1598 1411 y Fp(,)h(that)h(is)0 1486 y Fn(N)h Fo(3)14 b Fn(n)g Fo(7!)g Fp(\006)250 1493 y Ft(M)294 1486 y Fp(\()p Fn(\026)343 1460 y Ft(')369 1445 y Fk(\000)p Fj(1)416 1460 y Fs(\()p Ft(n)p Fs(\))343 1493 y Fm(\003)474 1486 y Fp(\),)j(is)f(in)g Fn(S)663 1493 y Ft(M)707 1486 y Fp(.)1017 b Fo(u)-33 b(t)0 1568 y Fg(\(1.8\))18 b(Lemma.)27 b Ff(L)m(et)17 b Fn(M)23 b Ff(b)m(e)17 b(an)h Fn(R)p Ff(-pr)m(enorme)m(d)h Fn(R)p Ff(-semimo)m(dule)f(with)f(left)g Fn(N)5 b Ff(-summation)0 1628 y Fp(\()p Fn(S)50 1635 y Ft(M)94 1628 y Fn(;)j Fp(\006)152 1635 y Ft(M)196 1628 y Fp(\))14 b Ff(and)f(denote)g(the)g Fn(N)5 b Ff(-summation)13 b(of)h Fn(R)f Ff(by)g Fp(\()p Fn(S)1082 1635 y Ft(R)1114 1628 y Fn(;)8 b Fp(\006)1172 1635 y Ft(R)1205 1628 y Fp(\))p Ff(.)14 b(If)f Fn(\013)1332 1635 y Fm(\003)1368 1628 y Fo(2)h Fn(S)1446 1635 y Ft(R)1491 1628 y Ff(and)g Fn(\026)1613 1610 y Fm(\003)1650 1628 y Fo(2)g Fn(S)1728 1635 y Ft(M)0 1688 y Ff(then)k Fn(\013)143 1695 y Fm(\003)166 1688 y Fn(\026)196 1669 y Fm(\003)218 1688 y Ff(,)h(de\014ne)m(d)g(as)f(the)g(map)g Fn(N)i Fo(3)14 b Fn(n)g Fo(7!)f Fn(\013)921 1695 y Ft(n)948 1688 y Fn(\026)978 1669 y Ft(n)1019 1688 y Fo(2)h Fn(M)5 b Ff(,)19 b(is)f(in)h Fn(S)1299 1695 y Ft(M)1342 1688 y Ff(.)0 1825 y(Pr)m(o)m(of.)24 b Fp(Since)d Fn(\013)320 1832 y Fm(\003)364 1825 y Fp(is)f(in)h Fn(S)512 1832 y Ft(R)544 1825 y Fp(,)g(it)g(follo)o(ws)f(from)h(\(1.3\),)g(\(o\),)h (that)f Fo(k)p Fn(\013)1329 1832 y Fm(\003)1352 1825 y Fo(k)g Fp(is)f(in)h Fn(S)1546 1832 y Ft(C)1579 1825 y Fp(,)g(whence)0 1885 y Fo(k)p Fn(\013)57 1892 y Ft(n)84 1885 y Fo(k)14 b(\024)g Fp(\006)212 1892 y Ft(C)245 1885 y Fo(k)p Fn(\013)302 1892 y Fm(\003)325 1885 y Fo(k)p Fp(,)j Fn(n)d Fo(2)g Fn(N)5 b Fp(,)17 b(on)f(accoun)o(t)h(of)g (\(1.2\).)g(Therefore)f Fo(k)p Fn(\013)1275 1892 y Fm(\003)1297 1885 y Fn(\026)1327 1867 y Fm(\003)1350 1885 y Fo(k)e(\024)g Fp(\(\006)1497 1892 y Ft(C)1531 1885 y Fo(k)p Fn(\013)1588 1892 y Fm(\003)1611 1885 y Fo(k)p Fp(\))p Fo(k)p Fn(\026)1710 1867 y Fm(\003)1733 1885 y Fo(k)p Fp(,)0 1945 y(and)i(\(1.1\))h(and)f (\(1.4\),)g(\(o\),)h(sho)o(w)f(that)h(the)f(righ)o(t)g(hand)f(side)h (of)h(this)f(inequalit)o(y)g(is)g(in)g Fn(S)1725 1952 y Ft(C)1758 1945 y Fp(.)0 2004 y(Th)o(us)f Fo(k)p Fn(\013)183 2011 y Fm(\003)206 2004 y Fn(\026)236 1986 y Fm(\003)259 2004 y Fo(k)h Fp(is)g(in)g Fn(S)439 2011 y Ft(C)489 2004 y Fp(due)g(to)h(\(1.1\),)g(\(ii\),)g(whence)f Fn(\013)1080 2011 y Fm(\003)1103 2004 y Fn(\026)1133 1986 y Fm(\003)1172 2004 y Fp(is)h(in)f Fn(S)1312 2011 y Ft(M)1372 2004 y Fp(b)o(y)g(\(1.4\),)h(\(o\).)511 2240 y Fq(2.)g(Elemen)n(tary)e (results)0 2381 y Fg(\(2.1\))j(Lemma.)27 b Ff(L)m(et)17 b Fn(M)23 b Ff(b)m(e)17 b(an)h Fn(R)p Ff(-pr)m(enorme)m(d)h Fn(R)p Ff(-semimo)m(dule)f(with)f(left)g Fn(N)5 b Ff(-summation)0 2441 y Fp(\()p Fn(S)50 2448 y Ft(M)94 2441 y Fn(;)j Fp(\006)152 2448 y Ft(M)196 2441 y Fp(\))p Ff(.)16 b(L)m(et)f(furthermor)m(e)g Fn(\026)630 2448 y Fm(\003)667 2441 y Fo(2)f Fn(S)745 2448 y Ft(M)803 2441 y Ff(and)i Fn(\027)922 2448 y Fm(\003)958 2441 y Fo(2)e Fn(M)1058 2423 y Ft(N)1111 2441 y Ff(b)m(e)h(such)g(that) g(ther)m(e)g(is)g(a)h(bije)m(ction)p 0 2474 33 2 v 0 2501 a Fn(')f Fp(:)h(supp)f Fn(\026)227 2508 y Fm(\003)266 2501 y Fo(\000)-8 b(!)15 b Fp(supp)h Fn(\027)507 2508 y Fm(\003)548 2501 y Ff(with)j Fn(\026)687 2508 y Ft(n)730 2501 y Fp(=)d Fn(\027)p 810 2489 27 2 v 9 x Ft(')p Fs(\()p Ft(n)p Fs(\))894 2501 y Ff(,)j(for)h(al)s(l)e Fn(n)e Fo(2)g Fp(supp)g Fn(\026)1326 2508 y Fm(\003)1348 2501 y Ff(.)k(Then)f Fn(\027)1539 2508 y Fm(\003)1580 2501 y Ff(is)h(in)f Fn(S)1728 2508 y Ft(M)0 2561 y Ff(and)g Fp(\006)133 2568 y Ft(M)177 2561 y Fp(\()p Fn(\027)221 2568 y Fm(\003)244 2561 y Fp(\))14 b(=)g(\006)366 2568 y Ft(M)410 2561 y Fp(\()p Fn(\026)459 2568 y Fm(\003)482 2561 y Fp(\))p Ff(.)0 2698 y(Pr)m(o)m(of.)22 b Fp(Extend)p 327 2671 33 2 v 19 w Fn(')c Fp(to)h(some)g(map)e Fn(')h Fp(:)f Fn(N)23 b Fo(\000)-9 b(!)18 b Fn(N)5 b Fp(.)19 b(Then)f(one)h(c)o(hec)o(ks)f(quic)o(kly)h(that)g Fn(\027)1693 2705 y Fm(\003)1733 2698 y Fp(=)0 2769 y Fn(\026)30 2745 y Ft(')56 2730 y Fk(\000)p Fj(1)30 2776 y Fm(\003)121 2769 y Fp(holds.)14 b(Hence)j(\(1.1\),)f(\(iii\),)g(sho)o(ws)e(that)i Fn(\027)928 2776 y Fm(\003)966 2769 y Fp(is)g(in)f Fn(S)1104 2776 y Ft(M)1164 2769 y Fp(and)g(that)h(\006)1403 2776 y Ft(M)1447 2769 y Fp(\()p Fn(\027)1491 2776 y Fm(\003)1514 2769 y Fp(\))e(=)g(\006)1636 2776 y Ft(M)1680 2769 y Fp(\()p Fn(\026)1729 2776 y Fm(\003)1752 2769 y Fp(\))0 2828 y(is)i(v)m(alid.)1570 b Fo(u)-33 b(t)p eop %%Page: 6 6 6 5 bop 0 42 a Fp(6)349 b Fv(Section)17 b(3:)61 b(B.)15 b(P)o(areigis,)i(D.)f(Pumpl)q(\177)-24 b(un,)15 b(and)f(H.)h(R\177)-23 b(ohrl)0 176 y Fg(\(2.2\))12 b(Corollary)-5 b(.)30 b Ff(L)m(et)13 b Fn(M)18 b Ff(b)m(e)13 b(an)h Fn(R)p Ff(-pr)m(enorme)m(d) g Fn(R)p Ff(-semimo)m(dule)g(with)f(left)f Fn(N)5 b Ff(-summation)0 235 y Fp(\()p Fn(S)50 242 y Ft(M)94 235 y Fn(;)j Fp(\006)152 242 y Ft(M)196 235 y Fp(\))p Ff(.)22 b(L)m(et)e(furthermor)m(e)h Fn(f)j Fp(:)19 b Fn(A)g Fo(\000)-9 b(!)18 b Fn(M)26 b Ff(b)m(e)21 b(a)g(map)g(with)g Fp(card)o(\(supp)16 b Fn(f)5 b Fp(\))19 b Fo(\024)g Fp(card)o Fn(N)5 b Ff(.)0 295 y(Then)26 b(ther)m(e)h(is)g(a)g(map)g Fn(\037)j Fp(:)f Fn(N)35 b Fo(\000)-9 b(!)29 b Fn(A)p Ff(,)e(a)g Fn(\026)952 302 y Fm(\003)1005 295 y Fo(2)i Fn(M)1120 277 y Ft(N)1159 295 y Ff(,)d(and)i(sets)e Fn(A)1449 277 y Fm(0)1490 295 y Ff(and)h Fn(N)1640 277 y Fm(0)1681 295 y Ff(with)0 355 y Fp(supp)15 b Fn(f)20 b Fo(\022)13 b Fn(A)252 337 y Fm(0)281 355 y Fo(\022)g Fn(A)p Ff(,)19 b Fp(supp)c Fn(\026)553 362 y Fm(\003)590 355 y Fo(\022)e Fn(N)687 337 y Fm(0)716 355 y Fo(\022)g Fn(N)5 b Ff(,)19 b(and)f Fp(card)p Fn(A)1075 337 y Fm(0)1103 355 y Fo(\024)13 b Fp(card)p Fn(N)23 b Ff(such)18 b(that)73 415 y Fp(a\))25 b Fn(A)179 397 y Fm(0)207 415 y Fo(\022)13 b Fn(\037)p Fp(\()p Fn(N)5 b Fp(\))20 b Ff(and)e Fn(A)526 397 y Fm(0)541 415 y Fo(j)p Fn(\037)p Fo(j)p Fn(N)645 397 y Fm(0)677 415 y Ff(is)g(a)h(bije)m(ction)70 474 y Fp(b\))25 b Fn(\026)172 481 y Ft(n)213 474 y Fp(=)13 b Fn(f)5 b Fp(\()p Fn(\037)p Fp(\()p Fn(n)p Fp(\)\))991 b Ff(,)18 b(for)h(al)s(l)e Fn(n)d Fo(2)g Fn(N)1742 456 y Fm(0)1756 474 y Ff(.)0 543 y(Mor)m(e)m(over,)h(if)p 274 516 32 2 v 14 w Fn(\037)q Ff(,)p 335 516 31 2 v 14 w Fn(\026)365 555 y Fm(\003)387 543 y Ff(,)p 416 503 38 2 v 14 w Fn(A)454 511 y Fm(0)481 543 y Ff(and)p 574 503 46 2 v 15 w Fn(N)619 511 y Fm(0)647 543 y Ff(is)f(another)h(set)e(of)i(data)f(satisfying)i(the)d(ab)m(ove)i (c)m(onditions,)0 603 y(for)26 b(al)s(l)f Fn(n)i Fo(2)h Fn(N)329 585 y Fm(0)344 603 y Ff(,)d(then)h Fn(\026)533 610 y Fm(\003)581 603 y Ff(is)g(in)g Fn(S)742 610 y Ft(M)811 603 y Ff(if)g(and)g(only)g(if)p 1143 575 31 2 v 26 w Fn(\026)1173 615 y Fm(\003)1221 603 y Ff(is)g(in)g Fn(S)1382 610 y Ft(M)1426 603 y Ff(,)f(in)h(which)h(c)m(ase)0 663 y Fp(\006)36 670 y Ft(M)80 663 y Fp(\()p Fn(\026)129 670 y Fm(\003)152 663 y Fp(\))15 b(=)e(\006)274 670 y Ft(M)318 663 y Fp(\()p 337 635 V Fn(\026)368 675 y Fm(\003)391 663 y Fp(\))18 b Ff(holds.)0 830 y(Pr)m(o)m(of.)23 b Fp(The)c(existence)h(of)g Fn(\037)p Fp(,)f Fn(\026)626 837 y Fm(\003)649 830 y Fp(,)h Fn(A)720 812 y Fm(0)734 830 y Fp(,)f(and)g Fn(N)912 812 y Fm(0)946 830 y Fp(is)g(ob)o(vious.)f (The)i(balance)e(of)i(\(2.2\))g(is)f(an)0 889 y(immediate)d (consequence)g(of)g(\(2.1\).)1041 b Fo(u)-33 b(t)0 1015 y Fg(\(2.3\))22 b(Remark.)c Fp(Supp)q(ose)g(w)o(e)h(are)g(in)g(the)h (situation)f(of)g(\(2.2\))h(but)f(are)g(dealing)f(with)i(a)0 1075 y(family)d Fo(f)p Fn(f)202 1082 y Ft(n)244 1075 y Fp(:)e Fn(n)f Fo(2)380 1062 y Fp(~)366 1075 y Fn(N)5 b Fo(g)17 b Fp(of)g(maps)f Fn(A)g Fo(\000)-9 b(!)15 b Fn(N)5 b Fp(,)17 b(whose)g(index)g(set)g(satis\014es)f(card)1519 1062 y(~)1504 1075 y Fn(N)21 b Fo(\024)14 b Fp(card)o Fn(N)5 b Fp(,)0 1134 y(instead)16 b(of)h(lo)q(oking)g(at)g(a)g(single)f (suc)o(h)f(map.)h(If)h(card)o(\(supp)f Fn(f)1190 1141 y Ft(n)1218 1134 y Fp(\))e Fo(\024)g Fp(card)o Fn(N)5 b Fp(,)18 b(for)e(all)h Fn(n)d Fo(2)1727 1122 y Fp(~)1712 1134 y Fn(N)6 b Fp(,)0 1194 y(then,)17 b(as)h(card)o Fn(N)330 1176 y Fs(2)369 1194 y Fp(=)d(card)o Fn(N)5 b Fp(,)19 b(card)o(\()708 1157 y Fi(S)750 1194 y Fo(f)p Fp(supp)c Fn(f)918 1201 y Ft(n)961 1194 y Fp(:)h Fn(n)g Fo(2)1100 1182 y Fp(~)1086 1194 y Fn(N)5 b Fo(g)p Fp(\))16 b Fo(\024)g Fp(card)o Fn(N)23 b Fp(holds.)17 b(Therefore,)0 1254 y(in)e(\(2.2\),)h Fn(\037)p Fp(,)f Fn(A)286 1236 y Fm(0)300 1254 y Fp(,)h(and)e Fn(N)470 1236 y Fm(0)500 1254 y Fp(can)h(b)q(e)h(c)o(hosen)e(suc)o(h)g(that)i(for)f(ev)o(ery)h Fn(f)1263 1261 y Ft(n)1290 1254 y Fp(,)f Fn(n)f Fo(2)1425 1241 y Fp(~)1410 1254 y Fn(N)6 b Fp(,)15 b(and)f(the)i(\(no)o(w)0 1314 y(uniquely)g(determined\))g Fn(\026)512 1321 y Fm(\003)p Ft(n)559 1314 y Fp(,)h Fn(n)c Fo(2)695 1301 y Fp(~)680 1314 y Fn(N)22 b Fp(,)17 b(the)g(conditions)e(in)h(\(2.2\))h(are)f (satis\014ed.)191 b Fo(u)-33 b(t)71 1439 y Fp(Due)16 b(to)h(\(2.2\))g(w)o(e)g(can)f(form)o(ulate)0 1565 y Fg(\(2.4\))i(De\014nition.)f Fp(Let)f Fn(M)22 b Fp(b)q(e)16 b(an)f Fn(R)p Fp(-prenormed)e Fn(R)p Fp(-semimo)q(dule)h(with)h(left)i Fn(N)5 b Fp(-summa-)0 1624 y(tion)17 b(\()p Fn(S)153 1631 y Ft(M)197 1624 y Fn(;)8 b Fp(\006)255 1631 y Ft(M)299 1624 y Fp(\).)18 b(Let)f(furthermore)f Fn(A)h Fp(b)q(e)g(an)o(y)g(set.) g(Then)g(w)o(e)f(de\014ne)h Fn(S)1405 1631 y Ft(M)r(;A)1505 1624 y Fp(as)g(the)g(set)h(of)0 1684 y(maps)f Fn(f)k Fp(:)16 b Fn(A)h Fo(\000)-9 b(!)16 b Fn(M)23 b Fp(suc)o(h)17 b(that)h(card\(supp)d Fn(f)5 b Fp(\))17 b Fo(\024)f Fp(card)o Fn(N)24 b Fp(and)17 b(for)g(some)g(data)h Fn(\037;)8 b(\026)1661 1691 y Fm(\003)1684 1684 y Fn(;)g(A)1743 1666 y Fm(0)1758 1684 y Fp(,)0 1744 y(and)16 b Fn(N)142 1726 y Fm(0)174 1744 y Fp(in)h(\(2.2\),)g Fn(\026)396 1751 y Fm(\003)433 1744 y Fo(2)e Fn(S)512 1751 y Ft(M)573 1744 y Fp(holds.)h(Moreo)o(v)o(er,)f(w)o(e)i(de\014ne)f(\006)1205 1751 y Ft(M)r(;A)1289 1744 y Fp(\()p Fn(f)5 b Fp(\))16 b(:=)f(\006)1476 1751 y Ft(M)1520 1744 y Fp(\()p Fn(\026)1569 1751 y Fm(\003)1592 1744 y Fp(\),)i(for)g(all)0 1804 y Fn(f)j Fo(2)14 b Fn(S)122 1811 y Ft(M)r(;A)205 1804 y Fp(,)i(to)h(obtain)f(a)h(map)f(\006)640 1811 y Ft(M)r(;A)737 1804 y Fp(:)e Fn(S)796 1811 y Ft(M)r(:A)893 1804 y Fo(\000)-8 b(!)13 b Fn(M)5 b Fp(.)71 1929 y(Note)25 b(that)f Fn(S)345 1936 y Ft(M)r(;N)460 1929 y Fp(=)i Fn(S)556 1936 y Ft(M)624 1929 y Fp(and)e(\006)765 1936 y Ft(M)r(;N)880 1929 y Fp(=)i(\006)981 1936 y Ft(M)1049 1929 y Fp(hold.)d(W)l(e)i(will)e (write)h(o)q(ccasionally)0 1989 y(\006)36 1996 y Ft(M)r(;A)120 1989 y Fo(f)p Fn(f)5 b Fp(\()p Fn(a)p Fp(\))15 b(:)f Fn(a)g Fo(2)g Fn(A)p Fo(g)j Fp(instead)f(of)h(\006)712 1996 y Ft(M)r(;A)796 1989 y Fp(\()p Fn(f)5 b Fp(\))p Fn(:)71 2049 y Fp(F)l(or)15 b Fn(f)20 b Fp(:)14 b Fn(A)g Fo(\000)-9 b(!)14 b Fn(M)22 b Fp(w)o(e)16 b(denote)h(b)o(y)f Fo(k)p Fn(f)5 b Fo(k)17 b Fp(the)g(map)f Fn(A)e Fo(3)g Fn(a)g Fo(7!)g(k)p Fn(f)5 b Fp(\()p Fn(a)p Fp(\))p Fo(k)15 b(2)f Fn(C)t Fp(.)0 2216 y Fg(\(2.5\))23 b(Lemma.)j Ff(F)l(or)21 b(every)g(set)g Fn(A)g Ff(and)h(every)f Fn(R)p Ff(-pr)m(enorme)m(d)i Fn(R)p Ff(-semimo)m(dule)f Fn(M)k Ff(with)0 2276 y(left)17 b Fn(N)5 b Ff(-summation,)19 b Fn(S)452 2283 y Ft(M)r(;A)554 2276 y Ff(is)f(an)h Fn(R)p Ff(-subsemimo)m(dule)f(of)h Fn(M)1185 2257 y Ft(A)1236 2276 y Ff(and)g Fp(\006)1369 2283 y Ft(M)r(;A)1467 2276 y Fp(:)14 b Fn(S)1526 2283 y Ft(M)r(;A)1624 2276 y Fo(\000)-9 b(!)14 b Fn(M)0 2335 y Ff(is)i(an)g Fn(R)p Ff(-homomorphism)i(of)f Fn(R)p Ff(-pr)m(enorme)m(d)g(semimo)m(dules;)g(in)f(p)m(articular,)h Fo(k)p Fp(\006)1542 2342 y Ft(M)r(;A)1626 2335 y Fp(\()p Fn(f)5 b Fp(\))p Fo(k)15 b(\024)0 2395 y Fp(\006)36 2402 y Ft(C;A)109 2395 y Fp(\()p Fo(k)p Fn(f)5 b Fo(k)p Fp(\))p Fn(:)0 2562 y Ff(Pr)m(o)m(of.)23 b Fp(Let)d Fn(f)26 b Fp(and)19 b Fn(g)i Fp(b)q(e)f(in)f Fn(S)602 2569 y Ft(M)r(;A)685 2562 y Fn(:)h Fp(Cho)q(ose)f(the)h(data)g(in)f(\(2.2\))h(to)g(serv)o(e) f(b)q(oth)h Fn(f)25 b Fp(and)19 b Fn(g)0 2622 y Fp(\(see)c(\(2.3\)\).)g (If)g Fn(\026)326 2629 y Fm(\003)363 2622 y Fp(corresp)q(onds)e(to)i Fn(f)20 b Fp(and)14 b Fn(\027)858 2629 y Fm(\003)895 2622 y Fp(corresp)q(onds)e(to)j Fn(g)h Fp(then,)f(ob)o(viously)l(,)d Fn(\026)1648 2629 y Fm(\003)1678 2622 y Fp(+)7 b Fn(\027)1749 2629 y Fm(\003)0 2682 y Fp(corresp)q(onds)15 b(to)i Fn(f)g Fp(+)11 b Fn(g)r Fp(.)16 b(Hence)h Fn(f)g Fp(+)11 b Fn(g)19 b Fp(b)q(elongs)d(to)h Fn(S)1035 2689 y Ft(M)r(;A)1118 2682 y Fp(.)g(Similarly)e(one)i(sho)o(ws)e(that)i Fn(r)q(f)5 b Fp(,)0 2742 y Fn(r)16 b Fo(2)e Fn(R)p Fp(,)i Fn(f)k Fo(2)14 b Fn(S)275 2749 y Ft(M)r(;A)358 2742 y Fp(,)j(also)f(b)q (elongs)g(to)g Fn(S)759 2749 y Ft(M)r(;A)843 2742 y Fp(.)g(Moreo)o(v)o (er,)92 2837 y(\006)128 2844 y Ft(M)r(;A)212 2837 y Fp(\()p Fn(f)h Fp(+)11 b Fn(g)r Fp(\))j(=)g(\006)470 2844 y Ft(M)514 2837 y Fp(\()p Fn(\026)563 2844 y Fm(\003)597 2837 y Fp(+)d Fn(\027)672 2844 y Fm(\003)694 2837 y Fp(\))k(=)e(\006)816 2844 y Ft(M)860 2837 y Fp(\()p Fn(\026)909 2844 y Fm(\003)933 2837 y Fp(\))e(+)g(\006)1049 2844 y Ft(M)1093 2837 y Fp(\()p Fn(\027)1137 2844 y Fm(\003)1160 2837 y Fp(\))j(=)g(\006)1282 2844 y Ft(M)r(;A)1366 2837 y Fp(\()p Fn(f)5 b Fp(\))12 b(+)f(\006)1531 2844 y Ft(M)r(;A)1615 2837 y Fp(\()p Fn(g)r Fp(\))p eop %%Page: 7 7 7 6 bop 1747 42 a Fp(7)0 167 y(and)361 260 y(\006)397 267 y Ft(M)r(;A)480 260 y Fp(\()p Fn(r)q(f)5 b Fp(\))17 b(=)c(\006)675 267 y Ft(M)719 260 y Fp(\()p Fn(r)760 267 y Fm(\003)784 260 y Fn(\026)814 267 y Fm(\003)837 260 y Fp(\))h(=)g Fn(r)q Fp(\006)982 267 y Ft(M)1027 260 y Fp(\()p Fn(\026)1076 267 y Fm(\003)1099 260 y Fp(\))g(=)g Fn(r)q Fp(\006)1244 267 y Ft(M)r(;A)1329 260 y Fp(\()p Fn(f)5 b Fp(\))p Fn(:)0 354 y Fp(Finally)l(,)330 447 y Fo(k)p Fp(\006)391 454 y Ft(M)r(;A)475 447 y Fp(\()p Fn(f)g Fp(\))p Fo(k)15 b Fp(=)f Fo(k)p Fp(\006)696 454 y Ft(M)740 447 y Fp(\()p Fn(\026)789 454 y Fm(\003)812 447 y Fp(\))p Fo(k)g(\024)g Fp(\006)959 454 y Ft(C)992 447 y Fp(\()p Fo(k)p Fn(\026)1066 454 y Fm(\003)1090 447 y Fo(k)p Fp(\))g(=)f(\006)1236 454 y Ft(C;A)1309 447 y Fp(\()p Fo(k)p Fn(f)5 b Fo(k)p Fp(\))p Fn(:)293 b Fe(\003)0 608 y Fg(\(2.6\))18 b(Lemma.)27 b Ff(L)m(et)17 b Fn(M)23 b Ff(b)m(e)17 b(an)h Fn(R)p Ff(-pr)m(enorme)m(d)h Fn(R)p Ff(-semimo)m(dule)f(with)f(left)g Fn(N)5 b Ff(-summation)0 668 y Fp(\()p Fn(S)50 675 y Ft(M)94 668 y Fn(;)j Fp(\006)152 675 y Ft(M)196 668 y Fp(\))p Ff(.)22 b(F)l(urthermor)m(e)e(let)g Fn(A)h Ff(b)m(e)g(any)g(set.)g(Then)f Fn(f)25 b Fo(2)19 b Fn(M)1217 650 y Ft(A)1271 668 y Ff(b)m(elongs)i(to)g Fn(S)1538 675 y Ft(M)r(;A)1642 668 y Ff(if)g(and)0 728 y(only)d(if)h Fo(k)p Fn(f)5 b Fo(k)14 b(2)g Fn(C)337 710 y Ft(A)386 728 y Ff(b)m(elongs)19 b(to)f Fn(S)648 735 y Ft(C;A)721 728 y Fn(:)0 889 y Ff(Pr)m(o)m(of.)i Fp(\(1.4\),\(o\),)d(and)f(\(2.3\).)1166 b Fo(u)-33 b(t)0 1050 y Fg(\(2.7\))18 b(Lemma.)27 b Ff(L)m(et)17 b Fn(M)23 b Ff(b)m(e)17 b(an)h Fn(R)p Ff(-pr)m(enorme)m(d)h Fn(R)p Ff(-semimo)m(dule)f(with)f(left)g Fn(N)5 b Ff(-summation)0 1110 y Fp(\()p Fn(S)50 1117 y Ft(M)94 1110 y Fn(;)j Fp(\006)152 1117 y Ft(M)196 1110 y Fp(\))p Ff(.)18 b(Then,)f(for)h(any)g(set)e Fn(A;)8 b(M)754 1092 y Fs(\()p Ft(A)p Fs(\))833 1110 y Fp(:=)14 b Fo(f)p Fn(f)19 b Fo(2)14 b Fn(M)1068 1092 y Ft(A)1115 1110 y Fp(:)g(supp)h Fn(f)23 b Ff(is)17 b(\014nite)p Fo(g)g Ff(is)h(c)m(ontaine)m(d)0 1169 y(in)k Fn(S)96 1176 y Ft(M)r(;A)202 1169 y Ff(and)h(for)g(al)s(l)f Fn(f)27 b Fo(2)22 b Fn(M)622 1151 y Fs(\()p Ft(A)p Fs(\))687 1169 y Fn(;)8 b Fp(\006)745 1176 y Ft(M)r(;A)829 1169 y Fp(\()p Fn(f)d Fp(\))23 b(=)980 1132 y Fi(P)1032 1144 y Fm(0)1046 1169 y Fo(f)p Fn(f)5 b Fp(\()p Fn(a)p Fp(\))24 b(:)d Fn(a)h Fo(2)g Fp(supp)38 b Fn(f)5 b Fo(g)p Ff(,)23 b(wher)m(e)1705 1132 y Fi(P)1758 1144 y Fm(0)0 1229 y Ff(stands)c(for)g(the)e(usual)h(sum)g(of)g(\014nitely)g(many)g (elements)f(in)h Fn(M)5 b Ff(.)0 1390 y(Pr)m(o)m(of.)20 b Fp(\(1.4\),)d(\(i\),)g(and)f(\(2.2\).)1160 b Fo(u)-33 b(t)0 1551 y Fg(\(2.8\))18 b(Lemma.)27 b Ff(L)m(et)17 b Fn(M)23 b Ff(b)m(e)17 b(an)h Fn(R)p Ff(-pr)m(enorme)m(d)h Fn(R)p Ff(-semimo)m(dule)f(with)f(left)g Fn(N)5 b Ff(-summation)0 1611 y Fp(\()p Fn(S)50 1618 y Ft(M)94 1611 y Fn(;)j Fp(\006)152 1618 y Ft(M)196 1611 y Fp(\))p Ff(.)18 b(F)l(urthermor)m(e)f(let)g Fn(A)g Ff(b)m(e)h(any)g(set.)f(If)g Fn(f)23 b Ff(is)18 b(in)g Fn(S)1150 1618 y Ft(M)r(;A)1250 1611 y Ff(and)g Fn(g)e Fo(2)e Fn(M)1486 1593 y Ft(A)1536 1611 y Ff(with)k Fo(k)p Fn(g)r Fo(k)13 b(\024)0 1671 y(k)p Fn(f)5 b Fo(k)18 b Ff(then)g Fn(g)i Ff(is)e(in)g Fn(S)397 1678 y Ft(M)r(;A)498 1671 y Ff(and)h Fo(k)p Fp(\006)656 1678 y Ft(M)r(;A)740 1671 y Fp(\()p Fn(g)r Fp(\))p Fo(k)14 b(\024)f Fp(\006)931 1678 y Ft(C;A)1005 1671 y Fp(\()p Fo(k)p Fn(f)5 b Fo(k)p Fp(\))p Fn(:)0 1832 y Ff(Pr)m(o)m(of.)20 b Fp(\(1.4\),\(ii\),)d(and)f (\(2.2\).)1163 b Fo(u)-33 b(t)0 1993 y Fg(\(2.9\))18 b(Lemma.)27 b Ff(L)m(et)17 b Fn(M)23 b Ff(b)m(e)17 b(an)h Fn(R)p Ff(-pr)m(enorme)m(d)h Fn(R)p Ff(-semimo)m(dule)f(with)f(left)g Fn(N)5 b Ff(-summation)0 2052 y Fp(\()p Fn(S)50 2059 y Ft(M)94 2052 y Fn(;)j Fp(\006)152 2059 y Ft(M)196 2052 y Fp(\))p Ff(.)19 b(F)l(urthermor)m(e)g(let)e Fn(A)i Ff(b)m(e)f(any)h(set.)f(If)h Fn(f)24 b Ff(is)19 b(in)g Fn(S)1162 2059 y Ft(M)r(;A)1263 2052 y Ff(and)g Fn( )e Fp(:)e Fn(A)g Fo(\000)-9 b(!)14 b Fn(A)19 b Ff(is)g(any)0 2121 y(map)f(then,)h(for)f(any)h Fn(a)14 b Fo(2)g Fn(A)p Ff(,)k(the)g(map)g Fn(f)789 2103 y Ft( )816 2088 y Fk(\000)p Fj(1)864 2103 y Fs(\()p Ft(a)p Fs(\))937 2121 y Ff(given)g(by)540 2242 y Fn(A)c Fo(3)g Fn(b)h Fo(7!)737 2172 y Fi(\032)783 2213 y Fn(f)5 b Fp(\()p Fn(b)p Fp(\))565 b Ff(,)18 b(if)h Fn( )r Fp(\()p Fn(b)p Fp(\))c(=)e Fn(a)p Ff(;)783 2273 y Fp(0)628 b Ff(,)18 b(otherwise.)0 2378 y(is)h(in)g Fn(S)147 2385 y Ft(M)r(;A)231 2378 y Ff(;)g(mor)m(e)m(over,)h(the)f (map)g Fn(f)719 2360 y Ft( )746 2345 y Fk(\000)p Fj(1)815 2378 y Ff(given)g(by)f Fn(A)e Fo(3)g Fn(a)g Fo(7!)f Fp(\006)1256 2385 y Ft(M)r(;A)1340 2378 y Fp(\()p Fn(f)1388 2360 y Ft( )1415 2345 y Fk(\000)p Fj(1)1463 2360 y Fs(\()p Ft(a)p Fs(\))1518 2378 y Fp(\))h Fo(2)g Fn(M)24 b Ff(is)19 b(in)0 2447 y Fn(S)31 2454 y Ft(M)r(;A)132 2447 y Ff(and)g Fp(\006)265 2454 y Ft(M)r(;A)349 2447 y Fp(\()p Fn(f)397 2429 y Ft( )424 2414 y Fk(\000)p Fj(1)474 2447 y Fp(\))c(=)e(\006)596 2454 y Ft(M)r(;A)680 2447 y Fp(\()p Fn(f)5 b Fp(\))p Ff(.)0 2608 y(Pr)m(o)m(of.)20 b Fp(\(1.4\),\(iii\),)d(and)f(\(2.2\).) 1149 b Fo(u)-33 b(t)0 2769 y Fg(\(2.10\))17 b(Corollary)-5 b(.)29 b Ff(L)m(et)17 b Fn(M)22 b Ff(b)m(e)16 b(an)h Fn(R)p Ff(-pr)m(enorme)m(d)h Fn(R)p Ff(-semimo)m(dule)f(with)g(left)f Fn(N)5 b Ff(-summa-)0 2828 y(tion)21 b Fp(\()p Fn(S)156 2835 y Ft(M)200 2828 y Fn(;)8 b Fp(\006)258 2835 y Ft(M)302 2828 y Fp(\))p Ff(.)21 b(F)l(urthermor)m(e)g(let)e Fn(A)i Ff(b)m(e)g(any)g(set)f(and)h Fn(\031)g Fp(:)d Fn(A)h Fo(\000)-8 b(!)18 b Fn(A)j Ff(b)m(e)f(any)h(bije)m(ction.)p eop %%Page: 8 8 8 7 bop 0 42 a Fp(8)349 b Fv(Section)17 b(3:)61 b(B.)15 b(P)o(areigis,)i(D.)f(Pumpl)q(\177)-24 b(un,)15 b(and)f(H.)h(R\177)-23 b(ohrl)0 176 y Ff(Then)21 b Fn(f)26 b Fo(2)21 b Fn(M)290 158 y Ft(A)345 176 y Ff(is)h(in)f Fn(S)497 183 y Ft(M)r(;A)602 176 y Ff(if)h(and)g(only)g(if)g Fn(f)947 158 y Ft(\031)995 176 y Fp(:=)e Fn(f)f Fo(\016)14 b Fn(\031)24 b Ff(is)e(in)f Fn(S)1354 183 y Ft(M)r(;A)1438 176 y Ff(,)g(in)h(which)h(c)m(ase)0 235 y Fp(\006)36 242 y Ft(M)r(;A)120 235 y Fp(\()p Fn(f)168 217 y Ft(\031)196 235 y Fp(\))14 b(=)g(\006)318 242 y Ft(M)r(;A)402 235 y Fp(\()p Fn(f)5 b Fp(\))p Fn(:)0 389 y Ff(Pr)m(o)m(of.)20 b Fp(Immediate)c(consequence)g(of)h(\(2.9\))901 b Fo(u)-33 b(t)0 542 y Fg(\(2.11\))14 b(Lemma.)26 b Ff(L)m(et)15 b Fn(M)k Ff(b)m(e)c(an)f Fn(R)p Ff(-pr)m(enorme)m(d)i Fn(R)p Ff(-semimo)m(dule)f(with)g(left)e Fn(N)5 b Ff(-summation)0 602 y Fp(\()p Fn(S)50 609 y Ft(M)94 602 y Fn(;)j Fp(\006)152 609 y Ft(M)196 602 y Fp(\))p Ff(.)19 b(F)l(urthermor)m(e)e(let)g Fn(A)h Ff(and)h Fn(B)h Ff(b)m(e)e(any)g(sets.)h(F)l(or)f Fn(f)h Fo(2)14 b Fn(S)1300 609 y Ft(M)r(;A)p Fm(\002)p Ft(B)1464 602 y Ff(de\014ne)k Fn(f)5 b Fp(\()p Fn(a;)j Ff(-)r Fp(\))14 b(:)0 662 y Fn(B)i Fo(\000)-8 b(!)13 b Fn(M)24 b Ff(r)m(esp.)19 b Fn(f)5 b Fp(\()p Ff(-)q Fn(;)j(b)p Fp(\))15 b(:)e Fn(A)i Fo(\000)-9 b(!)14 b Fn(M)23 b Ff(as)c(the)f(maps)273 753 y Fn(B)e Fo(3)e Fn(b)h Fo(7!)e Fn(f)5 b Fp(\()p Fn(a;)j(b)p Fp(\))16 b Fo(2)e Fn(M)108 b Ff(r)m(esp.)103 b Fn(A)15 b Fo(3)f Fn(a)g Fo(7!)f Fn(f)5 b Fp(\()p Fn(a;)j(b)p Fp(\))17 b Fo(2)d Fn(M)s(:)0 844 y Ff(Then)19 b Fn(f)5 b Fp(\()p Fn(a;)j Ff(-)q Fp(\))19 b Ff(in)g(in)g Fn(S)439 851 y Ft(M)r(;B)524 844 y Fn(;)g Ff(for)g(al)s(l)f Fn(a)e Fo(2)g Fn(A)p Ff(,)j(and)h Fn(f)5 b Fp(\()p Ff(-)q Fn(;)j(b)p Fp(\))19 b Ff(is)h(in)f Fn(S)1265 851 y Ft(M)r(;A)1348 844 y Fn(;)g Ff(for)g(al)s(l)f Fn(b)e Fo(2)f Fn(B)r Ff(,)20 b(and)0 904 y(the)e(maps)125 995 y Fn(A)c Fo(3)h Fn(a)f Fo(7!)f Fp(\006)363 1002 y Ft(M)r(;B)449 995 y Fp(\()p Fn(f)5 b Fp(\()p Fn(a;)j Ff(-)r Fp(\)\))14 b Fo(2)g Fn(M)108 b Ff(r)m(esp.)103 b Fn(B)16 b Fo(3)e Fn(b)h Fo(7!)e Fp(\006)1279 1002 y Ft(M)r(;A)1363 995 y Fp(\()p Fn(f)5 b Fp(\()p Ff(-)q Fn(;)j(b)p Fp(\)\))16 b Fo(2)e Fn(M)0 1086 y Ff(ar)m(e)19 b(in)f Fn(S)177 1093 y Ft(M)r(;A)278 1086 y Ff(r)m(esp.)h Fn(S)429 1093 y Ft(M)r(;B)532 1086 y Ff(and)g(satisfy)43 1178 y Fp(\006)79 1185 y Ft(M)r(;A)163 1178 y Fo(f)p Fp(\006)224 1185 y Ft(M)r(;B)309 1178 y Fo(f)p Fn(f)5 b Fp(\()p Fn(a;)j(b)p Fp(\))17 b(:)c Fn(b)h Fo(2)g Fn(B)r Fo(g)g Fp(:)g Fn(a)g Fo(2)g Fn(A)p Fo(g)g Fp(=)g(\006)955 1185 y Ft(M)r(;B)1041 1178 y Fo(f)p Fp(\006)1102 1185 y Ft(M)r(;A)1185 1178 y Fo(f)p Fn(f)5 b Fp(\()p Fn(a;)j(b)p Fp(\))17 b(:)c Fn(a)h Fo(2)g Fn(A)p Fo(g)h Fp(:)e Fn(b)h Fo(2)g Fn(B)r Fo(g)866 1253 y Fp(=)g(\006)955 1260 y Ft(M)r(;A)p Fm(\002)p Ft(B)1102 1253 y Fp(\()p Fn(f)5 b Fp(\))p Fn(:)0 1405 y Ff(Pr)m(o)m(of.)22 b Fp(By)d(\(2.2\),)f (\(2.11\))h(can)f(b)q(e)g(reduced)g(to)g(the)h(case)f Fn(A)f Fp(=)f Fn(B)j Fp(=)e Fn(N)5 b Fp(.)19 b(In)f(this)f(situation)0 1465 y(c)o(ho)q(ose)f(a)h(bijection)f Fn(N)448 1447 y Fs(2)485 1465 y Fo(\000)-9 b(!)14 b Fn(N)5 b Fp(,)17 b(use)f(\(2.1\),)g(and)g(apply)g(\(1.4\),)h(\(iii\),)g(t)o(wice.)261 b Fo(u)-33 b(t)71 1586 y Fp(As)16 b(a)h(sp)q(ecial)f(case)h(of)f (\(2.11\))h(w)o(e)g(obtain)0 1739 y Fg(\(2.12\))g(Corollary)-5 b(.)29 b Ff(L)m(et)17 b Fn(M)22 b Ff(b)m(e)16 b(an)h Fn(R)p Ff(-pr)m(enorme)m(d)h Fn(R)p Ff(-semimo)m(dule)f(with)g(left)f Fn(N)5 b Ff(-summa-)0 1799 y(tion)22 b Fp(\()p Fn(S)157 1806 y Ft(M)201 1799 y Fn(;)8 b Fp(\006)259 1806 y Ft(M)303 1799 y Fp(\))p Ff(.)22 b(If)g Fn(\013)447 1806 y Fm(\003)489 1799 y Fo(2)e Fn(S)573 1806 y Ft(R)627 1799 y Ff(and)i Fn(m)e Fo(2)g Fn(M)27 b Ff(then)21 b(the)g(map)h Fn(\013)1264 1806 y Fm(\003)1287 1799 y Fn(m)f Ff(given)g(by)g Fn(N)26 b Fo(3)20 b Fn(n)g Fo(7!)0 1859 y Fn(\013)32 1866 y Ft(n)59 1859 y Fn(m)15 b Fo(2)h Fn(M)24 b Ff(is)c(in)f Fn(S)387 1866 y Ft(M)449 1859 y Ff(and)h Fp(\006)583 1866 y Ft(M)627 1859 y Fp(\()p Fn(\013)678 1866 y Fm(\003)701 1859 y Fn(m)p Fp(\))c(=)f(\(\006)889 1866 y Ft(R)922 1859 y Fn(\013)954 1866 y Fm(\003)977 1859 y Fp(\))p Fn(m)p Ff(.)k(Similarly)g(if)g Fn(r)f Fo(2)d Fn(R)k Ff(and)h Fn(\026)1610 1866 y Fm(\003)1648 1859 y Fo(2)c Fn(S)1728 1866 y Ft(M)0 1919 y Ff(then)g(the)f(map)i Fn(r)q(\026)350 1926 y Fm(\003)389 1919 y Ff(given)f(by)g Fn(N)j Fo(3)14 b Fn(n)g Fo(7!)f Fn(r)q(\026)846 1926 y Ft(n)888 1919 y Fo(2)h Fn(M)22 b Ff(is)16 b(in)g Fn(S)1146 1926 y Ft(M)1205 1919 y Ff(and)h Fp(\006)1336 1926 y Ft(M)1380 1919 y Fp(\()p Fn(r)q(\026)1452 1926 y Fm(\003)1476 1919 y Fp(\))e(=)e Fn(r)q Fp(\(\006)1640 1926 y Ft(M)1686 1919 y Fn(\026)1716 1926 y Fm(\003)1738 1919 y Fp(\))p Fn(:)0 1978 y Fo(u)-33 b(t)0 2132 y Fg(\(2.13\))20 b(Corollary)-5 b(.)29 b Ff(L)m(et)19 b Fn(f)i Fo(2)16 b Fn(M)676 2114 y Ft(A)709 2132 y Ff(,)j Fn(')c Fp(:)g Fn(A)h Fo(\000)-9 b(!)16 b Fn(B)k Ff(a)g(map.)f(F)l(or)g Fn(b)d Fo(2)g Fn(B)k Ff(let)e Fo(k)p Fn(f)5 b Fo(k)1585 2114 y Ft(')1611 2099 y Fk(\000)p Fj(1)1659 2114 y Fs(\()p Ft(b)p Fs(\))1728 2132 y Ff(b)m(e)0 2192 y(the)18 b(map)512 2310 y Fn(A)d Fo(3)f Fn(a)g Fo(7!)715 2239 y Fi(\032)760 2281 y Fo(k)p Fn(f)5 b Fp(\()p Fn(a)p Fp(\))p Fo(k)565 b Ff(,)18 b(if)h Fn(')p Fp(\()p Fn(a)p Fp(\))14 b(=)g Fn(b)p Ff(;)760 2341 y Fp(0)683 b Ff(,)18 b(otherwise.)0 2443 y(Supp)m(ose)i(that)e Fo(k)p Fn(f)5 b Fo(k)371 2425 y Ft(')397 2410 y Fk(\000)p Fj(1)444 2425 y Fs(\()p Ft(b)p Fs(\))513 2443 y Ff(is)18 b(in)g Fn(S)658 2450 y Ft(C;A)749 2443 y Ff(for)g(every)g Fn(b)c Fo(2)g Fn(B)20 b Ff(and)f(that)f(the)g (map)406 2566 y Fn(B)e Fo(3)e Fn(b)h Fo(7!)606 2495 y Fi(\032)652 2543 y Fp(\006)688 2550 y Ft(C;A)761 2543 y Fp(\()p Fo(k)p Fn(f)5 b Fo(k)859 2525 y Ft(')885 2510 y Fk(\000)p Fj(1)933 2525 y Fs(\()p Ft(b)p Fs(\))984 2543 y Fp(\))450 b Ff(,)18 b(if)g Fn(b)c Fo(2)g Fn(')p Fp(\()p Fn(A)p Fp(\))p Ff(;)652 2603 y Fp(0)776 b Ff(,)18 b(otherwise;)0 2684 y(is)g(in)h Fn(S)146 2691 y Ft(C;B)220 2684 y Ff(.)f(Then)g Fn(f)24 b Ff(is)18 b(in)g Fn(S)576 2691 y Ft(M)r(;A)660 2684 y Ff(.)0 2837 y(Pr)m(o)m(of.)i Fp(This)c(is)g(an)g(immediate)g(consequence)g(of)g (\(1.1\),\(iii'\),\(1.4\),)h(\(o\),)h(and)d(\(2.2\).)111 b Fo(u)-33 b(t)p eop %%Page: 9 9 9 8 bop 1747 42 a Fp(9)71 167 y(If)18 b(should)e(b)q(e)h(p)q(oin)o(ted) g(out)h(that,)f(with)h(all)f(index)g(sets)h(assumed)d(to)j(ha)o(v)o(e)f (cardinalit)o(y)0 227 y Fo(\024)g Fp(card)p Fn(N)5 b Fp(,)19 b(there)g(is)f(a)h(corresp)q(ondence)e(b)q(et)o(w)o(een)i (certain)g(axioms)f(in)g([7],)g Fo(x)q Fp(6,)g(and)g(some)0 286 y(of)f(the)g(results)e(obtained)h(here.)g(This)f(corresp)q(ondence) g(is)h(as)g(follo)o(ws:)441 375 y(Equiv)m(alen)o(t)h(F)l(amilies)d (Axiom)31 b Fo(\021)f Fp(\(2.9\),)619 450 y(Unary)16 b(Sum)f(Axiom)31 b Fo(\021)f Fp(\(1.4\),)17 b(\(i\),)400 524 y(Generalized)f(P)o(artition)f(Axiom)31 b Fo(\021)f Fp(\(1.4\),)17 b(\(iii\),)463 599 y(W)l(eak)g(Double)f(Sum)f(Axiom)31 b Fo(\021)f Fp(\(2.11\).)364 852 y Fq(3.)16 b(The)h(closed)f(unit)g (ball)g(functor)0 972 y Fp(As)k(in)f([5])h(one)f(de\014nes)g(the)h (closed)f(unit)h(ball)f(functor)g Fo(B)1129 979 y Ft(N)1186 972 y Fp(:)g Fn(N)1259 979 y Ft(R)1292 972 y Fp(pnSmo)q(d)1471 949 y Fs(1)1512 972 y Fo(\000)-8 b(!)19 b Fp(Set.)h(Its)0 1031 y(v)m(alue)d(on)f(the)h(ob)s(ject)f Fn(M)23 b Fp(of)17 b Fn(N)602 1038 y Ft(R)634 1031 y Fp(pnSmo)q(d)813 1009 y Fs(1)852 1031 y Fp(is)540 1123 y Fo(B)573 1130 y Ft(N)610 1123 y Fp(\()p Fn(M)5 b Fp(\))16 b(:=)d Fo(f)p Fn(m)h Fo(2)g Fn(M)19 b Fp(:)14 b Fo(k)p Fn(m)p Fo(k)f(\024)h Fp(1)p Fo(g)p Fn(:)0 1277 y Fg(\(3.1\))19 b(Theorem.)27 b Fo(B)439 1284 y Ft(N)490 1277 y Fp(:)14 b Fn(N)558 1284 y Ft(R)590 1277 y Fp(pnSmo)q(d)769 1254 y Fs(1)805 1277 y Fo(\000)-9 b(!)14 b Fp(Set)k Ff(has)h(a)f(left)g(adjoint)h Fo(L)1407 1259 y Ft(N)1445 1277 y Ff(.)0 1431 y(Pr)m(o)m(of.)h Fp(W)l(e)d(put)f Fo(L)360 1413 y Ft(N)398 1431 y Fp(\()p Fo(;)p Fp(\))f(:=)e Fo(f)p Fp(0)p Fo(g)p Fp(.)j(If)h Fn(A)d Fo(6)p Fp(=)g Fo(;)i Fp(is)h(an)o(y)f(nonempt)o(y)f(set,)i(w)o (e)f(put)g({)h(as)f(a)h(set)f({)644 1522 y Fo(L)678 1501 y Ft(N)716 1522 y Fp(\()p Fn(A)p Fp(\))f(:=)e Fn(S)903 1529 y Ft(R;A)991 1522 y Fo(\022)h Fn(R)1082 1501 y Ft(A)1114 1522 y Fn(:)0 1613 y Fp(Due)i(to)h(\(2.5\),)g Fo(L)332 1595 y Ft(N)370 1613 y Fp(\()p Fn(A)p Fp(\))h(is)e(an)g Fn(R)p Fp(-subsemimo)q(dule)e(of)j Fn(R)1080 1595 y Ft(A)1112 1613 y Fp(.)71 1673 y(Next)h(w)o(e)e(de\014ne)g(a)g(prenorm)f Fo(jjj)h(jjj)d Fp(:)h Fo(L)826 1655 y Ft(N)864 1673 y Fp(\()p Fn(A)p Fp(\))h Fo(\000)-8 b(!)13 b Fn(C)20 b Fp(b)o(y)c(putting)0 1764 y(\(3)p Fn(:)p Fp(2\))242 b Fo(jjj)p Fn(f)5 b Fo(jjj)14 b Fp(:=)f(\006)573 1771 y Ft(C;A)646 1764 y Fp(\()p Fo(k)p Fn(f)5 b Fo(k)p Fp(\))15 b(=)f(\006)867 1771 y Ft(C;A)940 1764 y Fo(fk)p Fn(f)5 b Fp(\()p Fn(a)p Fp(\))p Fo(k)16 b Fp(:)d Fn(a)h Fo(2)g Fn(A)p Fo(g)171 b Fn(;)8 b(f)20 b Fo(2)14 b(L)1618 1743 y Ft(N)1656 1764 y Fp(\()p Fn(A)p Fp(\))p Fn(:)0 1855 y Fp(Since)20 b Fo(k)p Fn(f)g Fp(+)14 b Fn(g)r Fo(k)20 b(\024)h(k)p Fn(f)5 b Fo(k)14 b Fp(+)g Fo(k)p Fn(g)r Fo(k)p Fp(,)20 b Fn(f)26 b Fp(and)20 b Fn(g)j Fp(in)d Fo(L)937 1837 y Ft(N)975 1855 y Fp(\()p Fn(A)p Fp(\),)i(w)o(e)f(obtain) f(from)g(\(2.5\))h(and)f(\(2.8\))0 1915 y Fo(jjj)p Fn(f)h Fp(+)15 b Fn(g)r Fo(jjj)24 b(\024)h(jjj)p Fn(f)5 b Fo(jjj)16 b Fp(+)f Fo(jjj)p Fn(g)r Fo(jjj)p Fp(.)45 b Fo(jjj)p Fn(r)q(f)5 b Fo(jjj)26 b(\024)f(k)p Fn(r)q Fo(k)16 b(\001)f(jjj)p Fn(f)5 b Fo(jjj)p Fp(,)23 b Fn(r)k Fo(2)f Fn(R)p Fp(,)d Fn(f)31 b Fo(2)25 b(L)1473 1897 y Ft(N)1511 1915 y Fp(\()p Fn(A)p Fp(\),)g(follo)o(ws)0 1975 y(similarly)l(.)15 b(This)g(means)h(that)h Fo(L)622 1957 y Ft(N)660 1975 y Fp(\()p Fn(A)p Fp(\))h(is)e(an)g Fn(R)p Fp(-prenormed)e Fn(R)p Fp(-semimo)q(dule.)71 2035 y(It)f(remains)f(to)i(de\014ne)e(\()p Fn(S)552 2044 y Fm(L)580 2034 y Fd(N)612 2044 y Fs(\()p Ft(A)p Fs(\))675 2035 y Fn(;)c Fp(\006)733 2044 y Fm(L)761 2034 y Fd(N)793 2044 y Fs(\()p Ft(A)p Fs(\))857 2035 y Fp(\).)13 b(Let)h Fn(F)1021 2042 y Fm(\003)1058 2035 y Fo(2)g Fp(\()p Fn(R)1162 2016 y Ft(A)1195 2035 y Fp(\))1214 2016 y Ft(N)1252 2035 y Fp(.)f(Giv)o(en)g Fn(a)h Fo(2)g Fn(A)g Fp(w)o(e)f(denote)0 2094 y(the)19 b(map)f Fn(N)24 b Fo(3)18 b Fn(n)g Fo(7!)g Fn(F)464 2101 y Ft(n)491 2094 y Fp(\()p Fn(a)p Fp(\))h Fo(2)f Fn(R)h Fp(b)o(y)g Fn(F)786 2101 y Fm(\003)809 2094 y Fp(\()p Fn(a)p Fp(\).)h(Due)f(to)g(\(1.3\),)g (\(o\),)h Fn(F)1341 2101 y Fm(\003)1364 2094 y Fp(\()p Fn(a)p Fp(\))g(is)e(in)h Fn(S)1592 2101 y Ft(R)1643 2094 y Fp(if)g(and)0 2154 y(only)e(if)h Fo(k)p Fn(F)214 2161 y Fm(\003)236 2154 y Fp(\()p Fn(a)p Fp(\))p Fo(k)p Fp(,)h(that)e(is)g (the)h(map)e Fn(N)21 b Fo(3)15 b Fn(n)h Fo(7!)f(k)p Fn(F)992 2161 y Ft(n)1019 2154 y Fp(\()p Fn(a)p Fp(\))p Fo(k)h(2)f Fn(C)t Fp(,)i(is)f(in)h Fn(S)1383 2161 y Ft(C)1416 2154 y Fp(.)h(In)f(this)g(case)g(w)o(e)0 2214 y(can)f(form)g(the)h(map)f (\006)442 2221 y Ft(C)475 2214 y Fo(k)p Fn(F)532 2221 y Fm(\003)555 2214 y Fo(k)e Fp(:)f Fn(A)h Fo(\000)-8 b(!)14 b Fn(C)19 b Fp(that)e(is)f(giv)o(en)g(b)o(y)568 2305 y(\(\006)623 2312 y Ft(C)657 2305 y Fo(k)p Fn(F)714 2312 y Fm(\003)736 2305 y Fo(k)p Fp(\)\()p Fn(a)p Fp(\))g(:=)d(\006)962 2312 y Ft(C)995 2305 y Fp(\()p Fo(k)p Fn(F)1071 2312 y Fm(\003)1095 2305 y Fp(\()p Fn(a)p Fp(\))p Fo(k)p Fp(\))408 b Fn(;)8 b(a)14 b Fo(2)g Fn(A:)0 2396 y Fp(With)j(these)f(notations)g (w)o(e)g(ha)o(v)o(e)80 2475 y Fn(S)g Fp(:)e(=)f Fn(S)238 2485 y Fm(L)266 2475 y Fd(N)297 2485 y Fs(\()p Ft(A)p Fs(\))155 2558 y Fp(=)g Fo(f)p Fn(F)264 2565 y Fm(\003)301 2558 y Fo(2)h(L)382 2537 y Ft(N)420 2558 y Fp(\()p Fn(A)p Fp(\))495 2537 y Ft(N)548 2558 y Fp(:)f Fo(k)p Fn(F)632 2565 y Fm(\003)655 2558 y Fp(\()p Fn(a)p Fp(\))p Fo(k)i(2)f Fn(S)837 2565 y Ft(C)870 2558 y Fn(;)25 b Fp(for)16 b(all)g Fn(a)e Fo(2)h Fn(A;)25 b Fp(and)15 b(\006)1350 2565 y Ft(C)1384 2558 y Fo(k)p Fn(F)1441 2565 y Fm(\003)1464 2558 y Fo(k)f(2)g Fn(S)1581 2565 y Ft(C;A)1653 2558 y Fo(g)p Fn(:)0 2649 y Fp(In)f(order)g(to)h(de\014ne)f(\006)h(:=)f(\006) 538 2659 y Fm(L)566 2649 y Fd(N)598 2659 y Fs(\()p Ft(A)p Fs(\))661 2649 y Fp(,)h(let)g Fn(F)790 2656 y Fm(\003)826 2649 y Fp(b)q(e)g(in)f Fn(S)s Fp(.)g(Since)g Fo(k)p Fn(F)1191 2656 y Fm(\003)1214 2649 y Fp(\()p Fn(a)p Fp(\))p Fo(k)i Fp(is)e(in)g Fn(S)1451 2656 y Ft(C)1498 2649 y Fp(w)o(e)g(ha)o(v)o(e)g (that)0 2709 y Fn(F)32 2716 y Fm(\003)55 2709 y Fp(\()p Fn(a)p Fp(\),)k(that)f(is)f(the)i(map)e Fn(N)k Fo(3)14 b Fn(n)g Fo(7!)f Fn(F)747 2716 y Ft(n)774 2709 y Fp(\()p Fn(a)p Fp(\))i Fo(2)f Fn(R)p Fp(,)i(is)g(in)f Fn(S)1106 2716 y Ft(R)1138 2709 y Fp(.)h(Due)g(to)g(\(1.3\),)g(\(ii\),)h(w)o(e)e (obtain)0 2769 y Fo(k)p Fp(\006)61 2776 y Ft(R)94 2769 y Fn(F)126 2776 y Fm(\003)148 2769 y Fp(\()p Fn(a)p Fp(\))p Fo(k)j(\024)d Fp(\006)345 2776 y Ft(C)379 2769 y Fo(k)p Fn(F)436 2776 y Fm(\003)459 2769 y Fp(\()p Fn(a)p Fp(\))p Fo(k)p Fp(.)j(If)h(\006)668 2776 y Ft(R)700 2769 y Fn(F)732 2776 y Fm(\003)773 2769 y Fp(denotes)f(the)g(map)f Fn(A)f Fo(3)g Fn(a)h Fo(7!)f Fp(\006)1401 2776 y Ft(R)1433 2769 y Fn(F)1465 2776 y Fm(\003)1488 2769 y Fp(\()p Fn(a)p Fp(\))h Fo(2)g Fn(R)h Fp(then)0 2828 y(w)o(e)e(ha)o(v)o(e)g Fo(k)p Fp(\006)249 2835 y Ft(R)281 2828 y Fn(F)313 2835 y Fm(\003)336 2828 y Fo(k)e(\024)f Fp(\006)463 2835 y Ft(C)497 2828 y Fo(k)p Fn(F)554 2835 y Fm(\003)577 2828 y Fo(k)p Fp(.)j(Since)g(the)g(latter)h(function)e(is)h(in)g Fn(S)1316 2835 y Ft(C;A)1389 2828 y Fp(,)g(\(2.8\))h(sho)o(ws)d(that)p eop %%Page: 10 10 10 9 bop 0 42 a Fp(10)337 b Fv(Section)16 b(3:)61 b(B.)15 b(P)o(areigis,)j(D.)d(Pumpl)q(\177)-24 b(un,)15 b(and)g(H.)f(R\177)-23 b(ohrl)0 176 y Fo(k)p Fp(\006)61 183 y Ft(R)94 176 y Fn(F)126 183 y Fm(\003)148 176 y Fo(k)16 b Fp(is)f(in)g Fn(S)326 183 y Ft(C;A)399 176 y Fp(,)g(whence)h(\006)638 183 y Ft(R)670 176 y Fn(F)702 183 y Fm(\003)741 176 y Fp(is)f(in)g Fn(S)878 183 y Ft(R;A)968 176 y Fp(due)g(to)h(\(2.6\).)g (In)f(other)g(w)o(ords,)f(\006)1635 183 y Ft(R)1668 176 y Fn(F)1700 183 y Fm(\003)1738 176 y Fp(is)0 235 y(in)i Fo(L)92 217 y Ft(N)130 235 y Fp(\()p Fn(A)p Fp(\),)i(and)e(w)o(e)g(put) g(\006)p Fn(F)566 242 y Fm(\003)603 235 y Fp(:=)d(\006)705 242 y Ft(R)738 235 y Fn(F)770 242 y Fm(\003)793 235 y Fp(.)71 295 y(A)o(t)h(this)f(p)q(oin)o(t)g(w)o(e)h(ha)o(v)o(e)f(to)h (sho)o(w)f(that)h(\()p Fn(S;)8 b Fp(\006\))14 b(is)f(a)h(left)g Fn(N)5 b Fp(-summation)12 b(for)h Fo(L)1563 277 y Ft(N)1602 295 y Fp(\()p Fn(A)p Fp(\).)h(So,)0 355 y(let)i Fn(F)103 362 y Fm(\003)142 355 y Fp(and)f Fn(G)277 362 y Fm(\003)316 355 y Fp(b)q(e)h(in)g Fn(S)s Fp(.)f(Then)g Fn(F)665 362 y Fm(\003)698 355 y Fp(+)10 b Fn(G)786 362 y Fm(\003)825 355 y Fp(is)15 b(the)h(map)f Fn(N)k Fo(3)14 b Fn(n)g Fo(7!)g Fn(F)1315 362 y Ft(n)1352 355 y Fp(+)9 b Fn(G)1439 362 y Ft(n)1480 355 y Fo(2)14 b(L)1561 337 y Ft(N)1599 355 y Fp(\()p Fn(A)p Fp(\))j(and)0 415 y(hence)e Fn(F)169 422 y Fm(\003)200 415 y Fp(+)8 b Fn(G)286 422 y Fm(\003)322 415 y Fo(2)14 b(L)403 397 y Ft(N)441 415 y Fp(\()p Fn(A)p Fp(\))516 397 y Ft(N)555 415 y Fp(.)h(Moreo)o(v)o(er,)e(\()p Fn(F)866 422 y Ft(n)902 415 y Fp(+)8 b Fn(G)988 422 y Ft(n)1015 415 y Fp(\)\()p Fn(a)p Fp(\))15 b(=)f Fn(F)1198 422 y Ft(n)1225 415 y Fp(\()p Fn(a)p Fp(\))8 b(+)g Fn(G)1383 422 y Ft(n)1411 415 y Fp(\()p Fn(a)p Fp(\))16 b(for)f(all)f Fn(a)g Fo(2)g Fn(A)p Fp(.)0 474 y(Th)o(us,)h Fo(k)p Fp(\()p Fn(F)216 481 y Fm(\003)251 474 y Fp(+)c Fn(G)340 481 y Fm(\003)363 474 y Fp(\)\()p Fn(a)p Fp(\))p Fo(k)16 b(\024)e(k)p Fn(F)597 481 y Fm(\003)620 474 y Fp(\()p Fn(a)p Fp(\))p Fo(k)e Fp(+)f Fo(k)p Fn(G)835 481 y Fm(\003)858 474 y Fp(\()p Fn(a)p Fp(\))p Fo(k)p Fp(.)18 b(Since)e(b)q(oth)h Fo(k)p Fn(F)1284 481 y Fm(\003)1307 474 y Fp(\()p Fn(a)p Fp(\))p Fo(k)h Fp(and)e Fo(k)p Fn(G)1575 481 y Fm(\003)1598 474 y Fp(\()p Fn(a)p Fp(\))p Fo(k)i Fp(are)0 534 y(in)f Fn(S)90 541 y Ft(C)123 534 y Fp(,)h(\(1.1\))g(sho)o(ws)e(that)i Fo(k)p Fn(F)585 541 y Fm(\003)608 534 y Fp(\()p Fn(a)p Fp(\))p Fo(k)12 b Fp(+)g Fo(k)p Fn(G)824 541 y Fm(\003)847 534 y Fp(\()p Fn(a)p Fp(\))p Fo(k)19 b Fp(is)e(in)g Fn(S)1096 541 y Ft(C)1129 534 y Fp(.)g(Therefore)g(\(1.1\),)h(\(ii\),)g(implies)0 594 y(that)f Fo(k)p Fp(\()p Fn(F)184 601 y Fm(\003)218 594 y Fp(+)11 b Fn(G)307 601 y Fm(\003)330 594 y Fp(\)\()p Fn(a)p Fp(\))p Fo(k)18 b Fp(is)e(also)g(in)g Fn(S)695 601 y Ft(C)745 594 y Fp(and)g(that)30 690 y(\006)66 697 y Ft(C)99 690 y Fp(\()p Fo(k)p Fp(\()p Fn(F)194 697 y Fm(\003)229 690 y Fp(+)10 b Fn(G)317 697 y Fm(\003)341 690 y Fp(\)\()p Fn(a)p Fp(\))p Fo(k)p Fp(\))15 b Fo(\024)f Fp(\006)572 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853 y Fm(\003)903 846 y Fo(k)h Fp(are)f(in)g Fn(S)1117 853 y Ft(C)1150 846 y Fp(,)h(whence)f(\006)1391 853 y Ft(C)1425 846 y Fo(k)p Fn(F)1482 853 y Fm(\003)1504 846 y Fo(k)11 b Fp(+)g(\006)1626 853 y Ft(C)1660 846 y Fo(k)p Fn(G)1724 853 y Fm(\003)1747 846 y Fo(k)0 906 y Fp(is)i(in)g Fn(S)133 913 y Ft(C)166 906 y Fp(.)h(Th)o(us)e(\(1.1\),) i(\(ii\),)g(sho)o(ws)e(that)j(\006)823 913 y Ft(C)856 906 y Fo(k)p Fn(F)913 913 y Fm(\003)941 906 y Fp(+)5 b Fn(G)1024 913 y Fm(\003)1047 906 y Fo(k)14 b Fp(is)f(in)g Fn(S)1219 913 y Ft(C)1252 906 y Fp(.)h(Therefore)f Fn(F)1535 913 y Fm(\003)1563 906 y Fp(+)5 b Fn(G)1646 913 y Fm(\003)1683 906 y Fp(is)13 b(in)0 965 y Fn(S)s Fp(.)j(Similarly)l(,)e(but)j(more)e (simply)l(,)h(one)g(sho)o(ws)f(that)i Fn(F)1045 972 y Fm(\003)1082 965 y Fo(2)d Fn(S)19 b Fp(and)d Fn(r)g Fo(2)e Fn(R)j Fp(implies)e Fn(r)q(F)1640 972 y Fm(\003)1677 965 y Fo(2)f Fn(S)s Fp(.)0 1025 y(Th)o(us)h(w)o(e)i(ha)o(v)o(e)e(sho)o (wn)g(that)i Fn(S)i Fp(is)e(an)f Fn(R)p Fp(-subsemimo)q(dule)d(of)k Fo(L)1236 1007 y Ft(N)1274 1025 y Fp(\()p Fn(A)p Fp(\))1349 1007 y Ft(N)1388 1025 y Fp(.)71 1085 y(Next)k(w)o(e)f(need)g(to)h(pro)o (v)o(e)e(that)h(\006)g(is)g(a)g(homomorphism)d(of)j Fn(R)p Fp(-semimo)q(dules.)e(Again)0 1145 y(let)f Fn(F)104 1152 y Fm(\003)143 1145 y Fp(and)f Fn(G)279 1152 y Fm(\003)319 1145 y Fp(b)q(e)h(in)f Fn(S)s Fp(.)f(Then)h(\006)674 1152 y Ft(R)707 1145 y Fp(\()p Fn(F)758 1152 y Fm(\003)781 1145 y Fp(\()p Fn(a)p Fp(\))c(+)f Fn(G)946 1152 y Fm(\003)969 1145 y Fp(\()p Fn(a)p Fp(\)\))16 b(=)d(\006)1156 1152 y Ft(R)1189 1145 y Fn(F)1221 1152 y Fm(\003)1244 1145 y Fp(\()p Fn(a)p Fp(\))f(+)f(\006)1406 1152 y Ft(R)1438 1145 y Fn(G)1477 1152 y Fm(\003)1500 1145 y Fp(\()p Fn(a)p Fp(\))18 b(and)e(th)o(us)0 1204 y(\006)36 1211 y Ft(R)69 1204 y Fp(\()p Fn(F)120 1211 y Fm(\003)151 1204 y Fp(+)8 b Fn(G)237 1211 y Fm(\003)261 1204 y Fp(\))14 b(=)f(\006)382 1211 y Ft(R)415 1204 y Fn(F)447 1211 y Fm(\003)478 1204 y Fp(+)8 b(\006)561 1211 y Ft(R)594 1204 y Fn(G)633 1211 y Fm(\003)656 1204 y Fp(.)15 b(Moreo)o(v)o(er,)f(\(\006)972 1211 y Ft(R)1005 1204 y Fp(\()p Fn(F)1056 1211 y Fm(\003)1087 1204 y Fp(+)9 b Fn(G)1174 1211 y Fm(\003)1197 1204 y Fp(\)\))1235 1211 y Fm(\003)1272 1204 y Fp(=)14 b(\(\006)1380 1211 y Ft(R)1413 1204 y Fn(F)1445 1211 y Fm(\003)1468 1204 y Fp(\))1487 1211 y Fm(\003)1518 1204 y Fp(+)8 b(\(\006)1620 1211 y Ft(R)1653 1204 y Fn(G)1692 1211 y Fm(\003)1716 1204 y Fp(\))1735 1211 y Fm(\003)1758 1204 y Fp(,)0 1264 y(whence)193 1348 y(\006\()p Fn(F)280 1355 y Fm(\003)314 1348 y Fp(+)j Fn(G)403 1355 y Fm(\003)426 1348 y Fp(\))j(=)g(\006)548 1355 y Ft(R)581 1348 y Fp(\(\(\006)655 1355 y Ft(R)688 1348 y Fp(\()p Fn(F)739 1355 y Fm(\003)773 1348 y Fp(+)d Fn(G)862 1355 y Fm(\003)885 1348 y Fp(\)\))923 1355 y Fm(\003)947 1348 y Fp(\))j(=)g(\006)1069 1355 y Ft(R)1101 1348 y Fp(\(\(\006)1175 1355 y Ft(R)1209 1348 y Fn(F)1241 1355 y Fm(\003)1264 1348 y Fp(\))1283 1355 y Fm(\003)1317 1348 y Fp(+)d(\(\006)1422 1355 y Ft(R)1455 1348 y Fn(G)1494 1355 y Fm(\003)1517 1348 y Fp(\))1536 1355 y Fm(\003)1559 1348 y Fp(\))459 1423 y(=)j(\006)548 1430 y Ft(R)581 1423 y Fp(\(\(\006)655 1430 y Ft(R)688 1423 y Fn(F)720 1430 y Fm(\003)743 1423 y Fp(\))762 1430 y Fm(\003)785 1423 y Fp(\))e(+)f(\006)902 1430 y Ft(R)934 1423 y Fp(\(\(\006)1008 1430 y Ft(R)1042 1423 y Fn(G)1081 1430 y Fm(\003)1104 1423 y Fp(\))1123 1430 y Fm(\003)1146 1423 y Fp(\))j(=)g(\006)p Fn(F)1300 1430 y Fm(\003)1334 1423 y Fp(+)d(\006)p Fn(G)1459 1430 y Fm(\003)1482 1423 y Fn(:)0 1516 y Fp(Similarly)17 b(one)h(obtains)g(\006)p Fn(r)q(F)575 1523 y Fm(\003)616 1516 y Fp(=)f Fn(r)q Fp(\006)p Fn(F)763 1523 y Fm(\003)787 1516 y Fp(,)h(and)g(\006)h(is)f(recognized)g(as)g(a)h(homomorphis)o(m)d (of)0 1576 y Fn(R)p Fp(-semimo)q(dules.)71 1635 y(No)o(w)23 b(w)o(e)g(wish)g(to)h(v)o(erify)f(\(1.4\),)h(\(o\).)g(F)l(or)e(this,)h (let)h Fn(F)1171 1642 y Fm(\003)1218 1635 y Fp(b)q(e)f(in)g Fo(L)1391 1617 y Ft(N)1429 1635 y Fp(\()p Fn(A)p Fp(\))1504 1617 y Ft(N)1543 1635 y Fp(.)h(W)l(e)f(ha)o(v)o(e)0 1695 y(to)f(sho)o(w)e(that)i Fn(F)339 1702 y Fm(\003)384 1695 y Fo(2)g Fn(S)i Fp(is)d(equiv)m(alen)o(t)h(with)f Fo(jjj)p Fn(F)983 1702 y Fm(\003)1005 1695 y Fo(jjj)h(2)g Fn(S)1155 1702 y Ft(C)1188 1695 y Fp(,)f(where)g Fo(jjj)p Fn(F)1446 1702 y Fm(\003)1469 1695 y Fo(jjj)g Fp(is)g(the)h(map)0 1755 y Fn(N)g Fo(3)16 b Fn(n)h Fo(7!)f(jjj)p Fn(F)298 1762 y Ft(n)325 1755 y Fo(jjj)f(2)i Fn(C)t Fp(.)g(Let)i(\()p Fn(F)645 1762 y Fm(\003)668 1755 y Fp(\))g(denote)f(the)g(map)f Fn(A)c Fo(\002)f Fn(N)21 b Fo(3)c Fp(\()p Fn(a;)8 b(n)p Fp(\))17 b Fo(7!)g(k)p Fn(F)1536 1762 y Ft(n)1563 1755 y Fp(\()p Fn(a)p Fp(\))p Fo(k)g(2)g Fn(C)t Fp(.)0 1815 y(Supp)q(ose)f(that)i Fn(F)337 1822 y Fm(\003)375 1815 y Fo(2)d Fn(S)k Fp(holds.)d(Let)i Fn(')d Fp(:)f Fn(A)e Fo(\002)g Fn(N)20 b Fo(\000)-9 b(!)15 b Fn(A)j Fp(b)q(e)f(the)h(pro)s (jection)e(on)o(to)h(the)g(\014rst)0 1874 y(factor.)f(Then,)g(for)h(ev) o(ery)f Fn(a)e Fo(2)g Fn(A)p Fp(,)458 1998 y(\()p Fn(F)509 2005 y Fm(\003)532 1998 y Fp(\))551 1977 y Ft(')577 1962 y Fk(\000)p Fj(1)624 1977 y Fs(\()p Ft(a)p Fs(\))679 1998 y Fp(\()p Fn(b;)8 b(n)p Fp(\))15 b(=)858 1927 y Fi(\032)903 1969 y Fo(k)p Fn(F)960 1976 y Ft(n)988 1969 y Fp(\()p Fn(a)p Fp(\))p Fo(k)420 b Fp(,)17 b(if)g Fn(b)d Fp(=)f Fn(a)p Fp(,)903 2029 y(0)569 b(,)17 b(otherwise.)0 2136 y(Since)23 b Fo(k)p Fn(F)194 2143 y Fm(\003)217 2136 y Fp(\()p Fn(a)p Fp(\))p Fo(k)h Fp(is)g(in)f Fn(S)484 2143 y Ft(C)517 2136 y Fp(,)g(\()p Fn(F)605 2143 y Fm(\003)628 2136 y Fp(\))647 2118 y Ft(')673 2103 y Fk(\000)p Fj(1)720 2118 y Fs(\()p Ft(a)p Fs(\))799 2136 y Fp(b)q(elongs)g(to)h Fn(S)1084 2143 y Ft(C;A)p Fm(\002)p Ft(N)1247 2136 y Fp(due)f(to)h(\(2.1\).)g(Moreo)o(v)o(er,)0 2205 y(\006)36 2212 y Ft(C)69 2205 y Fo(k)p Fn(F)126 2212 y Fm(\003)149 2205 y Fp(\()p Fn(a)p Fp(\))p Fo(k)g Fp(=)e(\006)359 2212 y Ft(C;A)p Fm(\002)p Ft(N)498 2205 y Fp(\()p Fn(F)549 2212 y Fm(\003)572 2205 y Fp(\))591 2187 y Ft(')617 2172 y Fk(\000)p Fj(1)664 2187 y Fs(\()p 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Fp(=)d(\006)717 2448 y Ft(C;A)790 2441 y Fo(fk)p Fn(F)872 2448 y Ft(n)899 2441 y Fp(\()p Fn(a)p Fp(\))p Fo(k)i Fp(:)f Fn(a)g Fo(2)g Fn(A)p Fo(g)g Fp(=)g Fo(jjj)p Fn(F)1321 2448 y Ft(n)1347 2441 y Fo(jjj)213 b Fn(;)8 b(n)14 b Fo(2)g Fn(N)s(:)0 2537 y Fp(Hence)k(\(2.11\))h(sho)o (ws)d(that)i Fo(jjj)p Fn(F)622 2544 y Fm(\003)645 2537 y Fo(jjj)f Fp(is)g(in)h Fn(S)846 2544 y Ft(C)879 2537 y Fp(.)g(The)f(same)g(t)o(yp)q(e)i(of)f(argumen)o(t)e(sho)o(ws)g(that)0 2597 y Fo(jjj)p Fn(F)74 2604 y Fm(\003)96 2597 y Fo(jjj)e(2)g Fn(S)230 2604 y Ft(C)279 2597 y Fp(implies)i Fn(F)481 2604 y Fm(\003)517 2597 y Fo(2)e Fn(S)s Fp(.)71 2657 y(On)19 b(to)g(\(1.4\),)h(\(i\).)f(Here)h(w)o(e)f(are)g(dealing)f(with) h Fn(F)1046 2664 y Fm(\003)1087 2657 y Fo(2)g(L)1173 2639 y Ft(N)1211 2657 y Fp(\()p Fn(A)p Fp(\))1286 2639 y Ft(N)1344 2657 y Fp(with)g(\014nite)g(supp)q(ort.)0 2717 y(Hence)13 b Fo(k)p Fn(F)201 2724 y Fm(\003)223 2717 y Fp(\()p Fn(a)p Fp(\))p Fo(k)h Fp(has)d(\014nite)h(supp)q(ort)f (and)g(is)h(therefore)g(in)f Fn(S)1140 2724 y Ft(C)1173 2717 y Fp(,)h(for)g(all)g Fn(a)i Fo(2)g Fn(A)p Fp(.)f(In)e(particular,) 425 2830 y(\006)461 2837 y Ft(C)494 2830 y Fo(k)p Fn(F)551 2837 y Fm(\003)574 2830 y Fp(\()p Fn(a)p Fp(\))p Fo(k)k Fp(=)731 2782 y Fi(X)802 2790 y Fm(0)816 2830 y Fo(fk)p Fn(F)898 2837 y Ft(n)925 2830 y Fp(\()p Fn(a)p Fp(\))p Fo(k)g Fp(:)f Fn(n)g Fo(2)g Fp(supp)h Fn(F)1299 2837 y Fm(\003)1322 2830 y Fo(g)p eop %%Page: 11 11 11 10 bop 1722 42 a Fp(11)0 167 y(and)16 b(hence)483 258 y(\006)519 265 y Ft(C)552 258 y Fo(k)p Fn(F)609 265 y Fm(\003)632 258 y Fo(k)e Fp(=)724 211 y Fi(X)796 218 y Fm(0)810 258 y Fo(fk)p Fn(F)892 265 y Ft(n)919 258 y Fo(k)f Fp(:)h Fn(n)g Fo(2)g Fp(supp)h Fn(F)1227 265 y Fm(\003)1250 258 y Fo(g)p Fn(:)0 357 y Fp(Since)g Fn(S)160 364 y Ft(C;A)248 357 y Fp(is)g(an)h Fn(R)p Fp(-semimo)q(dule,)d(the)j (righ)o(t)e(hand)h(side)g(of)g(the)h(last)g(equation,)f(and)g(th)o(us)0 417 y(\006)36 424 y Ft(C)69 417 y Fo(k)p Fn(F)126 424 y Fm(\003)149 417 y Fo(k)p Fp(,)f(is)g(in)g Fn(S)337 424 y Ft(C;A)410 417 y Fp(.)g(Therefore)f Fn(F)693 424 y Fm(\003)731 417 y Fp(is)g(in)h Fn(S)s Fp(.)g(Moreo)o(v)o(er,)e(with)i Fn(F)1269 424 y Fm(\003)1307 417 y Fp(ha)o(ving)f(\014nite)h(supp)q (ort,)0 477 y(so)i(do)q(es)h Fn(F)206 484 y Fm(\003)228 477 y Fp(\()p Fn(a)p Fp(\),)h(for)e(all)g Fn(a)f Fo(2)f Fn(A)p Fp(,)i(and)475 585 y(\006)511 592 y Ft(R)544 585 y Fn(F)576 592 y Fm(\003)599 585 y Fp(\()p Fn(a)p Fp(\))f(=)730 538 y Fi(X)802 545 y Fm(0)816 585 y Fo(f)p Fn(F)873 592 y Ft(n)900 585 y Fp(\()p Fn(a)p Fp(\))g(:)f Fn(n)f Fo(2)h Fp(supp)i Fn(F)1249 592 y Fm(\003)1272 585 y Fo(g)314 b Fn(;)8 b(a)14 b Fo(2)g Fn(A:)0 684 y Fp(Consequen)o(tly)l(,)455 790 y(\006)p Fn(F)523 797 y Fm(\003)559 790 y Fp(=)g(\006)648 797 y Ft(R)681 790 y Fn(F)713 797 y Fm(\003)749 790 y Fp(=)802 743 y Fi(X)874 750 y Fm(0)888 790 y Fo(f)p Fn(F)945 797 y Ft(n)986 790 y Fp(:)f Fn(n)h Fo(2)g Fp(supp)h Fn(F)1255 797 y Fm(\003)1278 790 y Fo(g)p Fn(;)0 891 y Fp(and)h(\(1.4\),)h (\(i\),)g(is)f(satis\014ed.)71 951 y(Next)k(comes)f(\(1.4\),)g(\(ii\).) g(Let)h Fn(F)703 958 y Fm(\003)744 951 y Fo(2)e Fn(S)k Fp(and)c Fn(G)986 958 y Fm(\003)1027 951 y Fo(2)h(L)1113 933 y Ft(N)1151 951 y Fp(\()p Fn(A)p Fp(\))1226 933 y Ft(N)1284 951 y Fp(with)g Fo(jjj)p Fn(G)1481 958 y Fm(\003)1503 951 y Fo(jjj)f(\024)g(jjj)p Fn(F)1694 958 y Fm(\003)1716 951 y Fo(jjj)p Fp(.)0 1011 y(By)g(\(1.4\),)f(\(o\),)h Fo(jjj)p Fn(F)381 1018 y Fm(\003)404 1011 y Fo(jjj)c(2)h Fn(S)539 1018 y Ft(C)589 1011 y Fp(holds.)h(Due)h(to)h(\(1.1\),)g (\(ii\),)f Fo(jjj)p Fn(G)1213 1018 y Fm(\003)1236 1011 y Fo(jjj)f Fp(is)h(in)g Fn(S)1435 1018 y Ft(C)1468 1011 y Fp(.)g(By)h(\(1.4\),\(o\),)0 1071 y Fn(G)39 1078 y Fm(\003)79 1071 y Fp(is)f(in)g Fn(S)s Fp(.)g(Since)g(\(\006)440 1078 y Ft(R)473 1071 y Fn(G)512 1078 y Fm(\003)535 1071 y Fp(\)\()p Fn(a)p Fp(\))f(=)f(\006)724 1078 y Ft(R)757 1071 y Fp(\()p Fn(G)815 1078 y Fm(\003)838 1071 y Fp(\()p Fn(a)p Fp(\)\))i(=)e(\006)1028 1078 y Ft(R)1060 1071 y Fo(f)p Fn(G)1124 1078 y Ft(n)1152 1071 y Fp(\()p Fn(a)p Fp(\))h(:)f Fn(n)g Fo(2)g Fn(N)5 b Fo(g)p Fp(,)18 b(it)f(follo)o(ws)g (from)0 1131 y(\(1.3\),)g(\(ii\),)g(that)308 1222 y Fo(k)p Fp(\(\006)388 1229 y Ft(R)421 1222 y Fn(G)460 1229 y Fm(\003)483 1222 y Fp(\)\()p Fn(a)p Fp(\))p Fo(k)e(\024)f Fp(\006)695 1229 y Ft(C)728 1222 y Fo(fk)p Fn(G)817 1229 y Ft(n)844 1222 y Fp(\()p Fn(a)p Fp(\))p Fo(k)h Fp(:)f Fn(n)g Fo(2)g Fn(N)5 b Fo(g)14 b Fp(=)f(\006)1239 1229 y Ft(C)1273 1222 y Fo(k)p Fn(G)1337 1229 y Fm(\003)1360 1222 y Fp(\()p Fn(a)p Fp(\))p Fo(k)p Fn(:)0 1314 y Fp(Therefore)j(w)o (e)g(obtain,)g(as)g(in)g(the)h(pro)q(of)f(of)h(\(1.4\),)g(\(o\),)23 1405 y Fo(jjj)p Fp(\006)p Fn(G)140 1412 y Fm(\003)162 1405 y Fo(jjj)c Fp(=)h(\006)306 1412 y Ft(C;A)379 1405 y Fo(fk)p Fp(\(\006)484 1412 y Ft(R)517 1405 y Fn(G)556 1412 y Fm(\003)579 1405 y Fp(\)\()p Fn(a)p Fp(\))p Fo(k)h Fp(:)f Fn(a)g Fo(2)g Fn(A)p Fo(g)g(\024)g Fp(\006)982 1412 y Ft(C;A)1055 1405 y Fo(f)p Fp(\006)1116 1412 y Ft(C)1149 1405 y Fo(fk)p Fn(G)1238 1412 y Ft(n)1265 1405 y Fp(\()p Fn(a)p Fp(\))p Fo(k)h Fp(:)f Fn(n)g Fo(2)g Fn(N)5 b Fo(g)14 b Fp(:)g Fn(a)g Fo(2)g Fn(A)p Fo(g)217 1480 y Fp(=)g(\006)306 1487 y Ft(C;A)p Fm(\002)p Ft(N)446 1480 y Fp(\(\()p Fn(G)523 1487 y Fm(\003)546 1480 y Fp(\)\))h(=)f(\006) 688 1487 y Ft(C)721 1480 y Fo(jjj)p Fn(G)802 1487 y Fm(\003)825 1480 y Fo(jjj)0 1572 y Fp(and,)i(through)f(\(1.1\),)i(\(iii\),)511 1663 y Fo(jjj)p Fp(\006)p Fn(G)628 1670 y Fm(\003)651 1663 y Fo(jjj)c(\024)g Fp(\006)794 1670 y Ft(C)828 1663 y Fo(jjj)p Fn(G)909 1670 y Fm(\003)931 1663 y Fo(jjj)h(\024)f Fp(\006)1075 1670 y Ft(C)1109 1663 y Fo(jjj)p Fn(F)1183 1670 y Fm(\003)1205 1663 y Fo(jjj)p Fn(:)0 1755 y Fp(There)19 b(remains)e(\(1.4\),)i(\(iii\).)g(Let)h Fn(F)707 1762 y Fm(\003)748 1755 y Fo(2)e Fn(S)j Fp(and)e(let)g Fn(')f Fp(:)f Fn(N)24 b Fo(\000)-9 b(!)18 b Fn(N)25 b Fp(b)q(e)19 b(a)g(map.)f(By)h(\(1.7\),)0 1830 y Fn(F)39 1804 y Ft(')65 1789 y Fk(\000)p Fj(1)112 1804 y Fs(\()p Ft(n)p Fs(\))32 1837 y Fm(\003)188 1830 y Fp(is)e(in)g Fn(S)s Fp(,)h(for)f(all)g Fn(n)f Fo(2)g Fn(N)5 b Fp(,)19 b(and)e Fn(F)821 1806 y Ft(')847 1791 y Fk(\000)p Fj(1)814 1837 y Fm(\003)914 1830 y Fp(is)g(in)g Fn(S)s Fp(.)g(Moreo)o(v)o(er,)f(for)i(ev)o(ery)g Fn(a)e Fo(2)g Fn(A)p Fp(,)i(due)0 1889 y(to)f(\(1.3\),)g(\(iii\),)77 1991 y(\(\006)p Fn(F)164 1998 y Fm(\003)188 1991 y Fp(\)\()p Fn(a)p Fp(\))e(=)e(\006)374 1998 y Ft(R)407 1991 y Fp(\()p Fn(F)458 1998 y Fm(\003)481 1991 y Fp(\()p Fn(a)p Fp(\)\))j(=)d(\006) 668 1998 y Ft(R)701 1991 y Fp(\()p Fn(F)752 1998 y Fm(\003)775 1991 y Fp(\()p Fn(a)p Fp(\))839 1970 y Ft(')865 1955 y Fk(\000)p Fj(1)915 1991 y Fp(\))i(=)e(\006)1037 1998 y Ft(R)1070 1991 y Fo(f)p Fp(\(\006)1150 1998 y Ft(R)1183 1991 y Fp(\()p Fn(F)1234 1998 y Fm(\003)1257 1991 y Fp(\()p Fn(a)p Fp(\))1321 1970 y Ft(')1347 1955 y Fk(\000)p Fj(1)1395 1970 y Fs(\()p Ft(n)p Fs(\))1453 1991 y Fp(\)\))i(:)e Fn(n)h Fo(2)g Fn(N)5 b Fo(g)286 2086 y Fp(=)13 b(\006)374 2093 y Ft(R)407 2086 y Fo(f)p Fp(\(\006)487 2093 y Ft(R)520 2086 y Fp(\()p Fn(F)578 2060 y Ft(')604 2045 y Fk(\000)p Fj(1)651 2060 y Fs(\()p Ft(n)p Fs(\))571 2093 y Fm(\003)709 2086 y Fp(\)\()p Fn(a)p Fp(\)\))j(:)e Fn(n)f Fo(2)h Fn(N)5 b Fo(g)14 b Fp(=)g(\006)1118 2093 y Ft(R)1151 2086 y Fo(f)p Fp(\(\006)p Fn(F)1270 2060 y Ft(')1296 2045 y Fk(\000)p Fj(1)1343 2060 y Fs(\()p Ft(n)p Fs(\))1263 2093 y Fm(\003)1401 2086 y Fp(\)\()p Fn(a)p Fp(\))h(:)f Fn(n)f Fo(2)i Fn(N)5 b Fo(g)286 2181 y Fp(=)13 b(\(\006)p Fo(f)p Fp(\006)p Fn(F)493 2155 y Ft(')519 2140 y Fk(\000)p Fj(1)566 2155 y Fs(\()p Ft(n)p Fs(\))486 2188 y Fm(\003)638 2181 y Fp(:)h Fn(n)g Fo(2)g Fn(N)5 b Fo(g)p Fp(\)\()p Fn(a)p Fp(\))15 b(=)f(\(\006)p Fn(F)1072 2160 y Ft(')1098 2145 y Fk(\000)p Fj(1)1065 2193 y Fm(\003)1147 2181 y Fp(\)\()p Fn(a)p Fp(\))p Fn(;)0 2290 y Fp(and)i(th)o(us)f(\006\()p Fn(F)300 2266 y Ft(')326 2251 y Fk(\000)p Fj(1)293 2297 y Fm(\003)376 2290 y Fp(\))f(=)g(\006\()p Fn(F)549 2297 y Fm(\003)572 2290 y Fp(\),)j(as)f(had)g(to)h(b)q(e)f(sho)o(wn.)71 2350 y(A)o(t)g(this)f(p)q(oin)o(t)h(w)o(e)f(kno)o(w)g(that)i Fo(L)711 2332 y Ft(N)749 2350 y Fp(\()p Fn(A)p Fp(\))g(is)e(an)g Fn(R)p Fp(-prenormed)f Fn(R)p Fp(-semimo)q(dule)f(with)j(left)0 2410 y Fn(N)5 b Fp(-summation)15 b(\()p Fn(S)370 2420 y Fm(L)398 2410 y Fd(N)429 2420 y Fs(\()p Ft(A)p Fs(\))493 2410 y Fn(;)8 b Fp(\006)551 2420 y Fm(L)579 2410 y Fd(N)611 2420 y Fs(\()p Ft(A)p Fs(\))674 2410 y Fp(\).)71 2470 y(F)l(or)18 b Fn(a)i Fo(2)f Fn(A)h Fp(let)g Fn(\016)417 2452 y Ft(a)461 2470 y Fp(b)q(e)g(the)g(Dirac)f(function)g(at)h Fn(a)g Fp(on)g Fn(A)p Fp(,)g(that)g(is)f(the)h(map)f Fn(A)g Fo(\000)-8 b(!)18 b Fn(R)0 2529 y Fp(with)j Fn(\016)142 2511 y Ft(a)166 2529 y Fp(\()p Fn(a)p Fp(\))g(=)g(1)f(and)g Fn(\016)481 2511 y Ft(a)505 2529 y Fp(\()p Fn(b)p Fp(\))i(=)e(0,)h(for) f(all)g Fn(a)h Fo(6)p Fp(=)f Fn(b)h Fo(2)g Fn(A)p Fp(.)g(By)g(\(1.3\),) g(\(i\),)g Fn(\016)1462 2511 y Ft(a)1507 2529 y Fp(is)f(in)g Fo(L)1657 2511 y Ft(N)1696 2529 y Fp(\()p Fn(A)p Fp(\))0 2589 y(and)f Fo(jjj)p Fn(\016)166 2571 y Ft(a)189 2589 y Fo(jjj)f Fp(=)g(1)i(holds.)e(The)h(map)f Fn(A)h Fo(3)g Fn(a)g Fo(7!)f Fn(\016)961 2571 y Ft(a)1004 2589 y Fo(2)g(B)1088 2596 y Ft(N)1126 2589 y Fp(\()p Fo(L)1179 2571 y Ft(N)1217 2589 y Fp(\()p Fn(A)p Fp(\)\))j(is)e(denoted)g(b)o(y)g Fn(\016)i Fp(and)0 2649 y(is)e(called)g(the)h(Dirac)f(map)g(on)g Fn(A)p Fp(.)h(W)l(e)g(claim)e(that)i Fn(\016)h Fp(:)e Fn(A)g Fo(\000)-8 b(!)18 b(B)1280 2656 y Ft(N)1318 2649 y Fp(\()p Fo(L)1371 2631 y Ft(N)1409 2649 y Fp(\()p Fn(A)p Fp(\)\))j(is)f(univ)o(ersal)0 2709 y(with)g(resp)q(ect)h(to)f Fn(N)396 2716 y Ft(R)429 2709 y Fp(pnSmo)q(d)608 2687 y Fs(1)630 2709 y Fp(.)g(F)l(or)f(this)h(let)h Fn(M)26 b Fp(b)q(e)21 b(an)f Fn(R)p Fp(-prenormed)e Fn(R)p Fp(-semimo)q(dule)0 2769 y(with)f(left)i Fn(N)5 b Fp(-summation)15 b(\()p Fn(S)573 2776 y Ft(M)618 2769 y Fn(;)8 b Fp(\006)676 2776 y Ft(M)720 2769 y Fp(\))18 b(and)f(let)h Fn(h)d Fp(:)g Fn(A)h Fo(\000)-8 b(!)15 b(B)1183 2776 y Ft(N)1221 2769 y Fp(\()p Fn(M)5 b Fp(\))19 b(b)q(e)f(an)o(y)f(set)g(map.)g(Let)0 2828 y(furthermore)10 b Fn(f)20 b Fo(2)14 b(L)396 2810 y Ft(N)434 2828 y Fp(\()p Fn(A)p Fp(\))f(and)f(consider)e(the)j(map)e Fn(f)5 b(h)12 b Fp(giv)o(en)g(b)o(y)g Fn(A)i Fo(3)g Fn(a)g Fo(7!)g Fn(f)5 b Fp(\()p Fn(a)p Fp(\))p Fn(h)p Fp(\()p Fn(a)p Fp(\))16 b Fo(2)e Fn(M)5 b Fp(.)p eop %%Page: 12 12 12 11 bop 0 42 a Fp(12)337 b Fv(Section)16 b(3:)61 b(B.)15 b(P)o(areigis,)j(D.)d(Pumpl)q(\177)-24 b(un,)15 b(and)g(H.)f(R\177)-23 b(ohrl)0 176 y Fp(Then)24 b Fo(k)p Fn(f)5 b(h)p Fo(k)p Fp(\()p Fn(a)p Fp(\))28 b(=)f Fo(k)p Fn(f)5 b Fp(\()p Fn(a)p Fp(\))p Fn(h)p Fp(\()p Fn(a)p Fp(\))p Fo(k)29 b(\024)d(k)p Fn(f)5 b Fp(\()p Fn(a)p Fp(\))p Fo(kk)p Fn(h)p Fp(\()p Fn(a)p Fp(\))p Fo(k)29 b(\024)e(k)p Fn(f)5 b Fp(\()p Fn(a)p Fp(\))p Fo(k)p Fp(,)26 b(for)e(ev)o(ery)g Fn(a)j Fo(2)g Fn(A)p Fp(.)e(By)0 235 y(\(1.3\),)17 b(\(o\),)g Fn(f)j Fo(2)15 b Fn(S)350 242 y Ft(R;A)441 235 y Fp(implies)g Fo(k)p Fn(f)5 b Fo(k)15 b(2)f 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Fp(b)o(y)h(\(1.1\),)h(\(ii\),)g(and)f(hence)p 1283 641 V 17 w Fn(h)1312 649 y Ft(N)1350 681 y Fp(\()p Fn(F)1401 688 y Fm(\003)1424 681 y Fp(\))f Fo(2)f Fn(S)1536 688 y Ft(C)1586 681 y Fp(b)o(y)i(\(1.4\),)0 755 y(\(o\).)h(That)f(is,)p 282 714 V 16 w Fn(h)311 723 y Ft(N)349 755 y Fp(\()p Fn(S)s Fp(\))e Fo(\022)f Fn(S)518 762 y Ft(M)562 755 y Fp(.)j(No)o(w)h(consider)e Fn(F)932 762 y Fm(\003)971 755 y Fp(as)h(a)h(map)e Fn(A)c Fo(\002)g Fn(N)19 b Fo(\000)-8 b(!)13 b Fn(R)p Fp(.)k(Since)f Fo(jjj)p Fn(F)1708 762 y Fm(\003)1730 755 y Fo(jjj)0 815 y Fp(is)g(in)h Fn(S)140 822 y Ft(C)173 815 y Fp(,)f(it)i(follo)o(ws)d(from)h(\(2.12\))i(that)f Fn(F)820 822 y Fm(\003)860 815 y Fp(is)f(in)g Fn(S)999 822 y Ft(R;A)p Fm(\002)p Ft(N)1140 815 y Fp(.)g(Let)i Fn(p)c Fp(:)g Fn(A)e Fo(\002)f Fn(N)20 b Fo(\000)-9 b(!)14 b Fn(A)j Fp(b)q(e)g(the)0 880 y(\014rst)h(pro)s(jection)f(and)h(put)536 867 y(~)535 880 y Fn(h)f Fp(:=)f Fn(h)c Fo(\016)g Fn(p)p Fp(.)19 b(Then)f Fn(F)950 887 y Fm(\003)985 880 y Fo(\001)1011 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Fn(h)p Fp(\()p Fn(F)852 1465 y Ft(n)879 1458 y Fp(\))g(:)g Fn(n)g Fo(2)g Fn(N)5 b Fo(g)14 b Fp(=)g(\006)1204 1465 y Ft(M)1248 1458 y Fp(\()p 1267 1418 V Fn(h)1296 1426 y Ft(N)1334 1458 y Fp(\()p Fn(F)1385 1465 y Fm(\003)1408 1458 y Fp(\)\))p Fn(:)0 1568 y Fp(This)f(means)f(that)i(\006)401 1575 y Ft(M)445 1568 y Fp(\()p 464 1528 V Fn(h)493 1536 y Ft(N)531 1568 y Fp(\()p Fn(F)582 1575 y Fm(\003)605 1568 y Fp(\)\))h(=)p 710 1528 V 13 w Fn(h)p Fp(\(\006)p Fn(F)826 1575 y Fm(\003)849 1568 y Fp(\),)f(whic)o(h)e(is)h(the)h(form) o(ula)e(in)g(\(1.5\),)i(\(i\).)g(Finally)l(,)0 1642 y(as)i Fo(jjj)p 103 1601 V Fn(h)131 1609 y Ft(N)169 1642 y Fp(\()p Fn(F)220 1649 y Fm(\003)243 1642 y Fp(\))p Fo(jjj)e(\024)g(jjj)p Fn(F)445 1649 y Fm(\003)467 1642 y Fo(jjj)i Fp(has)g(b)q(een)h(sho)o (wn)e(b)q(efore,)h(\(1.4\),)h(\(ii\),)g(implies)358 1747 y Fo(k)p Fp(\006)419 1754 y Ft(M)p 463 1706 V 463 1747 a Fn(h)491 1714 y Ft(N)529 1747 y Fp(\()p Fn(F)580 1754 y Fm(\003)603 1747 y Fp(\))p Fo(k)e(\024)e Fp(\006)750 1754 y Ft(C)784 1747 y Fo(jjj)p 826 1706 V Fn(h)854 1714 y Ft(N)892 1747 y Fp(\()p Fn(F)943 1754 y Fm(\003)966 1747 y Fp(\))p Fo(jjj)h(\024)f Fp(\(\006)1148 1754 y Ft(C)1182 1747 y Fo(jjj)p Fn(F)1256 1754 y Fm(\003)1279 1747 y Fo(jjj)p Fp(\))d Fo(\001)h Fp(1)p Fn(;)0 1838 y Fp(sho)o(wing)k(\(1.5\),)i(\(ii\),)g(as)f(w)o(ell)g(as)g(pro)o(ving)f (that)p 931 1797 V 17 w Fn(h)h Fp(is)g(a)h(con)o(traction.)71 1897 y(The)j(v)o(ery)h(\014nal)e(step)h(is)g(no)o(w)g(to)h(sho)o(w)e (that)p 980 1857 V 21 w Fn(h)h Fp(is)g(unique.)g(So,)g(let)h Fn(h)1455 1879 y Fm(0)1489 1897 y Fp(:)f Fo(L)1557 1879 y Ft(N)1595 1897 y Fp(\()p Fn(A)p Fp(\))h Fo(\000)-8 b(!)0 1957 y Fn(M)24 b Fp(b)q(e)18 b(a)g(con)o(tractiv)o(e)f (homomorphism)d(of)k Fn(R)p Fp(-prenormed)e Fn(R)p Fp(-semimo)q(dules)f (with)j(left)h Fn(N)5 b Fp(-)0 2017 y(summation)17 b(suc)o(h)h(that)i Fn(h)e Fp(=)g Fn(h)619 1999 y Fm(0)646 2017 y Fo(\016)12 b Fn(\016)r Fp(.)19 b(Let)h Fn(f)k Fo(2)19 b(L)966 1999 y Ft(N)1004 2017 y Fp(\()p 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Fp(is)g(in)g Fn(S)j Fp(and)d(\006)p Fn(f)1335 2388 y Fm(\003)1372 2381 y Fp(=)i Fn(f)5 b Fp(.)13 b(Since)g Fn(h)1637 2362 y Fm(0)p Ft(N)1686 2381 y Fp(\()p Fn(f)1729 2388 y Fm(\003)1752 2381 y Fp(\))0 2440 y(is)j(the)h(map)330 2559 y Fn(N)j Fo(3)14 b Fn(n)f Fo(7!)544 2488 y Fi(\032)590 2534 y Fn(h)619 2515 y Fm(0)632 2534 y Fp(\()p Fn(f)5 b Fp(\()p Fn(\037)p Fp(\()p Fn(n)p Fp(\)\))p Fn(\016)841 2515 y Ft(\037)p Fs(\()p Ft(n)p Fs(\))927 2534 y Fp(\))14 b(=)g Fn(f)5 b Fp(\()p Fn(\037)p Fp(\()p Fn(n)p Fp(\)\))p Fn(h)1208 2515 y Fm(0)1224 2534 y Fp(\()p Fn(\016)1267 2515 y Ft(\037)p Fs(\()p Ft(n)p Fs(\))1351 2534 y Fp(\))135 b(,)17 b(if)f Fn(n)e Fo(2)g Fn(N)1717 2515 y Fm(0)1731 2534 y Fp(;)590 2593 y(0)890 b(,)17 b(otherwise;)0 2677 y(w)o(e)f(ha)o(v)o(e)227 2758 y Fn(h)256 2737 y Fm(0)269 2758 y Fp(\()p Fn(f)5 b Fp(\))16 b(=)d Fn(h)433 2737 y Fm(0)447 2758 y Fp(\(\006)p Fn(f)526 2765 y Fm(\003)550 2758 y Fp(\))h(=)f(\006)671 2765 y Ft(M)716 2758 y Fp(\()p Fn(h)764 2737 y Fm(0)p Ft(N)813 2758 y Fp(\()p Fn(f)856 2765 y Fm(\003)880 2758 y Fp(\)\))h(=)g(\006)1021 2765 y Ft(M)r(;A)1105 2758 y Fo(f)p Fn(f)5 b Fp(\()p Fn(a)p Fp(\))p Fn(h)1252 2737 y Fm(0)1267 2758 y Fp(\()p Fn(\016)1310 2737 y Ft(a)1335 2758 y Fp(\))14 b(:)f Fn(a)i Fo(2)f Fn(A)p Fo(g)352 2840 y Fp(=)f(\006)440 2847 y Ft(M)r(;A)524 2840 y Fo(f)p Fn(f)5 b Fp(\()p Fn(a)p Fp(\))p Fn(h)p Fp(\()p Fn(a)p Fp(\))17 b(:)c Fn(a)h Fo(2)g Fn(A)p Fo(g)h Fp(=)p 995 2799 V 13 w Fn(h)p Fp(\()p Fn(f)5 b Fp(\))p Fn(;)p eop %%Page: 13 13 13 12 bop 1722 42 a Fp(13)0 167 y(pro)o(ving)15 b(the)i(required)e (uniqueness.)1027 b Fo(u)-33 b(t)479 418 y Fq(4.)17 b Fc(N)p Fq(-con)n(v)n(exit)n(y)d(theories)0 596 y Fg(\(4.1\))36 b(De\014nition.)c Fp(Let)g Fn(R)f Fp(b)q(e)h(a)f(prenormed)e(semiring)g (with)i(left)h Fn(N)5 b Fp(-summation)0 656 y(\()p Fn(S)50 663 y Ft(R)83 656 y Fn(;)j Fp(\006)141 663 y Ft(R)173 656 y Fp(\).)22 b(By)g(a)f(left)h Fn(N)5 b Ff(-c)m(onvexity)24 b(the)m(ory)f(over)f Fn(R)f Fp(is)g(mean)o(t)g(a)g(subset)g(\000)h(of)f Fn(S)1622 663 y Ft(R)1676 656 y Fp(suc)o(h)0 716 y(that)53 775 y(\(o\))26 b Fo(k)p Fn(\013)199 782 y Fm(\003)221 775 y Fo(k)14 b(2)g Fn(S)338 782 y Ft(C)388 775 y Fp(and)i(\006)521 782 y Ft(C)554 775 y Fo(k)p Fn(\013)611 782 y Fm(\003)634 775 y Fo(k)e(\024)f Fp(1)685 b(,)17 b(for)f(all)g Fn(\013)1643 782 y Fm(\003)1680 775 y Fo(2)e Fp(\000,)64 835 y(\(i\))26 b Fn(\016)166 817 y Ft(n)164 847 y Fm(\003)207 835 y Fo(2)14 b Fp(\000)1161 b(,)16 b(for)g(all)h Fn(n)c Fo(2)h Fn(N)5 b Fp(,)50 895 y(\(ii\))26 b(for)12 b(all)f Fn(\013)310 902 y Fm(\003)333 895 y Fn(;)d(\014)386 877 y Ft(n)383 907 y Fm(\003)413 895 y Fp(,)k Fn(n)i Fo(2)g Fn(N)5 b Fp(,)13 b(in)e(\000)i(the)f(map)g Fo(h)p Fn(\013)938 904 y Fl(\003)971 895 y Fn(;)c(\014)1024 877 y Fl(\003)1021 907 y Fm(\003)1058 895 y Fo(i)13 b Fp(giv)o(en)e(b)o(y)h Fn(N)19 b Fo(3)14 b Fn(n)g Fo(7!)g Fp(\006)1530 902 y Ft(R)1563 895 y Fn(\013)1595 902 y Fm(\003)1617 895 y Fn(\014)1648 877 y Fm(\003)1645 907 y Ft(n)1686 895 y Fo(2)g Fn(R)142 955 y Fp(is)i(in)g(\000.)71 1073 y(It)g(is)f(a)g (simple)g(consequence)g(of)h(\(1.1\),)g(\(ii\))g({)f(\(iv\),)h(that)g Fo(h)p Fn(\013)1224 1082 y Fl(\003)1258 1073 y Fn(;)8 b(\014)1311 1055 y Fl(\003)1308 1085 y Fm(\003)1345 1073 y Fo(i)16 b Fp(satis\014es)e(\(4.1\),)i(\(o\).)71 1133 y(Let)g Fn(X)k Fp(b)q(e)c(an)o(y)f(set)g(and)g(denote)h(the)f(elemen)o (ts)g(of)h Fn(X)1100 1115 y Ft(N)1153 1133 y Fp(b)o(y)f(lo)o(w)o(er)g (case)g(letters)h(with)f(an)0 1192 y(upp)q(er)h(placeholder)f(sym)o(b)q (ol,)g(e.)i(g.)f Fn(x)726 1174 y Fm(\003)766 1192 y Fp(or)g Fn(x)855 1174 y Fl(\003)889 1192 y Fp(.)0 1311 y Fg(\(4.2\))j (De\014nition.)f Fp(Let)g(\000)f(b)q(e)g(a)g(left)g Fn(N)5 b Fp(-con)o(v)o(exit)o(y)17 b(theory)g(o)o(v)o(er)f Fn(R)p Fp(.)h(By)g(a)g(left)g(\000-con)o(v)o(ex)0 1370 y(mo)q(dule)f(is)g (mean)o(t)g(a)g(set)h Fn(X)h Fo(6)p Fp(=)c Fo(;)i Fp(together)h(with)f (a)h(map)509 1460 y(\000)11 b Fo(\002)g Fn(X)646 1440 y Ft(N)698 1460 y Fo(3)j Fp(\()p Fn(\013)796 1467 y Fm(\003)819 1460 y Fn(;)8 b(x)869 1440 y Fm(\003)893 1460 y Fp(\))14 b Fo(7!)f(h)p Fn(\013)1040 1467 y Fm(\003)1064 1460 y Fn(;)8 b(x)1114 1440 y Fm(\003)1137 1460 y Fo(i)14 b(2)g Fn(X)0 1550 y Fp(suc)o(h)h(that)64 1610 y(\(i\))26 b Fo(h)p Fn(\016)185 1591 y Ft(n)183 1622 y Fm(\003)212 1610 y Fn(;)8 b(x)262 1591 y Fm(\003)286 1610 y Fo(i)14 b Fp(=)g Fn(x)400 1591 y Ft(n)1043 1610 y Fp(,)i(for)h(all)f Fn(n)d Fo(2)h Fn(N)5 b Fp(,)17 b Fn(\013)1417 1617 y Fm(\003)1454 1610 y Fo(2)d Fp(\000,)j Fn(x)1591 1591 y Fm(\003)1628 1610 y Fo(2)d Fn(X)1720 1591 y Ft(N)1758 1610 y Fp(,)50 1669 y(\(ii\))26 b Fo(h)p Fn(\013)193 1678 y Fl(\003)227 1669 y Fn(;)8 b Fo(h)p Fn(\014)299 1651 y Fl(\003)296 1682 y Fm(\003)333 1669 y Fn(;)g(x)383 1651 y Fm(\003)406 1669 y Fo(ii)15 b Fp(=)e Fo(hh)p Fn(\013)581 1678 y Fl(\003)615 1669 y Fn(;)8 b(\014)668 1651 y Fl(\003)665 1682 y Fm(\003)702 1669 y Fo(i)p Fn(;)g(x)771 1651 y Fm(\003)795 1669 y Fo(i)171 b Fp(,)16 b(for)h(all)f Fn(\013)1193 1676 y Fm(\003)1229 1669 y Fo(2)e Fp(\000,)j Fn(\014)1369 1651 y Fl(\003)1366 1682 y Fm(\003)1416 1669 y Fo(2)d Fp(\000)1494 1651 y Ft(N)1532 1669 y Fp(,)j Fn(x)1591 1651 y Fm(\003)1628 1669 y Fo(2)d Fn(X)1720 1651 y Ft(N)1758 1669 y Fp(.)0 1788 y Fg(\(4.3\))i(De\014nition.)g Fp(Let)f(\000)g(b)q (e)g(a)f(left)h Fn(N)5 b Fp(-con)o(v)o(exit)o(y)14 b(theory)l(.)g(By)h (a)g(homomorphism)10 b(of)15 b(left)0 1847 y(\000-con)o(v)o(ex)h(mo)q (dules)f Fn(X)j Fo(\000)-8 b(!)13 b Fn(X)604 1829 y Fm(0)635 1847 y Fp(is)j(mean)o(t)g(a)h(map)e Fn(f)20 b Fp(:)14 b Fn(X)k Fo(\000)-9 b(!)14 b Fn(X)1258 1829 y Fm(0)1289 1847 y Fp(suc)o(h)h(that)317 1937 y Fn(f)5 b Fp(\()p Fo(h)p Fn(\013)416 1944 y Fm(\003)440 1937 y Fn(;)j(x)490 1916 y Fm(\003)514 1937 y Fo(i)p Fp(\))14 b(=)g Fo(h)p Fn(\013)670 1944 y Fm(\003)693 1937 y Fn(;)8 b(f)744 1916 y Ft(N)783 1937 y Fp(\()p Fn(x)830 1916 y Fm(\003)854 1937 y Fp(\))p Fo(i)318 b Fn(;)24 b Fp(for)16 b(all)h Fn(\013)1426 1944 y Fm(\003)1462 1937 y Fo(2)d Fp(\000)p Fn(;)8 b(x)1590 1919 y Fm(\003)1628 1937 y Fo(2)14 b Fn(X)1720 1919 y Ft(N)1758 1937 y Fn(:)71 2027 y Fp(Let)g(\000)f(b)q(e) g(an)g Fn(N)5 b Fp(-con)o(v)o(exit)o(y)12 b(theory)l(.)h(Then)g(the)g (totalit)o(y)g(of)h(left)f(\000-con)o(v)o(ex)g(mo)q(dules)e(and)0 2086 y(their)18 b(homomorphism)o(s,)d(with)k(comp)q(osition)e(the)h (set-theoretical)g(one,)h(form)e(a)h(category)0 2146 y(\000)p Fn(C)t Fp(,)g(the)h(category)g(of)g(left)g(\000-con)o(v)o(ex)g (mo)q(dules.)e(Clearly)l(,)h(\000)p Fn(C)k Fp(is)d(an)f(algebraic)g (category)l(.)0 2206 y(Since)h(it)h(has)f(a)h(rank)f(\([2],)g(p.56\),)h (it)g(has)f(free)g(ob)s(jects)h(on)f(an)o(y)h(set.)f(Ho)o(w)o(ev)o(er,) g(w)o(e)g(w)o(an)o(t)0 2266 y(to)i(construct)f(suc)o(h)f(free)i(ob)s (jects)f(explicitly)l(.)g(First,)g(three)g(tec)o(hnical)g(statemen)o (ts)g(ab)q(out)0 2326 y(\000-con)o(v)o(ex)i(mo)q(dules.)f(They)i (corresp)q(ond)e(to)i([4],)f(\(2.4\),)h(\(iii\),)f(\(iv\),)i(and)e (\(viii\),)h(and)e(the)0 2385 y(pro)q(ofs)16 b(there)g(carry)g(o)o(v)o (er)g(to)h(the)g(curren)o(t)e(situation)h(with)g(nominal)f(c)o(hanges)g (only)l(.)0 2472 y Fg(\(4.4\))27 b(Lemma.)f Ff(L)m(et)e Fn(X)29 b Ff(b)m(e)24 b(a)h(left)e Fp(\000)p Ff(-c)m(onvex)i(mo)m (dule,)g(let)e Fn(\013)1258 2479 y Fm(\003)1306 2472 y Fo(2)i Fp(\000)f Ff(with)h Fp(supp)15 b Fn(\013)1685 2479 y Fm(\003)1733 2472 y Fo(\022)0 2532 y Fn(N)40 2539 y Fs(0)89 2532 y Fo(\022)27 b Fn(N)5 b Ff(,)26 b(and)g(let)e Fn(y)448 2514 y Fm(\003)471 2532 y Fn(;)8 b(z)518 2514 y Fm(\003)569 2532 y Fo(2)27 b Fn(X)674 2514 y Ft(N)737 2532 y Ff(b)m(e)f(such)f(that)g Fn(y)1061 2514 y Ft(n)1116 2532 y Fp(=)h Fn(z)1206 2514 y Ft(n)1234 2532 y Ff(,)f(for)h(al)s(l)f Fn(n)i Fo(2)g Fn(N)1597 2539 y Fs(0)1619 2532 y Ff(.)f(Then)0 2592 y Fo(h)p Fn(\013)51 2599 y Fm(\003)74 2592 y Fn(;)8 b(y)122 2574 y Fm(\003)145 2592 y Fo(i)15 b Fp(=)e Fo(h)p Fn(\013)282 2599 y Fm(\003)305 2592 y Fn(;)8 b(z)352 2574 y Fm(\003)376 2592 y Fo(i)p Ff(.)1328 b Fo(u)-33 b(t)0 2679 y Fg(\(4.5\))17 b(Lemma.)27 b Ff(L)m(et)16 b Fn(X)21 b Ff(b)m(e)16 b(a)h(left)f Fp(\000)p Ff(-c)m(onvex)h(mo)m (dule,)g(let)f Fn(\013)1187 2686 y Fm(\003)1224 2679 y Fo(2)e Fp(\000)i Ff(and)h Fn(x)1441 2661 y Fm(\003)1479 2679 y Fo(2)d Fn(X)t Ff(.)i(F)l(or)h(any)0 2739 y(bije)m(ction)i Fn(\033)d Fp(:)e Fn(N)19 b Fo(\000)-9 b(!)14 b Fn(N)23 b Ff(de\014ne)633 2721 y Ft(\033)660 2739 y Fn(\013)692 2746 y Fm(\003)733 2739 y Ff(r)m(esp.)853 2721 y Ft(\033)880 2739 y Fn(x)908 2721 y Fm(\003)949 2739 y Ff(as)c(the)f(maps)280 2828 y Fn(N)i Fo(3)14 b Fn(n)f Fo(7!)h Fn(\013)526 2837 y Ft(\033)q Fs(\()p Ft(n)p Fs(\))622 2828 y Fo(2)g Fn(R)120 b Ff(r)m(esp.)h Fn(N)20 b Fo(3)14 b Fn(n)f Fo(7!)h Fn(x)1291 2808 y Ft(\033)q Fs(\()p Ft(n)p Fs(\))1388 2828 y Fo(2)g Fn(X)q(:)p eop %%Page: 14 14 14 13 bop 0 42 a Fp(14)337 b Fv(Section)16 b(3:)61 b(B.)15 b(P)o(areigis,)j(D.)d(Pumpl)q(\177)-24 b(un,)15 b(and)g(H.)f(R\177)-23 b(ohrl)0 176 y Ff(Then)18 b Fo(h)p Fn(\013)181 183 y Fm(\003)204 176 y Fn(;)8 b(x)254 158 y Fm(\003)277 176 y Fo(i)15 b Fp(=)e Fo(h)382 158 y Ft(\033)409 176 y Fn(\013)441 183 y Fm(\003)464 176 y Fn(;)486 158 y Ft(\033)513 176 y Fn(x)541 158 y Fm(\003)565 176 y Fo(i)p Ff(.)1139 b Fo(u)-33 b(t)0 256 y Fg(\(4.6\))19 b(Lemma.)26 b Ff(L)m(et)18 b Fn(X)k Ff(b)m(e)c(a)g(left)f Fp(\000)p Ff(-c)m(onvex)i(mo)m(dule,)f Fn(\013)1129 263 y Fm(\003)1170 256 y Ff(and)g Fn(\014)1294 263 y Fm(\003)1335 256 y Ff(in)g Fp(\000)p Ff(,)g(and)h Fn(x)1585 238 y Fm(\003)1626 256 y Ff(and)g Fn(y)1749 238 y Fm(\003)0 316 y Ff(in)f Fn(X)t Ff(.)g(L)m(et)g(furthermor)m(e)h Fn(')13 b Fp(:)h Fn(N)19 b Fo(\000)-8 b(!)13 b Fn(N)23 b Ff(b)m(e)c(an)f(inje)m(ction)h(and)g(assume)213 403 y Fn(\014)241 410 y Ft(n)282 403 y Fp(=)13 b Fn(\013)366 412 y Ft(')392 402 y Fk(\000)p Fj(1)439 412 y Fs(\()p Ft(n)p Fs(\))548 403 y Fn(;)8 b(n)14 b Fo(2)g Fn(')p Fp(\()p Fn(N)5 b Fp(\))p Fn(;)128 b Ff(and)121 b Fn(\014)1146 410 y Ft(n)1187 403 y Fp(=)14 b(0)51 b Fn(;)8 b(n)19 b(=)-31 b Fo(2)14 b Fn(')p Fp(\()p Fn(N)5 b Fp(\);)213 493 y Fn(x)241 472 y Ft(n)282 493 y Fp(=)14 b Fn(y)361 472 y Ft(')387 457 y Fk(\000)p Fj(1)433 472 y Fs(\()p Ft(n)p Fs(\))543 493 y Fn(;)8 b(n)13 b Fo(2)h Fn(')p Fp(\()p Fn(N)5 b Fp(\))p Fn(:)0 583 y Ff(Then)18 b Fo(h)p Fn(\013)181 590 y Fm(\003)204 583 y Fn(;)8 b(y)252 564 y Fm(\003)275 583 y Fo(i)14 b Fp(=)g Fo(h)p Fn(\014)408 590 y Fm(\003)431 583 y Fn(;)8 b(x)481 564 y Fm(\003)505 583 y Fo(i)p Ff(.)1199 b Fo(u)-33 b(t)71 663 y Fp(Next)18 b(w)o(e)e(ha)o(v)o(e)0 778 y Fg(\(4.7\))23 b(Theorem.)18 b Fp(\(see)j([3],)e(5.4\))h Ff(The)h(for)m(getful)h(functor)f Fo(V)1205 785 y Fs(\000)1251 778 y Fp(:)f(\000)p Fn(C)j Fo(\000)-9 b(!)20 b Fp(Set)h Ff(has)i(a)e(left)0 838 y(adjoint)e Fo(F)210 819 y Fs(\000)237 838 y Ff(.)0 952 y(Pr)m(o)m(of.)25 b Fp(Let)d Fn(R)g Fp(b)q(e)g(the)f(prenormed)f (semiring)f(with)j(left)g Fn(N)5 b Fp(-summation)20 b(\()p Fn(S)1516 959 y Ft(R)1549 952 y Fn(;)8 b Fp(\006)1607 959 y Ft(R)1639 952 y Fp(\))22 b(that)0 1012 y(app)q(ears)17 b(in)h(the)h(de\014nition)e(\(4.1\))i(of)f(\000.)h(Giv)o(en)f(an)o(y)f (set)i Fn(A)g Fp(w)o(e)f(de\014ne)g Fo(F)1439 994 y Fs(\000)1465 1012 y Fp(\()p Fn(A)p Fp(\))i(as)e(the)g(set)0 1072 y(of)h(maps)f Fn(f)23 b Fp(:)18 b Fn(A)g Fo(\000)-8 b(!)17 b Fn(R)j Fp(suc)o(h)d(that)j(card)o(\(supp)c Fn(f)5 b Fp(\))19 b Fo(\024)e Fp(card)o Fn(N)25 b Fp(and)18 b(for)h(some)f(data)h Fn(\037)p Fp(,)g Fn(\014)1735 1079 y Fm(\003)1758 1072 y Fp(,)0 1132 y Fn(A)37 1114 y Fm(0)51 1132 y Fp(,)c(and)g Fn(N)221 1114 y Fm(0)251 1132 y Fp(in)f(\(2.2\),)i Fn(\014)467 1139 y Fm(\003)504 1132 y Fo(2)e Fp(\000)h(holds.)f(In)h(order)f(to)h (mak)o(e)g Fo(F)1159 1114 y Fs(\000)1186 1132 y Fp(\()p Fn(A)p Fp(\))h(a)f(left)h(\000-con)o(v)o(ex)f(mo)q(dule)0 1192 y(w)o(e)k(ha)o(v)o(e)f(to)i(de\014ne)e Fo(h)p Fn(\013)454 1199 y Fm(\003)478 1192 y Fn(;)8 b(f)529 1173 y Fm(\003)552 1192 y Fo(i)p Fp(,)20 b(for)f(all)f Fn(\013)787 1199 y Fm(\003)828 1192 y Fo(2)h Fp(\000)g(and)f Fn(f)1058 1173 y Fm(\003)1100 1192 y Fo(2)h(F)1193 1173 y Fs(\000)1220 1192 y Fp(\()p Fn(A)p Fp(\))1295 1173 y Ft(N)1333 1192 y Fp(.)g(According)g(to)g(\(2.3\))0 1251 y(w)o(e)e(can)g(c)o(ho)q(ose)g Fn(\037)p Fp(,)h Fn(A)425 1233 y Fm(0)456 1251 y Fp(and)f Fn(N)599 1233 y Fm(0)631 1251 y Fp(suc)o(h)f(that)i(for)f(ev)o(ery)g Fn(f)1091 1233 y Ft(n)1134 1251 y Fo(2)f(F)1224 1233 y Fs(\000)1251 1251 y Fp(\()p Fn(A)p Fp(\))g Fo(\022)f Fn(S)1427 1258 y Ft(R;A)1501 1251 y Fp(,)i Fn(n)e Fo(2)h Fn(N)5 b Fp(,)17 b(the)0 1311 y(conditions)f(in)g(\(2.2\))h(are)g (satis\014ed.)e(Supp)q(ose)h(that)h Fn(\014)1040 1293 y Ft(n)1037 1323 y Fm(\003)1084 1311 y Fp(is)f(asso)q(ciated)h(to)g Fn(f)1462 1293 y Ft(n)1507 1311 y Fp(via)g(the)g(data)0 1371 y Fn(\037)p Fp(,)k Fn(A)103 1353 y Fm(0)138 1371 y Fp(and)e Fn(N)283 1353 y Fm(0)298 1371 y Fp(.)h(Then)g Fn(\014)497 1353 y Ft(n)494 1383 y Fm(\003)545 1371 y Fo(2)h Fp(\000,)f(for)g(all)h Fn(n)f Fo(2)h Fn(N)5 b Fp(.)21 b(Due)f(to)h(\(4.1\),)g(\(ii\),)g Fo(h)p Fn(\013)1464 1380 y Fl(\003)1498 1371 y Fn(;)8 b(\014)1551 1353 y Fl(\003)1548 1383 y Fm(\003)1584 1371 y Fo(i)21 b Fp(is)f(in)h(\000)0 1431 y(and,)16 b(as)g(is)h(seen)f(easily)l(,)h(supp)e Fo(h)p Fn(\013)649 1439 y Fl(\003)683 1431 y Fn(;)8 b(\014)736 1413 y Fl(\003)733 1443 y Fm(\003)769 1431 y Fo(i)15 b(\022)856 1393 y Fi(S)898 1431 y Fo(f)p Fp(supp)g Fn(\014)1073 1413 y Ft(n)1070 1443 y Fm(\003)1114 1431 y Fp(:)f Fn(n)h Fo(2)f Fn(N)5 b Fo(g)15 b(\022)f Fn(N)1417 1413 y Fm(0)1431 1431 y Fp(.)j(Hence)g(there)g(is)0 1490 y(a)i(unique)g Fn(f)k Fo(2)c(F)349 1472 y Fs(\000)376 1490 y Fp(\()p Fn(A)p Fp(\))h(with)f(supp)c Fn(f)24 b Fo(\022)18 b Fn(A)848 1472 y Fm(0)882 1490 y Fp(and)g Fn(f)5 b Fp(\()p Fn(\037)p Fp(\()p Fn(n)p Fp(\)\))21 b(=)d Fo(h)p Fn(\013)1276 1499 y Fl(\003)1309 1490 y Fn(;)8 b(\014)1362 1472 y Fl(\003)1359 1503 y Fm(\003)1396 1490 y Fo(i)p Fp(,)19 b(for)g(all)g Fn(n)f Fo(2)g Fn(N)1743 1472 y Fm(0)1758 1490 y Fp(.)0 1550 y(Denote)h(this)f Fn(f)24 b Fp(b)o(y)18 b Fo(h)p Fn(\013)442 1557 y Fm(\003)465 1550 y Fn(;)8 b(f)516 1532 y Fm(\003)540 1550 y Fo(i)p Fp(.)18 b(By)h(\(2.2\),)g Fn(f)24 b Fp(do)q(es)18 b(not)h(dep)q(end)e(on)h(the)h(data)f(c)o (hosen.)f(It)i(is)0 1610 y(clear)c(from)f(the)h(construction)f(that)i (\(4.2\),)f(\(i\),)h(is)e(satis\014ed.)g(In)h(order)f(to)h(v)o(erify)g (\(4.2\),)g(\(ii\),)0 1670 y(it)i(su\016ces)e(to)i(sho)o(w)e(the)i (existence)g(and)f(equalit)o(y)h(of)f(the)h(t)o(w)o(o)f(terms)417 1759 y Fo(h)p Fn(\013)468 1768 y Fl(\003)502 1759 y Fn(;)8 b Fo(h)p Fn(\014)574 1739 y Fl(\003)571 1772 y Fm(\003)608 1759 y Fn(;)g(\015)659 1739 y Fm(\003)656 1772 y Fs(\001)691 1759 y Fo(ii)117 b Fp(and)e Fo(hh)p Fn(\013)1112 1768 y Fl(\003)1146 1759 y Fn(;)8 b(\014)1199 1739 y Fl(\003)1196 1772 y Fm(\003)1233 1759 y Fo(i)p Fn(;)g(\015)1303 1739 y Fm(\003)1300 1772 y Fs(\001)1336 1759 y Fo(i)0 1849 y Fp(for)16 b(all)h Fn(\013)178 1858 y Fl(\003)225 1849 y Fo(2)d Fp(\000,)j Fn(\014)365 1831 y Fl(\003)362 1861 y Fm(\003)413 1849 y Fo(2)d Fp(\000)491 1831 y Ft(N)529 1849 y Fp(,)i Fn(\015)588 1831 y Fm(\003)585 1863 y Fs(\001)634 1849 y Fo(2)f Fp(\000)713 1831 y Ft(N)750 1849 y Fp(.)i(Ho)o(w)o(ev)o (er,)f(the)h(existence)g(of)g(these)f(expressions)f(is)0 1909 y(an)f(immediate)f(consequence)g(of)i(\(4.1\),)f(\(ii\).)g(As)h (for)e(equalit)o(y)l(,)h(consider)f(the)h(map)f Fn(\013)1622 1918 y Fl(\003)1656 1909 y Fn(\014)1687 1891 y Fl(\003)1684 1921 y Fm(\003)1720 1909 y Fn(\015)1749 1891 y Fm(\003)1746 1921 y Ft(t)0 1969 y Fp(giv)o(en)j(b)o(y)542 2058 y Fn(N)h Fo(\002)10 b Fn(N)20 b Fo(3)14 b Fp(\()p Fn(n;)8 b(p)p Fp(\))14 b Fo(7!)g Fn(\013)980 2065 y Ft(n)1007 2058 y Fn(\014)1038 2038 y Ft(n)1035 2071 y(p)1065 2058 y Fn(\015)1094 2034 y Ft(p)1091 2071 y(t)1130 2058 y Fo(2)g Fn(R:)0 2148 y Fp(Since)g(b)o(y)g(\(1.2\),)g Fo(k)p Fn(\013)382 2155 y Ft(n)409 2148 y Fn(\014)440 2130 y Ft(n)437 2160 y(p)467 2148 y Fn(\015)496 2124 y Ft(p)493 2160 y(t)519 2148 y Fo(k)f(\024)h(k)p Fn(\013)667 2155 y Ft(n)694 2148 y Fo(k)p Fp(,)g(it)h(follo)o(ws)e(from)g(\(1.1\),)i(\(o\))g(and)e (\(ii\),)i(and)f(\(1.4\),)g(\(o\),)0 2208 y(that)20 b(the)g(map)f Fn(N)24 b Fo(3)19 b Fn(n)g Fo(7!)g Fn(\013)580 2215 y Ft(n)607 2208 y Fn(\014)638 2190 y Ft(n)635 2220 y(p)665 2208 y Fn(\015)694 2184 y Ft(p)691 2220 y(t)735 2208 y Fo(2)g Fn(R)h Fp(is)f(in)g Fn(S)990 2215 y Ft(R)1023 2208 y Fp(,)g(whence)g(\006)1269 2215 y Ft(R)1302 2208 y Fo(f)p Fn(\013)1359 2215 y Ft(n)1386 2208 y Fn(\014)1417 2190 y Ft(n)1414 2220 y(p)1444 2208 y Fn(\015)1473 2184 y Ft(p)1470 2220 y(t)1514 2208 y Fp(:)g Fn(n)g Fo(2)g Fn(N)5 b Fo(g)20 b Fp(is)0 2268 y(de\014ned.)15 b(Since,)h(due)g(to)h (\(1.2\))g(and)f(\(1.3\),)h(\(ii\),)288 2357 y Fo(k)p Fp(\006)349 2364 y Ft(R)381 2357 y Fo(f)p Fn(\013)438 2364 y Ft(n)465 2357 y Fn(\014)496 2337 y Ft(n)493 2370 y(p)523 2357 y Fn(\015)552 2333 y Ft(p)549 2369 y(t)589 2357 y Fp(:)c Fn(n)h Fo(2)g Fn(N)5 b Fo(gk)14 b(\024)g Fp(\006)905 2364 y Ft(C)938 2357 y Fo(fk)p Fn(\013)1020 2364 y Ft(n)1047 2357 y Fo(kk)p Fn(\014)1128 2337 y Ft(n)1125 2370 y(p)1155 2357 y Fo(kk)p Fn(\015)1234 2333 y Ft(p)1231 2369 y(t)1256 2357 y Fo(k)g Fp(:)g Fn(n)f Fo(2)h Fn(N)5 b Fo(g)816 2439 y(\024)14 b Fp(\006)905 2446 y Ft(C)938 2439 y Fo(fk)p Fn(\013)1020 2446 y Ft(n)1047 2439 y Fo(kk)p Fn(\014)1128 2418 y Ft(n)1125 2451 y(p)1155 2439 y Fo(k)g Fp(:)f Fn(n)h Fo(2)g Fn(N)5 b Fo(g)p Fn(;)0 2530 y Fp(and)16 b(since)h(b)o(y)f(\(1.1\),)i(\(iv\),)g(the)f(righ)o(t)f(hand)g(side)g (of)h(this)g(inequalit)o(y)f(\(as)h(a)g(function)g(of)g Fn(p)p Fp(\))0 2590 y(is)f(in)g Fn(S)139 2597 y Ft(C)172 2590 y Fp(,)h(it)g(follo)o(ws)e(from)h(\(1.3\),)h(\(o\),)g(that)499 2679 y Fn(N)j Fo(3)14 b Fn(p)g Fo(7!)f Fp(\006)744 2686 y Ft(R)777 2679 y Fo(f)p Fn(\013)834 2686 y Ft(n)861 2679 y Fn(\014)892 2659 y Ft(n)889 2692 y(p)919 2679 y Fn(\015)948 2655 y Ft(p)945 2692 y(t)984 2679 y Fp(:)h Fn(n)g Fo(2)g Fn(N)5 b Fo(g)14 b(2)g Fn(R)0 2777 y Fp(is)e(in)h Fn(S)132 2784 y Ft(R)164 2777 y Fp(.)g(Hence)g(\(1.1\),)g(\(iii'\),)g (and)f(the)h(use)f(of)h(a)g(bijection)f Fn(N)1177 2759 y Fs(2)1214 2777 y Fo(\000)-8 b(!)13 b Fn(N)18 b Fp(sho)o(w)12 b(that)h Fn(\013)1622 2786 y Fl(\003)1656 2777 y Fn(\014)1687 2759 y Fl(\003)1684 2790 y Fm(\003)1720 2777 y Fn(\015)1749 2759 y Fm(\003)1746 2790 y Ft(t)0 2837 y Fp(is)19 b(in)f Fn(S)144 2844 y Ft(R;N)t Fm(\002)p Ft(N)290 2837 y Fp(.)h(Therefore)g (\(2.11\))g(leads)g(to)g(the)g(desired)f(equalit)o(y)l(.)h(Whic)o(h)f (means,)g(that)p eop %%Page: 15 15 15 14 bop 1722 42 a Fp(15)0 167 y Fo(F)41 149 y Fs(\000)68 167 y Fp(\()p Fn(A)p Fp(\))17 b(is)e(a)g(left)i(\000-con)o(v)o(ex)e(mo) q(dule.)f(The)i(Dirac)f(map)g Fn(\016)h Fp(:)d Fn(A)i Fo(\000)-9 b(!)14 b(F)1332 149 y Fs(\000)1358 167 y Fp(\()p Fn(A)p Fp(\))j(assigns)d(to)i(eac)o(h)0 227 y Fn(a)i Fo(2)f Fn(A)i Fp(the)g(Dirac)f(function)h Fn(\016)597 209 y Ft(a)639 227 y Fp(at)g Fn(a)p Fp(.)g(W)l(e)g(claim)f(that)h Fn(\016)h Fp(:)d Fn(A)g Fo(\000)-8 b(!)17 b(F)1360 209 y Fs(\000)1387 227 y Fp(\()p Fn(A)p Fp(\))j(is)e(a)h(univ)o(ersal)0 286 y(arro)o(w.)e(Let)j Fn(X)k Fp(b)q(e)19 b(a)g(left)h(\000-con)o(v)o (ex)f(mo)q(dule)f(and)h(let)g Fn(h)f Fp(:)g Fn(A)h Fo(\000)-9 b(!)18 b Fn(X)23 b Fp(b)q(e)d(a)f(set)g(map.)f(W)l(e)0 346 y(w)o(an)o(t)e(to)g(de\014ne)g(an)g(appropriate)e(map)i Fn(h)803 328 y Fm(0)830 346 y Fp(:)e Fo(F)899 328 y Fs(\000)926 346 y Fp(\()p Fn(A)p Fp(\))h Fo(\000)-9 b(!)14 b Fn(X)t Fp(.)i(Let)h Fn(f)j Fo(2)14 b(F)1406 328 y Fs(\000)1433 346 y Fp(\()p Fn(A)p Fp(\).)j(Cho)q(ose)f(for)0 406 y Fn(f)22 b Fp(the)17 b(data)g Fn(\037)p Fp(,)f Fn(\014)335 413 y Fm(\003)358 406 y Fp(,)h Fn(A)426 388 y Fm(0)440 406 y Fp(,)g(and)e Fn(N)612 388 y Fm(0)643 406 y Fp(and)h(denote)h(b)o (y)f Fn(h)999 388 y Fm(\003)1038 406 y Fp(the)h(map)516 530 y Fn(N)j Fo(3)14 b Fn(n)f Fo(7!)730 460 y Fi(\032)776 502 y Fn(h)p Fp(\()p Fn(\037)p Fp(\()p Fn(n)p Fp(\)\))563 b(,)17 b(if)f Fn(n)e Fo(2)g Fn(N)1717 484 y Fm(0)1731 502 y Fp(;)776 561 y Fn(x)804 568 y Fs(0)1505 561 y Fp(,)j(if)f Fn(n)k(=)-31 b Fo(2)14 b Fn(N)1717 543 y Fm(0)1731 561 y Fp(;)0 657 y(where)20 b Fn(x)176 664 y Fs(0)219 657 y Fp(is)f(some)h(elemen)o(t)f(of)i Fn(X)t Fp(.)f(It)h(follo)o(ws)e (from)g(\(4.4\))i({)f(\(4.6\))h(that)f Fo(h)p Fn(\014)1523 664 y Fm(\003)1547 657 y Fn(;)8 b(h)1598 639 y Fm(\003)1620 657 y Fo(i)21 b Fp(is)f(in-)0 717 y(dep)q(enden)o(t)c(of)h Fn(x)326 724 y Fs(0)366 717 y Fp(and)f(of)i(the)f(data)g(c)o(hosen,)e (and)i(w)o(e)f(put)h Fn(h)1186 699 y Fm(0)1200 717 y Fp(\()p Fn(f)5 b Fp(\))16 b(:=)e Fo(h)p Fn(\014)1397 724 y Fm(\003)1420 717 y Fn(;)8 b(h)1471 699 y Fm(\003)1494 717 y Fo(i)p Fp(.)17 b(Ob)o(viously)l(,)0 777 y Fn(h)29 759 y Fm(0)43 777 y Fp(\()p Fn(\016)86 759 y Ft(a)110 777 y Fp(\))f(=)g Fn(h)p Fp(\()p Fn(a)p Fp(\),)j Fn(a)d Fo(2)g Fn(A)p Fp(,)i(whence)f Fn(h)690 759 y Fm(0)716 777 y Fo(\016)12 b Fn(\016)18 b Fp(=)d Fn(h)j Fp(holds.)e(Next)j(w)o(e) f(c)o(hec)o(k)f(that)h Fn(h)1510 759 y Fm(0)1542 777 y Fp(is)f(a)h(homo-)0 836 y(morphism)e(of)k(left)g(\000-con)o(v)o(ex)e (mo)q(dules.)g(Let)i Fn(\013)934 843 y Fm(\003)975 836 y Fo(2)e Fp(\000)h(and)g Fn(f)1205 818 y Fm(\003)1246 836 y Fo(2)g(F)1339 818 y Fs(\000)1366 836 y Fp(\()p Fn(A)p Fp(\))1441 818 y Ft(N)1480 836 y Fp(.)g(By)g(\(2.3\))h(w)o(e)0 896 y(can)g(c)o(ho)q(ose)f(data)h(so)g(that)g(they)h(serv)o(e)e(for)h (all)g Fn(f)975 878 y Ft(n)1003 896 y Fp(,)g Fn(n)f Fo(2)h Fn(N)5 b Fp(.)21 b(If)f Fn(\014)1303 878 y Ft(n)1300 909 y Fm(\003)1350 896 y Fp(is)f(the)i(elemen)o(t)e(of)i(\000)0 956 y(asso)q(ciated)16 b(with)h Fn(f)380 938 y Ft(n)407 956 y Fp(,)g Fn(n)d Fo(2)g Fn(N)5 b Fp(,)17 b(via)f(suc)o(h)g(data)g (then)569 1053 y Fn(h)598 1033 y Fm(0)612 1053 y Fp(\()p Fo(h)p Fn(\013)682 1060 y Fm(\003)705 1053 y Fn(;)8 b(f)756 1033 y Fm(\003)780 1053 y Fo(i)p Fp(\))15 b(=)e Fo(hh)p Fn(\013)955 1062 y Fl(\003)990 1053 y Fn(;)8 b(\014)1043 1033 y Fl(\003)1040 1065 y Fm(\003)1076 1053 y Fo(i)p Fn(;)g(h)1146 1033 y Fm(\003)1169 1053 y Fo(i)p Fn(;)0 1187 y(w)q(hil)q(e)430 b Fo(h)p Fn(\013)603 1194 y Fm(\003)626 1187 y Fn(;)8 b(h)677 1167 y Fm(0)p Ft(N)726 1187 y Fp(\()p Fn(f)774 1167 y Fm(\003)798 1187 y Fp(\))p Fo(i)14 b Fp(=)g Fo(h)p Fn(\013)954 1196 y Fl(\003)988 1187 y Fn(;)8 b Fo(h)p Fn(\014)1060 1167 y Fl(\003)1057 1200 y Fm(\003)1094 1187 y Fn(;)g(h)1145 1167 y Fm(\003)1168 1187 y Fo(ii)p Fn(;)0 1285 y Fp(whence)18 b(the)h(equalit)o(y)g(of)g(the)g(left)g (hand)f(sides)f(of)i(the)g(last)f(t)o(w)o(o)h(equations)f(follo)o(ws)g (from)0 1344 y(\(4.2\),)i(\(ii\),)g(sho)o(wing)f(that)h Fn(h)570 1326 y Fm(0)604 1344 y Fp(is)f(a)h(homomorphism)c(of)k(left)h (\000-con)o(v)o(ex)e(mo)q(dules.)g(As)h(for)0 1404 y(uniqueness)13 b(of)h(the)h(required)e(factorization,)g(let)951 1391 y(~)950 1404 y Fn(h)h Fp(:)f Fo(F)1061 1386 y Fs(\000)1088 1404 y Fp(\()p Fn(A)p Fp(\))i Fo(\000)-8 b(!)13 b Fn(X)19 b Fp(b)q(e)14 b(a)g(homomorphism)0 1464 y(of)24 b(left)h(\000-con)o(v)o (ex)e(mo)q(dules)g(with)701 1451 y(~)701 1464 y Fn(h)15 b Fo(\016)h Fn(\016)28 b Fp(=)e Fn(h)p Fp(.)e(F)l(or)f Fn(f)31 b Fo(2)c(F)1221 1446 y Fs(\000)1248 1464 y Fp(\()p Fn(A)p Fp(\))e(c)o(ho)q(ose)e(the)h(data)g(as)0 1524 y(ab)q(o)o(v)o(e)18 b(to)h(obtain)f Fn(\014)391 1531 y Fm(\003)431 1524 y Fo(2)f Fp(\000)h(asso)q(ciated)h(with)f Fn(f)5 b Fp(.)19 b(Denote)g(furthermore)d(b)o(y)i Fn(\016)1492 1506 y Ft(\037)p Fs(\()p Fm(\003)p Fs(\))1590 1524 y Fp(the)h(map)0 1583 y Fn(N)g Fo(3)14 b Fn(n)g Fo(7!)g Fn(\016)238 1565 y Ft(\037)p Fs(\()p Ft(n)p Fs(\))335 1583 y Fo(2)g(F)423 1565 y Fs(\000)450 1583 y Fp(\()p Fn(A)p Fp(\).)j(An)g(easy)f(computation)g(sho)o(ws)f(that)i Fn(f)j Fp(=)13 b Fo(h)p Fn(\014)1432 1590 y Fm(\003)1456 1583 y Fn(;)8 b(\016)1502 1565 y Ft(\037)p Fs(\()p Fm(\003)p Fs(\))1581 1583 y Fo(i)p Fp(.)17 b(Hence)259 1667 y(~)259 1681 y Fn(h)p Fp(\()p Fn(f)5 b Fp(\))15 b(=)423 1667 y(~)422 1681 y Fn(h)p Fp(\()p Fo(h)p Fn(\014)517 1688 y Fm(\003)541 1681 y Fn(;)8 b(\016)587 1660 y Ft(\037)p Fs(\()p Fm(\003)p Fs(\))666 1681 y Fo(i)p Fp(\))15 b(=)e Fo(h)p Fn(\014)818 1688 y Fm(\003)842 1681 y 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Fn(\026)740 2808 y Fm(\003)763 2828 y Fo(k)p Fp(\))h Fo(\024)f Fp(\006)909 2835 y Ft(C)943 2828 y Fp(\()p Fo(k)p Fn(\013)1019 2835 y Fm(\003)1042 2828 y Fo(kk)p Fn(\026)1122 2808 y Fm(\003)1145 2828 y Fo(k)p Fp(\))h Fo(\024)f Fp(\006)1291 2835 y Ft(C)1325 2828 y Fp(\()p Fo(k)p Fn(\013)1401 2835 y Fm(\003)1424 2828 y Fo(k)p Fp(\))h Fo(\024)g Fp(1)p Fn(;)p eop %%Page: 16 16 16 15 bop 0 42 a Fp(16)337 b Fv(Section)16 b(3:)61 b(B.)15 b(P)o(areigis,)j(D.)d(Pumpl)q(\177)-24 b(un,)15 b(and)g(H.)f(R\177)-23 b(ohrl)0 176 y Fp(whence)16 b Fo(h)p Fn(\013)225 183 y Fm(\003)248 176 y Fn(;)8 b(\026)300 158 y Fm(\003)324 176 y Fo(i)14 b Fp(=)f(\006)445 183 y Ft(M)490 176 y Fp(\()p Fn(\013)541 183 y Fm(\003)564 176 y Fn(\026)594 158 y Fm(\003)617 176 y Fp(\))k(is)f(in)g Fo(O)801 183 y Fs(\000)828 176 y Fp(\()p Fn(M)5 b Fp(\).)17 b(By)h(\(1.4\),)e (\(i\),)339 272 y Fo(h)p Fn(\016)382 252 y Ft(n)380 285 y Fm(\003)410 272 y Fn(;)8 b(\026)462 252 y Fm(\003)485 272 y Fo(i)14 b Fp(=)f(\006)606 279 y 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Ft(n)1243 437 y Fo(kk)p Fn(\014)1324 419 y Ft(n)1321 449 y(p)1351 437 y Fo(k)p Fp(,)g(for)g(all)g Fn(n;)8 b(p)19 b Fo(2)h Fn(N)5 b Fp(.)0 497 y(By)19 b(\(1.1\),)g(\(iv\),)g(the) g(righ)o(t)e(hand)h(side)g(of)g(these)h(inequalities,)f(as)g(a)g(map)g Fn(N)g Fo(\002)12 b Fn(N)22 b Fo(\000)-8 b(!)17 b Fn(C)t Fp(,)0 557 y(is)f(in)g Fn(S)139 564 y Ft(C;N)t Fm(\002)p Ft(N)283 557 y Fp(.)g(Therefore)g Fo(k)p Fn(\013)596 565 y Fl(\003)629 557 y Fn(\014)660 539 y Fl(\003)657 569 y Fm(\003)693 557 y Fn(\026)723 539 y Fm(\003)746 557 y Fo(k)h Fp(is)e(in)h Fn(S)926 564 y Ft(C;N)t Fm(\002)p Ft(N)1087 557 y Fp(b)o(y)g(\(1.1\),)g(\(ii\),)h(and)e Fn(\013)1513 565 y Fl(\003)1547 557 y Fn(\014)1578 539 y Fl(\003)1575 569 y Fm(\003)1611 557 y Fn(\026)1641 539 y Fm(\003)1680 557 y Fp(is)h(in)0 616 y Fn(S)31 623 y Ft(M)r(;N)t Fm(\002)p Ft(N)203 616 y Fp(b)o(y)g(\(1.4\),)h(\(o\).)g (Hence)g(\(2.11\))g(and)f(\(2.12\))h(imply)236 719 y(\006)272 726 y Ft(M)r(;N)t Fm(\002)p Ft(N)428 719 y Fp(\()p Fn(\013)479 728 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Ff(satis\014es)h(the)f(fol)s(lowing)g(c)m(onditions)99 2069 y Fp(CO:)25 b Fn(C)c Ff(is)d(c)m(omplete)g(\(in)g(the)g(sense)h (of)f Fp([5],)e Fo(x)p Fp(1)p Ff(\);)128 2129 y Fp(IS:)25 b Ff(for)18 b(every)g Fn(T)j Fo(\022)13 b Fn(S)557 2136 y Ft(C)590 2129 y Ff(,)19 b Fp(inf)s Fo(f)p Fp(\006)745 2136 y Ft(C)779 2129 y Fp(\()p Fn(t)816 2136 y Fm(\003)839 2129 y Fp(\))14 b(:)g Fn(t)918 2136 y Fm(\003)954 2129 y Fo(2)g Fn(T)7 b Fo(g)14 b Fp(=)g(\006)1165 2136 y Ft(C)1198 2129 y Fp(\(inf)t Fo(f)p Fn(t)1321 2136 y Fm(\003)1358 2129 y Fp(:)g Fn(t)1404 2136 y Fm(\003)1440 2129 y Fo(2)g Fn(T)7 b Fo(g)p Fp(\))p Ff(;)105 2188 y Fp(LD:)25 b Ff(for)e(every)g Fo(;)h(6)p Fp(=)f Fn(U)28 b Fo(\022)23 b Fn(C)t Ff(,)g Fn(u)776 2195 y Fs(0)821 2188 y Fp(:=)g(inf)s Fo(f)p Fn(u)g Fp(:)g Fn(u)g Fo(2)h Fn(U)5 b Fo(g)p Ff(,)23 b(and)h(every)f Fn(t)h Fo(2)f Fn(C)j Ff(with)213 2248 y Fn(u)242 2255 y Fs(0)277 2248 y Fn(<)14 b(u)359 2255 y Fs(0)392 2248 y Fp(+)d Fn(t)17 b Ff(ther)m(e)i(is)f(a)g Fn(u)727 2255 y Fs(1)763 2248 y Fo(2)c Fn(U)23 b Ff(with)c Fn(u)1005 2255 y Fs(0)1040 2248 y Fo(\024)14 b Fn(u)1122 2255 y Fs(1)1157 2248 y Fo(\024)g Fn(u)1239 2255 y Fs(0)1272 2248 y Fp(+)d Fn(t)p Ff(;)79 2308 y Fp(LIM:)25 b Ff(for)18 b(every)g Fn(U)h Fo(\022)14 b Fn(C)21 b Ff(and)e(every)e Fn(c)d Fo(2)g Fn(C)t Ff(,)j Fp(inf)t Fo(f)p Fn(cu)12 b Fp(:)i Fn(u)f Fo(2)h Fn(U)5 b Fo(g)15 b Fp(=)e Fn(c)8 b Fp(inf)t Fo(f)p Fn(u)13 b Fp(:)h Fn(u)f Fo(2)h Fn(U)5 b Fo(g)p Ff(;)101 2368 y Fp(OP:)25 b Ff(for)18 b(every)g Fn(c)446 2375 y Fm(\003)482 2368 y Fo(2)c Fn(S)560 2375 y Ft(C)611 2368 y Ff(ther)m(e)k(is)h(a)f Fn(d)858 2375 y Fm(\003)895 2368 y Fo(2)c Fn(C)982 2350 y Ft(N)1036 2368 y Ff(such)19 b(that)337 2427 y Fp(\(o\))25 b Fn(c)447 2434 y Fm(\003)481 2427 y Fp(+)11 b Fn(d)557 2434 y Fm(\003)593 2427 y Fo(2)j Fn(S)671 2434 y Ft(C)704 2427 y Ff(;)348 2487 y Fp(\(i\))25 b Ff(if)16 b Fn(c)493 2494 y Ft(n)536 2487 y Ff(is)g(not)g(a)g(maximal)g(element)f(of)h Fn(C)j 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Fn(S)f Fp(:=)e Fo(f)p Fp(\()p Fn(\016)353 248 y Fm(h)p Ft(\013)395 253 y Fk(\003)416 248 y Ft(;x)451 233 y Fk(\003)471 248 y Fm(i)490 269 y Fn(;)8 b Fo(h)p Fn(\013)563 276 y Fm(\003)586 269 y Fn(;)g(\016)632 248 y Ft(x)655 233 y Fk(\003)678 269 y Fo(i)p Fp(\))15 b(:)e Fn(\013)790 276 y Fm(\003)827 269 y Fo(2)h Fp(\000)p Fn(;)8 b(x)955 248 y Fm(\003)992 269 y Fo(2)14 b Fn(X)1084 248 y Ft(N)1122 269 y Fo(g)g(\022)g(L)1248 248 y Ft(N)1286 269 y Fp(\()p Fn(X)t Fp(\))e Fo(\002)f(L)1465 248 y Ft(N)1503 269 y Fp(\()p Fn(X)t Fp(\))p Fn(;)0 371 y Fp(where)21 b Fn(\016)173 353 y Ft(x)196 338 y Fk(\003)240 371 y Fp(is)f(the)i(map)e Fn(N)27 b Fo(3)22 b Fn(n)f Fo(7!)g Fn(\016)768 353 y Ft(x)791 338 y Fd(n)840 371 y Fo(2)g(L)928 353 y Ft(N)966 371 y Fp(\()p Fn(X)t Fp(\).)h(If)g Fn(h)f Fp(:)g Fo(L)1259 353 y Ft(N)1297 371 y Fp(\()p Fn(X)t Fp(\))i Fo(\000)-9 b(!)22 b Fn(M)27 b Fp(is)20 b(a)h(con-)0 431 y(tracting)g(homomorphism)d(of)j Fn(R)p Fp(-prenormed)e Fn(R)p Fp(-semimo)q(dules)g(with)i Fn(N)5 b Fp(-summation)20 b(w)o(e)0 490 y(sa)o(y)f(that)h Fn(h)f Fp(is)g Fn(S)s Fp(-compatible)e(if)j Fn(S)688 497 y Ft(h)731 490 y Fp(:=)f Fo(f)p Fp(\()p Fn(f)s(;)8 b(f)925 472 y Fm(0)940 490 y Fp(\))19 b(:)f Fn(h)p Fp(\()p Fn(f)5 b Fp(\))20 b(=)e Fn(h)p Fp(\()p Fn(f)1260 472 y Fm(0)1275 490 y Fp(\))p Fo(g)h(\022)f(L)1429 472 y Ft(N)1467 490 y Fp(\()p Fn(X)t Fp(\))c Fo(\002)f(L)1650 472 y Ft(N)1688 490 y Fp(\()p Fn(X)t Fp(\))0 550 y(con)o(tains)j Fn(S)s Fp(.)h(Clearly)l(,)f(there)h (are)g(suc)o(h)f(con)o(tracting)h(homomorphism)o(s,)d(e.g.)j(the)g (zero)g(ho-)0 610 y(momorphism.)j(Let)k Fo(\030)f Fp(b)q(e)g(the)h(in)o (tersection)e(of)h(all)g(these)g Fn(S)1223 617 y Ft(h)1248 610 y Fp(.)g(Then)g Fo(\030)g Fp(is)g(an)g(equiv)m(a-)0 670 y(lence)c(relation,)g Fo(S)361 652 y Fs(\000)388 670 y Fp(\()p Fn(X)t Fp(\))h(:=)e Fo(L)596 652 y Ft(N)634 670 y Fp(\()p Fn(X)t Fp(\))p Fn(=)i Fo(\030)f Fp(is)g(an)g Fn(R)p Fp(-semimo)q(dule,)f(and)g(the)i(quotien)o(t)g(map)0 729 y Fn(q)c Fp(:)e Fo(L)100 711 y Ft(N)138 729 y Fp(\()p Fn(X)t Fp(\))g Fo(\000)-8 b(!)14 b(S)364 711 y Fs(\000)391 729 y Fp(\()p Fn(X)t Fp(\))j(is)f(a)h(homomorphis)o(m)d(of)i Fn(R)p Fp(-semimo)q(dules)f(satisfying)331 831 y Fn(q)r Fp(\()p Fn(\016)398 811 y Fm(h)p Ft(\013)440 816 y Fk(\003)462 811 y Ft(;x)497 796 y Fk(\003)517 811 y Fm(i)536 831 y Fp(\))f(=)f Fn(q)r Fp(\()p Fo(h)p Fn(\013)715 838 y Fm(\003)739 831 y Fn(;)8 b(\016)785 811 y Ft(x)808 796 y Fk(\003)831 831 y Fo(i)p Fp(\))349 b(,)16 b(for)g(all)h Fn(\013)1426 838 y Fm(\003)1462 831 y Fo(2)d Fp(\000)p Fn(;)8 b(x)1590 813 y Fm(\003)1628 831 y Fo(2)14 b Fn(X)1720 813 y Ft(N)1758 831 y Fn(:)0 933 y Fp(No)o(w)i(de\014ne)g Fo(jjj)g(jjj)e Fp(:)g Fo(S)433 915 y Fs(\000)460 933 y Fp(\()p Fn(X)t Fp(\))g Fo(\000)-8 b(!)13 b Fn(C)20 b Fp(b)o(y)569 1035 y Fo(jjj)p Fn(s)p Fo(jjj)13 b Fp(:=)h(inf)s Fo(fjjj)p Fn(f)5 b Fo(jjj)14 b Fp(:)f Fn(q)r Fp(\()p Fn(f)5 b Fp(\))16 b(=)d Fn(s)p Fo(g)305 b Fn(;)8 b(s)14 b Fo(2)g(S)1647 1017 y Fs(\000)1674 1035 y Fp(\()p Fn(X)t Fp(\))p 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Fp(\()p Fn(f)262 1389 y Fm(\003)286 1382 y Fp(\))f(=)f Fn(q)403 1364 y Ft(N)441 1382 y Fp(\()p Fn(f)489 1364 y Fm(0)484 1394 y(\003)507 1382 y Fp(\),)i(that)g(is)g Fn(q)r Fp(\()p Fn(f)789 1389 y Ft(n)816 1382 y Fp(\))f(=)f Fn(q)r Fp(\()p Fn(f)981 1364 y Fm(0)976 1394 y Ft(n)1004 1382 y Fp(\))i(for)g(all)f Fn(n)f Fo(2)h Fn(N)5 b Fp(.)19 b(Then)f(\()p Fn(f)1543 1389 y Ft(n)1571 1382 y Fn(;)8 b(f)1622 1364 y Fm(0)1617 1394 y Ft(n)1644 1382 y Fp(\))18 b Fo(2)h(\030)0 1442 y Fp(and)11 b(hence)h(\()p Fn(f)269 1449 y Ft(n)297 1442 y Fn(;)c(f)348 1424 y Fm(0)343 1454 y Ft(n)371 1442 y Fp(\))14 b Fo(2)g Fn(S)482 1449 y Ft(h)520 1442 y Fp(for)d(all)h(con)o (tracting)f(homomorphism)o(s)e Fn(h)14 b Fp(:)f Fo(L)1384 1424 y Ft(N)1423 1442 y Fp(\()p Fn(X)t Fp(\))h Fo(\000)-8 b(!)13 b Fn(M)18 b Fp(that)0 1501 y(are)e Fn(S)s Fp(-compatible,)f(for) h(all)g Fn(n)e Fo(2)g Fn(N)5 b Fp(.)17 b(This)e(sho)o(ws)h(that)g Fn(h)1107 1483 y Ft(N)1145 1501 y Fp(\()p Fn(f)1188 1508 y Fm(\003)1212 1501 y Fp(\))e(=)f Fn(h)1326 1483 y Ft(N)1364 1501 y Fp(\()p Fn(f)1412 1483 y Fm(0)1407 1514 y(\003)1431 1501 y Fp(\))k(and)e(therefore)229 1603 y Fn(h)p Fp(\(\006)313 1613 y Fm(L)341 1603 y Fd(N)372 1613 y Fs(\()p Ft(X)s Fs(\))441 1603 y Fn(f)465 1610 y Fm(\003)488 1603 y Fp(\))g(=)e(\006) 610 1610 y Ft(M)654 1603 y Fp(\()p Fn(h)702 1583 y Ft(N)740 1603 y Fp(\()p Fn(f)783 1610 y Fm(\003)807 1603 y Fp(\)\))i(=)e(\006) 948 1610 y Ft(M)992 1603 y Fp(\()p Fn(h)1040 1583 y Ft(N)1078 1603 y Fp(\()p Fn(f)1126 1583 y Fm(0)1121 1616 y(\003)1145 1603 y Fp(\)\))h(=)g Fn(h)p Fp(\(\006)1334 1613 y Fm(L)1362 1603 y Fd(N)1394 1613 y Fs(\()p Ft(X)s Fs(\))1463 1603 y Fn(f)1492 1583 y Fm(0)1487 1616 y(\003)1510 1603 y Fp(\))p Fn(;)0 1705 y Fp(that)19 b(is)e(w)o(e)h(obtain)g(the)g (relation)g(\(\006)720 1715 y Fm(L)748 1705 y Fd(N)780 1715 y Fs(\()p Ft(X)s Fs(\))849 1705 y Fn(f)873 1712 y Fm(\003)896 1705 y Fn(;)8 b Fp(\006)954 1715 y Fm(L)982 1705 y Fd(N)1014 1715 y Fs(\()p Ft(X)s Fs(\))1083 1705 y Fn(f)1112 1687 y Fm(0)1107 1718 y(\003)1130 1705 y Fp(\))17 b Fo(2)g Fn(S)1247 1712 y Ft(h)1290 1705 y Fp(for)h(all)g(suc) o(h)f Fn(h)p Fp(,)h(whence)0 1765 y(\(\006)55 1775 y Fm(L)83 1765 y Fd(N)115 1775 y Fs(\()p Ft(X)s Fs(\))184 1765 y Fn(f)208 1772 y Fm(\003)231 1765 y Fn(;)8 b Fp(\006)289 1775 y Fm(L)317 1765 y Fd(N)349 1775 y Fs(\()p Ft(X)s Fs(\))418 1765 y Fn(f)447 1747 y Fm(0)442 1777 y(\003)465 1765 y Fp(\))14 b Fo(2\030)j Fp(holds.)e(Th)o(us)g(w)o(e)i(can)f (de\014ne)588 1867 y(\006)624 1877 y Fm(S)651 1867 y Fj(\000)674 1877 y Fs(\()p Ft(X)s Fs(\))743 1867 y Fp(\()p Fn(s)785 1874 y Fm(\003)809 1867 y Fp(\))e(:=)g Fn(q)r Fp(\(\006)988 1877 y Fm(L)1016 1867 y Fd(N)1048 1877 y Fs(\()p Ft(X)s Fs(\))1117 1867 y Fn(f)1141 1874 y Fm(\003)1164 1867 y Fp(\))296 b Fn(;)8 b(s)1524 1874 y Fm(\003)1561 1867 y Fo(2)14 b Fn(S)1639 1877 y Fm(S)1666 1867 y Fj(\000)1689 1877 y Fs(\()p Ft(X)s Fs(\))1758 1867 y Fn(;)0 1975 y Fp(where)d Fn(f)163 1982 y Fm(\003)200 1975 y Fo(2)j Fn(S)281 1956 y Ft(N)278 1988 y Fm(L)318 1975 y Fp(\()p Fn(X)t Fp(\))f(is)e(c)o(hosen)f(suc)o(h)g(that)i Fn(g)849 1956 y Ft(N)886 1975 y Fp(\()p Fn(f)929 1982 y Fm(\003)953 1975 y Fp(\))i(=)f Fn(s)1061 1982 y Fm(\003)1085 1975 y Fp(.)e(In)g(particular,)f Fn(q)r Fp(\(\006)1485 1984 y Fm(L)1513 1974 y Fd(N)1545 1984 y Fs(\()p Ft(X)s Fs(\))1614 1975 y Fp(\()p Fn(f)1657 1982 y Fm(\003)1680 1975 y Fp(\)\))15 b(=)0 2040 y(\006)36 2050 y Fm(S)63 2040 y Fj(\000)86 2050 y Fs(\()p Ft(X)s Fs(\))155 2040 y Fp(\()p Fn(q)198 2022 y Ft(N)236 2040 y Fp(\()p Fn(f)279 2047 y Fm(\003)303 2040 y Fp(\)\))26 b(for)g(all)f Fn(f)555 2047 y Fm(\003)607 2040 y Fo(2)30 b Fn(S)701 2050 y Fm(L)729 2040 y Fd(N)760 2050 y Fs(\()p Ft(X)s Fs(\))829 2040 y Fp(.)25 b(One)h(c)o(hec)o(ks)f (easily)g(that)h Fn(S)1443 2022 y Fs(\000)1440 2054 y Fm(S)1470 2040 y Fp(\()p Fn(X)t Fp(\))g(is)g(an)f Fn(R)p Fp(-)0 2105 y(subsemimo)q(dule)16 b(of)j Fo(S)443 2087 y Fs(\000)470 2105 y Fp(\()p Fn(X)t Fp(\))553 2087 y Ft(N)611 2105 y Fp(and)f(that)h(\006)856 2115 y Fm(S)883 2105 y Fj(\000)906 2115 y Fs(\()p Ft(X)s Fs(\))993 2105 y Fp(:)f Fn(S)1056 2115 y Fm(S)1083 2105 y Fj(\000)1105 2115 y Fs(\()p Ft(X)s Fs(\))1192 2105 y Fo(\000)-8 b(!)17 b(S)1324 2087 y Fs(\000)1351 2105 y Fp(\()p Fn(X)t Fp(\))j(is)e(a)h (homomor-)0 2165 y(phism)c(of)i Fn(R)p Fp(-semimo)q(dules.)71 2225 y(Next)h(w)o(e)e(v)o(erify)h(\(1.4\),)g(\(o\),)g(for)g Fo(S)743 2207 y Fs(\000)770 2225 y Fp(\()p Fn(X)t Fp(\).)h(Let)f Fn(s)997 2232 y Fm(\003)1034 2225 y Fo(2)e Fn(S)1113 2235 y Fm(S)1140 2225 y Fj(\000)1163 2235 y Fs(\()p Ft(X)s Fs(\))1248 2225 y Fp(and)h(c)o(ho)q(ose)h Fn(f)1528 2232 y Fm(\003)1565 2225 y Fo(2)e Fn(S)1644 2235 y Fm(L)1672 2225 y Fd(N)1703 2235 y Fs(\()p Ft(X)s Fs(\))0 2290 y Fp(with)23 b Fn(q)144 2272 y Ft(N)182 2290 y Fp(\()p Fn(f)225 2297 y Fm(\003)248 2290 y Fp(\))i(=)f Fn(s)378 2297 y Fm(\003)401 2290 y Fp(.)f(Then)f Fo(jjj)p Fn(s)639 2297 y Ft(n)666 2290 y Fo(jjj)i(\024)g(jjj)p Fn(f)861 2297 y Ft(n)888 2290 y Fo(jjj)p Fp(,)e Fn(n)i Fo(2)h Fn(N)5 b Fp(.)23 b(Since)f Fo(jjj)p Fn(f)1362 2297 y Fm(\003)1385 2290 y Fo(jjj)g Fp(is)h(in)f Fn(S)1601 2297 y Ft(C)1634 2290 y Fp(,)h(so)f(is)0 2350 y Fo(jjj)p Fn(s)65 2357 y Fm(\003)88 2350 y Fo(jjj)d Fp(due)g(to)h(\(1.1\),)g(\(ii\).)g (Con)o(v)o(ersely)l(,)d(assume)i Fn(s)1009 2357 y Fm(\003)1051 2350 y Fo(2)g(S)1137 2332 y Fs(\000)1164 2350 y Fp(\()p Fn(X)t Fp(\))1247 2332 y Ft(N)1305 2350 y Fp(and)g Fo(jjj)p Fn(s)1470 2357 y Fm(\003)1493 2350 y Fo(jjj)f(2)h Fn(S)1636 2357 y Ft(C)1669 2350 y Fp(.)g(W)l(e)0 2410 y(apply)k(\(OP\))g(to)g Fn(c)367 2417 y Fm(\003)415 2410 y Fp(:=)h Fo(jjj)p Fn(s)557 2417 y Fm(\003)580 2410 y Fo(jjj)e Fp(and)h(obtain)f Fn(d)935 2417 y Fm(\003)981 2410 y Fp(with)h(the)g(prop)q(erties)f (stated)i(there.)e(If)0 2470 y Fo(jjj)p Fn(s)65 2477 y Ft(n)92 2470 y Fo(jjj)g Fp(is)h(a)g(maximal)e(elemen)o(t)i(of)g Fn(C)t Fp(,)f(c)o(ho)q(ose)g Fn(f)988 2477 y Fm(\003)1036 2470 y Fo(2)j(L)1128 2452 y Ft(N)1166 2470 y Fp(\()p Fn(X)t Fp(\))f(suc)o(h)d(that)j Fn(q)r Fp(\()p Fn(f)1573 2477 y Ft(n)1600 2470 y Fp(\))h(=)f Fn(s)1730 2477 y Ft(n)1758 2470 y Fp(.)0 2529 y(Then)17 b Fo(jjj)p Fn(f)197 2536 y Ft(n)224 2529 y Fo(jjj)f Fp(=)f Fo(jjj)p Fn(s)401 2536 y Ft(n)428 2529 y Fo(jjj)p Fp(.)i(If)i Fo(jjj)p Fn(s)618 2536 y Ft(n)645 2529 y Fo(jjj)e Fp(is)g(not)h(a)g(maximal)e (elemen)o(t)i(of)g Fn(C)t Fp(,)e(\(LD\))j(implies)d(the)0 2589 y(existence)i(of)g(an)g Fn(f)366 2596 y Ft(n)409 2589 y Fo(2)f(L)493 2571 y Ft(N)531 2589 y Fp(\()p Fn(X)t Fp(\))i(with)e Fn(q)r Fp(\()p Fn(f)814 2596 y Ft(n)842 2589 y Fp(\))g(=)e Fn(s)955 2596 y Ft(n)1001 2589 y Fp(and)i Fo(jjj)p Fn(s)1164 2596 y Ft(n)1191 2589 y Fo(jjj)f(\024)f(jjj)p Fn(f)1369 2596 y Ft(n)1396 2589 y Fo(jjj)h(\024)g(jjj)p Fn(s)1574 2596 y Ft(n)1601 2589 y Fo(jjj)11 b Fp(+)h Fn(d)1731 2596 y Ft(n)1758 2589 y Fp(.)0 2649 y(Hence)17 b Fn(f)172 2656 y Fm(\003)212 2649 y Fp(is)g(in)f Fo(L)355 2631 y Ft(N)393 2649 y Fp(\()p Fn(X)t Fp(\))i(and)e Fo(jjj)p Fn(f)657 2656 y Fm(\003)679 2649 y Fo(jjj)e(\024)g(jjj)p Fn(s)853 2656 y Fm(\003)876 2649 y Fo(jjj)c Fp(+)h Fn(d)1004 2656 y Fm(\003)1041 2649 y Fo(2)k Fn(S)1120 2656 y Ft(C)1153 2649 y Fp(,)h(whence)h Fo(jjj)p Fn(f)1424 2656 y Fm(\003)1447 2649 y Fo(jjj)f Fp(is)g(in)g Fn(S)1644 2656 y Ft(C)1694 2649 y Fp(due)0 2709 y(to)j(\(1.1\),)g(\(ii\),)f(and)g(th)o(us)g Fn(f)531 2716 y Fm(\003)573 2709 y Fp(is)g(in)g Fn(S)716 2719 y Fm(L)744 2709 y Fd(N)775 2719 y Fs(\()p Ft(X)s Fs(\))863 2709 y Fp(b)o(y)g(\(1.4\),)h(\(o\).)g(Therefore)e Fn(s)1415 2716 y Fm(\003)1456 2709 y Fp(=)g Fn(q)1536 2691 y Ft(N)1573 2709 y Fp(\()p Fn(f)1616 2716 y Fm(\003)1640 2709 y Fp(\))i(is)f(in)0 2769 y Fn(S)31 2778 y Fm(S)58 2768 y Fj(\000)81 2778 y Fs(\()p Ft(X)s Fs(\))150 2769 y Fp(.)71 2828 y(\(1.4\),)f(\(i\),)g(is)f(trivially)g(satis\014ed)g(in) g(the)h(curren)o(t)e(situation.)p eop %%Page: 18 18 18 17 bop 0 42 a Fp(18)337 b Fv(Section)16 b(3:)61 b(B.)15 b(P)o(areigis,)j(D.)d(Pumpl)q(\177)-24 b(un,)15 b(and)g(H.)f(R\177)-23 b(ohrl)71 176 y Fp(On)16 b(to)g(\(1.4\),)h(\(ii\).)f(It)h(follo)o(ws)e (from)h(\(1.5\))h(that)f Fn(s)1027 183 y Fm(\003)1064 176 y Fo(2)e Fn(S)1142 185 y Fm(S)1169 175 y Fj(\000)1192 185 y Fs(\()p Ft(X)s Fs(\))1277 176 y Fp(and)i Fn(t)1392 183 y Fm(\003)1429 176 y Fo(2)e(S)1510 158 y Fs(\000)1537 176 y Fp(\()p Fn(X)t Fp(\))1620 158 y Ft(N)1675 176 y Fp(with)0 241 y Fo(jjj)p Fn(t)60 248 y Fm(\003)82 241 y Fo(jjj)g(\024)f(jjj)p Fn(s)255 248 y Fm(\003)278 241 y Fo(jjj)g Fp(implies)f Fn(t)517 248 y Fm(\003)554 241 y Fo(2)i Fn(S)632 251 y Fm(S)659 241 y Fj(\000)681 251 y Fs(\()p Ft(X)s Fs(\))750 241 y Fp(.)g(Hence)g(there)f(are)g Fn(g)1151 248 y Fm(\003)1187 241 y Fo(2)h Fn(S)1265 251 y Fm(L)1293 241 y Fd(N)1324 251 y Fs(\()p Ft(X)s Fs(\))1407 241 y Fp(with)f Fn(t)1535 248 y Fm(\003)1572 241 y Fp(=)g Fn(q)1648 223 y Ft(N)1686 241 y Fp(\()p Fn(g)1729 248 y Fm(\003)1752 241 y Fp(\))0 301 y(and)215 397 y Fo(jjj)p Fp(\006)293 407 y Fm(S)320 397 y Fj(\000)342 407 y Fs(\()p Ft(X)s Fs(\))411 397 y Fn(t)429 404 y Fm(\003)452 397 y Fo(jjj)g Fp(=)h Fo(jjj)p Fn(q)r Fp(\(\006)681 407 y Fm(L)709 397 y Fd(N)740 407 y Fs(\()p Ft(X)s Fs(\))809 397 y Fn(g)833 404 y Fm(\003)856 397 y Fp(\))p Fo(jjj)g(\024)f(jjj)p Fp(\006)1061 407 y Fm(L)1089 397 y Fd(N)1121 407 y Fs(\()p Ft(X)s Fs(\))1190 397 y Fn(g)1214 404 y Fm(\003)1236 397 y Fo(jjj)h(\024)f Fp(\006)1380 404 y Ft(C)1414 397 y Fo(jjj)p Fn(g)1480 404 y Fm(\003)1502 397 y Fo(jjj)p Fn(;)0 499 y Fp(due)j(to)h(\(1.4\),)g(\(ii\),)g(for)f Fo(L)495 481 y Ft(N)533 499 y Fp(\()p Fn(X)t Fp(\).)h(Hence)g(w)o(e)g (ha)o(v)o(e)254 597 y Fo(jjj)p Fp(\006)332 607 y Fm(S)359 597 y Fj(\000)382 607 y Fs(\()p Ft(X)s Fs(\))451 597 y Fn(t)469 604 y Fm(\003)492 597 y Fo(jjj)c(\024)h Fp(inf)s Fo(f)p Fp(\006)721 604 y Ft(C)755 597 y Fo(jjj)p Fn(g)821 604 y Fm(\003)843 597 y Fo(jjj)f Fp(:)h Fn(q)950 577 y Ft(N)988 597 y Fp(\()p Fn(g)1031 604 y Fm(\003)1054 597 y Fp(\))g(=)f Fn(t)1157 604 y Fm(\003)1180 597 y Fo(g)547 680 y Fp(=)h(\006)636 687 y Ft(C)669 680 y Fp(\(inf)t Fo(fjjj)p Fn(g)840 687 y Fm(\003)862 680 y Fo(jjj)f Fp(:)h Fn(q)969 659 y Ft(N)1007 680 y Fp(\()p Fn(g)1050 687 y Fm(\003)1073 680 y Fp(\))g(=)g Fn(t)1177 687 y Fm(\003)1200 680 y Fo(g)p Fp(\))g(=)f(\006)1346 687 y Ft(C)1380 680 y Fo(jjj)p Fn(t)1440 687 y Fm(\003)1462 680 y Fo(jjj)p Fn(:)71 776 y Fp(Finally)e(\(1.4\),)h(\(iii\).)g(Let)h Fn(')h Fp(:)f Fn(N)20 b Fo(\000)-9 b(!)14 b Fn(N)j Fp(b)q(e)c(a)f(map)f (and)g(let)i Fn(s)1230 783 y Fm(\003)1267 776 y Fo(2)h Fn(S)1345 786 y Fm(S)1372 776 y Fj(\000)1395 786 y Fs(\()p Ft(X)s Fs(\))1464 776 y Fp(.)e(Then)f(there)h(is)0 857 y(an)17 b Fn(f)94 864 y Fm(\003)131 857 y Fo(2)e Fn(S)210 867 y Fm(L)238 857 y Fd(N)269 867 y Fs(\()p Ft(X)s Fs(\))355 857 y Fp(with)i Fn(s)492 864 y Fm(\003)529 857 y Fp(=)e Fn(q)607 839 y Ft(N)644 857 y Fp(\()p Fn(f)687 864 y Fm(\003)711 857 y Fp(\).)i(Since)g Fn(s)915 831 y Ft(')941 816 y Fk(\000)p Fj(1)988 831 y Fs(\()p Ft(n)p Fs(\))915 864 y Fm(\003)1060 857 y Fp(=)d Fn(q)1137 839 y Ft(N)1175 857 y Fp(\()p Fn(f)1223 831 y Ft(')1249 816 y Fk(\000)p Fj(1)1297 831 y Fs(\()p Ft(n)p Fs(\))1218 864 y Fm(\003)1355 857 y Fp(\))j(and)g(since)f Fn(f)1640 831 y Ft(')1666 816 y Fk(\000)p Fj(1)1713 831 y Fs(\()p Ft(n)p Fs(\))1635 864 y Fm(\003)0 938 y Fp(is)i(in)f Fn(S)142 948 y Fm(L)170 938 y Fd(N)201 948 y Fs(\()p Ft(X)s Fs(\))270 938 y Fp(,)h(due)g(to)g (\(1.4\),)h(\(iii\),)f(applied)f(to)h Fo(L)979 920 y Ft(N)1017 938 y Fp(\()p Fn(X)t Fp(\),)h(w)o(e)f(ha)o(v)o(e)f Fn(s)1347 912 y Ft(')1373 897 y Fk(\000)p Fj(1)1420 912 y Fs(\()p Ft(n)p Fs(\))1347 945 y Fm(\003)1495 938 y Fo(2)f Fn(S)1575 948 y Fm(S)1602 938 y Fj(\000)1625 948 y Fs(\()p Ft(X)s Fs(\))1712 938 y Fp(for)0 1019 y(ev)o(ery)j Fn(n)e Fo(2)h Fn(N)5 b Fp(.)19 b(Moreo)o(v)o(er,)e(b)o(y)i (de\014nition,)e(\006)890 1029 y Fm(S)917 1019 y Fj(\000)940 1029 y Fs(\()p Ft(X)s Fs(\))1009 1019 y Fn(s)1032 993 y Ft(')1058 978 y Fk(\000)p Fj(1)1105 993 y Fs(\()p Ft(n)p Fs(\))1032 1026 y Fm(\003)1181 1019 y Fp(=)h Fn(q)r Fp(\(\006)1317 1029 y Fm(L)1345 1019 y Fd(N)1376 1029 y Fs(\()p Ft(X)s Fs(\))1445 1019 y Fn(f)1474 993 y Ft(')1500 978 y Fk(\000)p Fj(1)1548 993 y Fs(\()p Ft(n)p Fs(\))1469 1026 y Fm(\003)1606 1019 y Fp(\).)h(Since)0 1100 y Fn(f)29 1076 y Ft(')55 1061 y Fk(\000)p Fj(1)24 1107 y Fm(\003)105 1100 y Fp(,)e(that)h(is)f (the)g(map)g Fn(N)j Fo(3)15 b Fn(n)g Fo(7!)g Fp(\006)748 1110 y Fm(L)776 1100 y Fd(N)808 1110 y Fs(\()p Ft(X)s Fs(\))877 1100 y Fn(f)906 1074 y Ft(')932 1059 y Fk(\000)p Fj(1)979 1074 y Fs(\()p Ft(n)p Fs(\))901 1107 y Fm(\003)1052 1100 y Fo(2)h(L)1135 1082 y Ft(N)1173 1100 y Fp(\()p Fn(X)t Fp(\),)i(is)f(in)g Fn(S)1429 1110 y Fm(L)1457 1100 y Fd(N)1488 1110 y Fs(\()p Ft(X)s Fs(\))1574 1100 y Fp(it)h(follo)o(ws)0 1179 y(that)f Fn(s)131 1155 y Ft(')157 1140 y Fk(\000)p Fj(1)131 1186 y Fm(\003)223 1179 y Fp(is)f(in)g Fn(S)362 1189 y Fm(S)389 1179 y Fj(\000)412 1189 y Fs(\()p Ft(X)s Fs(\))481 1179 y Fp(.)g(Finally)248 1291 y(\006)284 1301 y Fm(S)311 1291 y Fj(\000)334 1301 y Fs(\()p Ft(X)s Fs(\))403 1291 y 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Fp(except)i(that)f(the)f(requiremen)o(t)0 2359 y(\\existence)21 b(of)f(the)g(suprem)o(um)e(of)i(ev)o(ery)g(subset)g Fn(Y)31 b Fp(with)20 b(card)o Fn(Y)31 b Fo(\024)20 b Fp(card)o Fn(N)5 b Fp(")20 b(has)g(to)g(b)q(e)0 2419 y(replaced)e(b)o(y)g(the)h (requiremen)o(t)e(\\existence)j(of)f(the)g(suprem)o(um)c(of)k(ev)o(ery) g(subset)f Fn(Y)30 b Fp(with)0 2478 y(card)o Fn(Y)25 b(<)14 b Fp(card)o Fn(N)5 b Fp(".)71 2538 y(Instead)20 b(of)g(\000)342 2545 y Fb(D)390 2538 y Fp(one)g(could)g(tak)o(e)h(the)f (set)h(\000)936 2545 y Fb(D)977 2538 y Fh(r)14 b Fo(f)p Fp(0)1080 2545 y Fm(\003)1102 2538 y Fo(g)p Fp(.)20 b(Again)h(it)f(is)g (easy)g(to)h(see)f(that)0 2598 y(\000)31 2605 y Fb(D)t Ft(;sc)125 2598 y Fp(:=)e(\000)227 2605 y Fb(D)268 2598 y Fh(r)12 b Fo(f)p Fp(0)369 2605 y Fm(\003)392 2598 y Fo(g)19 b Fp(is)g(an)f Fn(N)5 b Fp(-con)o(v)o(exit)o(y)19 b(theory)g(o)o(v)o(er)g Fh(D)8 b Fp(.)22 b(Again,)c(as)h(b)q(efore,)g (the)g(same)0 2658 y(results)j(remain)f(in)i(force,)f(except)i(that)f (the)g(subsets)f Fn(Y)34 b Fp(in)23 b(question)f(no)o(w)g(ha)o(v)o(e)g (to)h(b)q(e)0 2718 y(nonempt)o(y)l(.)18 b(It)h(should)f(b)q(e)h(p)q (oin)o(ted)g(out,)g(that)g(\000)961 2725 y Fb(D)t Ft(;sc)1057 2718 y Fp(is)f(the)i Fh(D)8 b Fp(-analog)20 b(to)g(the)f(sup)q(ercon-)0 2777 y(v)o(exit)o(y)j(theory)f(\012)350 2784 y Ft(sc)412 2777 y Fp(:=)g Fo(f)p Fn(\013)543 2784 y Fm(\003)588 2777 y Fo(2)h Fh(R)682 2759 y Fb(N)729 2777 y Fp(:)g Fn(\013)797 2784 y Ft(n)846 2777 y Fo(\025)g Fp(0)p Fn(;)j Fp(for)16 b(all)g Fn(n)22 b Fo(2)g Fh(N)p Fn(;)k Fp(and)16 b(\006)1431 2784 y Fb(N)1459 2777 y Fn(\013)1491 2784 y Ft(n)1540 2777 y Fp(=)22 b(1)p Fo(g)p Fp(.)f(The)0 2837 y Fh(D)8 b Fp(-analog)20 b(to)e(the)g(classical)f(con)o(v)o(exit)o (y)h(theory)g(\012)971 2844 y Ft(c)1008 2837 y Fp(:=)e Fo(f)p Fn(\013)1134 2844 y Fm(\003)1173 2837 y Fo(2)g Fp(\012)1258 2844 y Ft(sc)1314 2837 y Fp(:)g(supp)f Fn(\013)1495 2844 y Fm(\003)1534 2837 y Fp(is)h(\014nite)p Fo(g)i Fp(is)p eop %%Page: 21 21 21 20 bop 1722 42 a Fp(21)0 167 y(then)17 b(\000)145 174 y Fb(D)5 b Ft(;c)218 167 y Fp(:=)14 b Fo(f)p Fn(\013)342 174 y Fm(\003)380 167 y Fo(2)h Fp(\000)459 174 y Fb(D)5 b Ft(;sc)551 167 y Fp(:)14 b(supp)i Fn(\013)731 174 y Fm(\003)770 167 y Fp(is)g(\014nite)p Fo(g)p Fp(;)h(results)f(similar)g (to)h(the)h(ab)q(o)o(v)o(e)f(hold)f(for)0 227 y(\000)31 234 y Fb(D)t Ft(;c)88 227 y Fp(.)724 406 y Fa(References)-23 521 y Fv([1])23 b(Birkho\013,)15 b(G.:)g Fz(L)n(attic)n(e)h(The)n(ory.) f Fv(AMS)g(Coll.)i(Publ.)f(XXV)e(\(1967\).)-23 575 y([2])23 b(Manes,)15 b(E.)h(G.:)f Fz(A)o(lgebr)n(aic)i(The)n(ories.)d Fv(Springer)i(V)l(erlag)g(\(1976\).)-23 630 y([3])23 b(P)o(areigis,)16 b(B.,)d(and)g(R\177)-23 b(ohrl,)14 b(H.:)f Fz(L)n(eft)i(line)n(ar)g(the)n(ories.)e Fv(Sem.b)q(er.)i(F)l (ern)o(univ.)f(Hagen)f(44,)h(276-302)49 685 y(\(1992\).)-23 740 y([4])23 b(Pumpl)q(\177)-24 b(un,)19 b(D.,)h(and)f(R\177)-23 b(ohrl,)20 b(H.:)f Fz(Banach)h(sp)n(ac)n(es)f(and)g(total)r(ly)h(c)n (onvex)h(sp)n(ac)n(es)e(I.)g Fv(Comm.)i(in)49 795 y(Alg.)15 b(12,)g(953{1019)h(\(1984\).)-23 849 y([5])23 b(R\177)-23 b(ohrl,)17 b(H.:)f Fz(Convexity)j(the)n(ories)f(0.)g(F)l(oundations.)d Fv(Sem.b)q(er.)j(F)l(ern)o(univ.)f(Hagen)g(44,)g(333{376)49 904 y(\(1992\).)-23 959 y([6])23 b(Semadeni,)16 b(Z.:)f Fz(Banach)g(sp)n(ac)n(es)g(I.)g Fv(P)o(olish)i(Sci.)f(Publ.)g (\(1971\).)-23 1014 y([7])23 b(W)l(einert,)f(H.)e(J.:)g Fz(Gener)n(alize)n(d)g(semialgebr)n(as)g(over)i(semirings.)f Fv(Springer)g(Lecture)g(Notes)g(in)49 1068 y(Mathematics)c(1320,)e (380{416)h(\(1986\).)p eop %%Page: 22 22 22 21 bop 0 42 a Fp(22)337 b Fv(Section)16 b(3:)61 b(B.)15 b(P)o(areigis,)j(D.)d(Pumpl)q(\177)-24 b(un,)15 b(and)g(H.)f(R\177)-23 b(ohrl)15 176 y(Bo)q(do)15 b(P)o(areigis)15 230 y(Math.)g(Inst.)g (Univ.)g(M)q(\177)-24 b(unc)o(hen)15 285 y(Theresienstr.)18 b(39)15 340 y(80333)e(M)q(\177)-24 b(unc)o(hen)15 395 y(German)o(y)584 176 y(Dieter)17 b(Pumpl)q(\177)-24 b(un)584 230 y(F)l(ac)o(h)o(b.)14 b(Math.)h(F)l(ern)o(univ.)h(Hagen)584 285 y(58084)f(Hagen)584 340 y(German)o(y)1215 176 y(Helm)o(ut)h(R\177) -23 b(ohrl)1215 230 y(9322)16 b(La)f(Jolla)h(F)l(arms)h(Road.)1215 285 y(La)e(Jolla,)i(CA)e(92037)1215 340 y(USA)1230 395 y(and)1215 450 y(Math.)g(Inst.)f(TU)i(M)q(\177)-24 b(unc)o(hen)1215 504 y(Arcisstr.)16 b(21)1215 559 y(80333)g(M)q(\177)-24 b(unc)o(hen)1264 614 y(German)o(y)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF