%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: Left_lin.dvi %%Pages: 28 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -O 0cm,2cm Left_lin %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1995.05.31:1950 %%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 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y(addition)k(deriv)o(ed)d(from)i(the)g(giv)o(en)f(op)q (erations.)35 b(W)l(e)20 b(obtain)h(that)g(certain)f(prop)q(erties)p eop %%Page: 4 4 4 3 bop 0 265 a Fn(carry)17 b(o)o(v)o(er)g(from)f(the)h(left)f(linear)h (theory)g Fi(L)h Fn(to)g(left)e Fi(L)p Fn(-mo)q(dules)h(suc)o(h)g(as)h (zero)f(elemen)o(ts)0 325 y(or)h(some)f(kind)g(of)i(additiv)o(e)d (structure:)24 b(for)18 b(example,)e(if)h Fi(L)i Fn(b)q(ecomes)d(an)j (ab)q(elian)f(group)0 385 y(b)o(y)i(its)g(op)q(eration,)i(then)e(ev)o (ery)f(mo)q(dule)g(o)o(v)o(er)g Fi(L)i Fn(carries)f(the)g(structure)g (of)h(an)f(ab)q(elian)0 445 y(group)f(as)g(w)o(ell.)27 b(It)18 b(is)g(in)o(teresting)f(to)i(see)f(ho)o(w)h(free)e(mo)q(dules)h (arise)g(and)h(ho)o(w)g(they)e(can)0 505 y(b)q(e)e(em)o(b)q(edded)d(in) o(to)i Fi(L)h Fn(\(section)f(5\).)21 b(In)15 b(fact)f(w)o(e)g(will)g (see,)g(that)h(the)f(category)h(of)g(mo)q(dules)0 566 y(is)h(an)h(algebraic)g(category)l(.)22 b(In)16 b(the)g(\014nal)h 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b(4.14)h(w)o(e)e(sho)o(w)i(this)f(functor)g(to)g(b)q(e)g(an)g(equiv) m(alence)0 2491 y(of)g(categories.)0 2576 y(e\))f(F)l(or)g(an)o(y)g (index)g(set)g Fm(I)k Fn(the)c(pro)q(duct)h Fi(L)787 2558 y Fj(I)823 2576 y Fn(is)f(an)h Fi(L)p Fn(-bimo)q(dule.)p eop %%Page: 10 10 10 9 bop 0 265 a Fn(f)s(\))22 b(F)l(or)h Fm(N)k Fn(the)22 b(set)g(of)g(natural)h(n)o(um)o(b)q(ers)d(and)j Fi(L)g Fn(:=)g Fm(R)1101 247 y Fk(\()p Fj(N)t Fk(\))1185 265 y Fn(the)e(set)h Fm(X)27 b Fn(of)22 b(in\014nite)f(se-)0 325 y(quences)d(of)h(p)q(o)o(w)o(er)f(series)518 292 y Fh(P)562 335 y Fj(i)584 325 y Fm(r)607 301 y Fj(j)606 336 y(i)626 325 y Fm(x)654 307 y Fj(i)686 325 y Fn(\()p Fm(X)23 b Fi(\022)17 b Fm(R)p Fn([[)p Fm(x)p Fn(]])945 307 y Fj(N)977 325 y Fn(\))i(suc)o(h)f(that)h(\()p Fm(r)1277 301 y Fj(j)1276 336 y(i)1296 325 y Fn(\))1315 332 y Fj(j)r Fl(2)p Fj(N)1407 325 y Fn(has)g(\014nite)f(sup-)0 385 y(p)q(ort)i(for)f(all)f Fm(i)g Fi(2)g Fm(N)24 b Fn(is)19 b(an)g Fi(L)p Fn(-bimo)q(dule.)28 b(The)19 b(left)f(m)o(ultiplic)o (ation)f(is)h(de\014ned)h(b)o(y)f Fm(l)q(\030)j Fn(=)0 445 y(\()p Fm(\013)50 427 y Fj(k)72 445 y Fn(\)\()110 412 y Fh(P)153 456 y Fj(i)176 445 y Fm(r)199 422 y Fj(j)198 458 y(ik)231 445 y Fm(x)259 427 y Fj(i)273 445 y Fn(\))k(:=)g(\()413 412 y Fh(P)457 456 y Fj(i;k)508 445 y Fm(\013)539 427 y Fj(k)561 445 y Fm(r)584 422 y Fj(j)583 458 y(ik)616 445 y Fm(x)644 427 y Fj(i)658 445 y Fn(\))677 452 y Fj(j)695 445 y Fn(.)42 b(The)23 b(righ)o(t)f(m)o(ultiplicati)o(on)f(is)i(\()1377 412 y Fh(P)1420 456 y Fj(i)1443 445 y Fm(r)1466 422 y Fj(j)1465 457 y(i)1484 445 y Fm(x)1512 427 y Fj(i)1526 445 y Fn(\)\()p Fm(\013)1595 427 y Fj(k)1595 458 y(j)1617 445 y Fn(\))i(:=)0 514 y(\()19 481 y Fh(P)63 524 y Fj(i;j)111 514 y Fm(r)134 490 y Fj(j)133 525 y(i)153 514 y Fm(\013)184 496 y Fj(k)184 526 y(j)205 514 y Fm(x)233 496 y Fj(i)247 514 y Fn(\))266 521 y Fj(k)q Fl(2)p Fj(N)343 514 y Fn(.)f(In)17 b(fact)g(there)g(is)g(a)g(bimo)q(dule)f(isomorphism)f Fm(X)1317 500 y Fi(\030)1317 516 y Fn(=)1371 514 y Fi(L)1405 496 y Fj(N)1456 514 y Fn(b)o(y)i(\()1544 481 y Fh(P)1588 524 y Fj(i)1610 514 y Fm(r)1633 490 y Fj(j)1632 525 y(i)1652 514 y Fm(x)1680 496 y Fj(i)1694 514 y Fn(\))0 582 y Fi(7!)d Fn(\(\()p Fm(r)125 559 y Fj(j)124 594 y(i)143 582 y Fn(\))162 589 y Fj(j)r Fl(2)p Fj(N)235 582 y Fn(\))254 589 y Fj(i)p Fl(2)p Fj(N)324 582 y Fn(.)0 667 y(g\))h(Let)h Fi(L)f Fn(b)q(e)h(the)e(unital)h(sequen)o(tial)f(left)g(linear)h(theory)g(of)g (1.4)g(example)e(i\))i(consisting)g(of)0 728 y(sequences)i(in)g Fm(R)318 709 y Fk(\()p Fj(I)l(N)s Fk(\))408 728 y Fn(with)g(only)g(one) h(non-zero)g(co)q(e\016cien)o(t.)23 b(Then)17 b Fm(X)j Fn(:=)c Fm(R)1471 722 y Fn(_)1461 728 y Fi([)p Fm(R=)p Fn(\(0)i(=)d(0\))0 788 y(is)g(a)h(left)f Fi(L)p Fn(-mo)q(dule,)g(where) g(\(0)p Fm(;)8 b(:)g(:)g(:)g(;)g Fn(0)p Fm(;)g(\013)779 795 y Fj(i)793 788 y Fm(;)g Fn(0)p Fm(;)g(:)g(:)g(:)o Fn(\)\()p Fm(x)984 795 y Fj(j)1002 788 y Fn(\))1021 795 y Fj(j)r Fl(2)p Fj(I)l(N)1120 788 y Fn(=)13 b Fm(\013)1202 795 y Fj(i)1216 788 y Fm(x)1244 795 y Fj(i)1274 788 y Fn(view)o(ed)h(as)i(elemen)o(t)d(in)0 848 y(the)j(same)f(comp)q(onen)o (t)h(of)g(X)g(as)h Fm(x)652 855 y Fj(i)682 848 y Fn(is)f(from.)585 1058 y Fo(3.)52 b(Zer)o(o)23 b(Elements)0 1168 y Fw(3.1)18 b(De\014nition:)24 b Fn(Let)16 b Fi(L)h Fn(b)q(e)f(a)h(left)e(linear)g (theory)l(.)0 1228 y(a\))21 b(Let)f Fm(X)25 b Fn(b)q(e)c(a)g(left)e Fi(L)p Fn(-mo)q(dule.)34 b Fm(X)25 b Fg(has)c(a)g(righ)o(t)f(zero)g Fn(if)g(there)g(is)g(a)h(unique)f(elemen)o(t)0 1288 y(0)14 b Fi(2)g Fm(X)21 b Fn(suc)o(h)16 b(that)296 1398 y Fi(8)p Fm(l)d Fi(2)h(L)g Fn(:)f Fm(l)q Fn(0)515 1377 y Fj(N)563 1398 y Fn(=)h(0)49 b(\(where)16 b(0)872 1377 y Fj(N)922 1398 y Fn(is)g(the)g(constan)o(t)h(family\))m Fm(:)241 b Fn(\()p Fm(i)p Fn(\))0 1508 y(b\))20 b(Let)h Fm(X)k Fn(b)q(e)20 b(a)h(righ)o(t)f Fi(L)p Fn(-mo)q(dule.)33 b Fm(X)24 b Fg(has)d(a)g(left)f(zero)g Fn(if)f(there)h(is)g(a)h(unique) f(elemen)o(t)0 1568 y(0)14 b Fi(2)g Fm(X)21 b Fn(suc)o(h)16 b(that)665 1628 y Fi(8)p Fm(\025)d Fi(2)h(L)816 1608 y Fj(N)864 1628 y Fn(:)f(0)p Fm(\025)i Fn(=)f(0)p Fm(:)593 b Fn(\()p Fm(ii)p Fn(\))0 1716 y(c\))15 b(Let)h Fm(X)k Fn(b)q(e)15 b(an)h Fi(L)p Fn(-bimo)q(dule.)21 b Fm(X)e Fg(has)e(a)f(zero)f Fn(if)g(there)g(is)g(an)h(elemen)o(t)d(0)h Fi(2)g Fm(X)20 b Fn(suc)o(h)15 b(that)90 1786 y(\(i\))59 b Fi(8)p Fm(l)13 b Fi(2)h(L)g Fn(:)g Fm(l)q Fn(0)421 1768 y Fj(N)468 1786 y Fn(=)g(0)p Fm(;)77 1857 y Fn(\(ii\))58 b Fi(8)p Fm(\025)13 b Fi(2)h(L)352 1839 y Fj(N)400 1857 y Fn(:)f(0)p Fm(\025)i Fn(=)e(0)p Fm(:)0 1978 y Fw(3.2)18 b(Lemma:)43 b Fg(Let)17 b Fm(X)j Fg(b)q(e)d(an)f Fi(L)p Fg(-bimo)q(dule.)-4 2038 y(\(a\))25 b(If)16 b Fm(X)k Fg(has)d(a)g(righ)o(t)f(zero)g(then)g Fi(8)p Fm(\025)d Fi(2)h(L)803 2020 y Fj(N)851 2038 y Fn(:)f(0)p Fm(\025)i Fn(=)f(0)p Fm(:)-6 2098 y Fg(\(b\))24 b(If)16 b Fm(X)k Fg(has)d(a)g(left)e(zero)h(then)g Fi(8)p Fm(l)e Fi(2)g(L)g Fn(:)f Fm(l)q Fn(0)839 2080 y Fj(N)887 2098 y Fn(=)g(0)p Fm(:)0 2159 y Fu(Pr)o(oof:)33 b Fn(\(a\))23 b(By)f(asso)q(ciativit)o(y) g(w)o(e)h(ha)o(v)o(e)f Fi(8)p Fm(l)i Fi(2)h(L)p Fm(;)8 b(\025)25 b Fi(2)g(L)1185 2141 y Fj(N)1244 2159 y Fn(:)f Fm(l)q Fn(\(0)p Fm(\025)p Fn(\))1388 2141 y Fj(N)1447 2159 y Fn(=)h Fm(l)q Fn(\(0)1569 2141 y Fj(N)1602 2159 y Fm(\025)p Fn(\))h(=)0 2219 y(\()p Fm(l)q Fn(0)59 2201 y Fj(N)93 2219 y Fn(\))p Fm(\025)14 b Fn(=)g(0)p Fm(\025)j Fn(hence)e(b)o(y)h(uniqueness)g(0)p Fm(\025)e Fn(=)g(0.)0 2279 y(\(b\))k(By)f(asso)q(ciativit)o(y)h(w)o(e)g(ha)o(v)o(e)f Fi(8)p Fm(l)f Fi(2)h(L)p Fm(;)8 b(\025)18 b Fi(2)f(L)927 2261 y Fj(N)978 2279 y Fn(:)f(\()p Fm(l)q Fn(0)1067 2261 y Fj(N)1101 2279 y Fn(\))p Fm(\025)h Fn(=)g Fm(l)q Fn(\(0)1279 2261 y Fj(N)1313 2279 y Fm(\025)p Fn(\))g(=)g Fm(l)q Fn(0)1472 2261 y Fj(N)1524 2279 y Fn(hence)g(b)o(y)0 2339 y(uniqueness)f Fm(l)q Fn(0)286 2321 y Fj(N)333 2339 y Fn(=)e(0.)p 1689 2313 24 2 v 1689 2338 2 25 v 1711 2338 V 1689 2340 24 2 v 0 2449 a Fw(3.3)19 b(Corollary:)50 b Fg(Let)18 b Fm(X)j Fg(b)q(e)d(an)f Fi(L)p Fg(-bimo)q(dule.)24 b(If)17 b Fm(X)k Fg(has)d(a)g(righ)o(t)f(zero)g(or)g(a)h(left)e(zero,)0 2509 y(then)g Fm(X)21 b Fg(has)c(a)f(zero.)0 2619 y Fw(3.4)i(Lemma:)43 b Fg(The)17 b(follo)o(wing)f(are)g(equiv)m(alen)o(t)e(for)j(the)f Fi(L)p Fg(-bimo)q(dule)f Fi(L)p Fg(:)p eop %%Page: 11 11 11 10 bop -4 265 a Fg(\(a\))25 b Fi(L)17 b Fg(has)g(a)f(righ)o(t)g (zero.)-6 325 y(\(b\))24 b Fi(L)17 b Fg(has)g(a)f(left)g(zero.)-1 385 y(\(c\))24 b Fi(L)17 b Fg(has)g(a)f(zero.)0 467 y Fu(Pr)o(oof:)27 b Fn(\(c\))19 b Fi(\))g Fn(\(a\):)28 b(W)l(e)19 b(ha)o(v)o(e)f(to)i(sho)o(w)g(uniqueness)f(of)h(the)f(zero.) 30 b(Let)20 b(0)1482 449 y Fl(0)1513 467 y Fi(2)f(L)h Fn(suc)o(h)0 527 y(that)d Fi(8)p Fm(l)c Fi(2)h(L)g Fn(:)f Fm(l)q Fn(0)325 509 y Fl(0)p Fj(N)382 527 y Fn(=)h(0)458 509 y Fl(0)470 527 y Fn(,)i(then)g(0)635 509 y Fl(0)661 527 y Fn(=)e(00)761 509 y Fl(0)p Fj(N)819 527 y Fn(=)f(0)k(b)o(y)f (\(ii\).)0 609 y(\(c\))23 b Fi(\))h Fn(\(b\):)36 b(W)l(e)23 b(ha)o(v)o(e)g(to)h(sho)o(w)h(uniqueness)e(of)h(the)f(zero.)44 b(Let)23 b(0)1349 591 y Fl(0)1388 609 y Fi(2)k(L)d Fn(suc)o(h)f(that)0 669 y Fi(8)p Fm(\025)13 b Fi(2)h(L)151 651 y Fj(N)199 669 y Fn(:)f(0)250 651 y Fl(0)262 669 y Fm(\025)i Fn(=)e(0)380 651 y Fl(0)392 669 y Fn(,)j(then)g(0)557 651 y Fl(0)583 669 y Fn(=)e(0)659 651 y Fl(0)671 669 y Fn(0)695 651 y Fj(N)743 669 y Fn(=)g(0)i(b)o(y)g(\(i\).)0 751 y(The)g(con)o(v)o (erse)f(is)h(the)g(previous)g(corollary)l(.)p 1689 724 24 2 v 1689 749 2 25 v 1711 749 V 1689 751 24 2 v 0 854 a Fw(3.5)i(Lemm)o(a:)43 b Fg(Let)16 b Fi(L)g Fg(ha)o(v)o(e)f(a)h(zero)f Fn(0)h Fg(and)g(let)f Fm(X)20 b Fg(b)q(e)c(a)g(left)f Fi(L)p Fg(-mo)q(dule.)20 b(Then)c(w)o(e)f(ha)o(v)o(e)628 952 y Fi(8)p Fm(\030)r(;)8 b(\021)15 b Fi(2)f Fm(X)832 931 y Fj(N)880 952 y Fn(:)f(0)p Fm(\030)k Fn(=)d(0)p Fm(\021)r(:)0 1050 y Fu(Pr)o(oof:)20 b Fn(Let)d Fm(\030)r(;)8 b(\021)16 b Fi(2)e Fm(X)457 1032 y Fj(N)507 1050 y Fn(and)j Fm(j)g Fi(2)d Fm(N)5 b Fn(.)22 b(De\014ne)16 b Fm(\020)i Fi(2)c Fm(X)1047 1032 y Fj(N)1097 1050 y Fn(b)o(y)547 1176 y Fm(\020)t Fn(\()p Fm(i)p Fn(\))g(:=)706 1103 y Fh(\()760 1146 y Fm(\030)r Fn(\()p Fm(j)s Fn(\))p Fm(;)91 b Fn(for)16 b Fm(i)e Fn(=)g Fm(j;)760 1206 y(\021)r Fn(\()p Fm(i)p Fn(\))p Fm(;)94 b Fn(for)16 b Fm(i)e Fi(6)p Fn(=)g Fm(j:)0 1309 y Fn(Then)19 b(with)g Fm(k)i Fi(6)p Fn(=)e Fm(j)j Fn(w)o(e)d(get)h(0)p Fm(\030)i Fn(=)c(\(0)p Fm(\016)r Fn(\()p Fm(j)s Fn(\))800 1291 y Fj(N)834 1309 y Fn(\))p Fm(\030)j Fn(=)e(0\()p Fm(\016)r Fn(\()p Fm(j)s Fn(\))1080 1291 y Fj(N)1114 1309 y Fm(\030)r Fn(\))g(=)g(0)p Fm(\030)r Fn(\()p Fm(j)s Fn(\))1340 1291 y Fj(N)1394 1309 y Fn(=)f(0\()p Fm(\016)r Fn(\()p Fm(j)s Fn(\))1578 1291 y Fj(N)1612 1309 y Fm(\020)t Fn(\))h(=)0 1369 y(\(0)p Fm(\016)r Fn(\()p Fm(j)s Fn(\))128 1351 y Fj(N)161 1369 y Fn(\))p Fm(\020)f Fn(=)c(\(0)p Fm(\016)r Fn(\()p Fm(k)r Fn(\))403 1351 y Fj(N)436 1369 y Fn(\))p Fm(\020)k Fn(=)c(0\()p Fm(\016)r Fn(\()p Fm(k)r Fn(\))678 1351 y Fj(N)711 1369 y Fm(\020)t Fn(\))g(=)g(0)p Fm(\021)r Fn(\()p Fm(k)r Fn(\))936 1351 y Fj(N)984 1369 y Fn(=)g(0\()p Fm(\016)r Fn(\()p Fm(k)r Fn(\))1168 1351 y Fj(N)1201 1369 y Fm(\021)r Fn(\))g(=)f(\(0)p Fm(\016)r Fn(\()p Fm(k)r Fn(\))1443 1351 y Fj(N)1477 1369 y Fn(\))p Fm(\021)j Fn(=)d(0)p Fm(\021)r Fn(.)p 1689 1342 V 1689 1367 2 25 v 1711 1367 V 1689 1369 24 2 v 0 1472 a Fw(3.6)18 b(Corollary:)46 b Fg(Let)16 b Fi(L)h Fg(ha)o(v)o(e)e(a)i(zero)f Fn(0)p Fg(.)-4 1533 y(\(a\))25 b(If)e Fm(X)k Fg(is)d(a)f(non-empt)o(y)f(left)g Fi(L)p Fg(-mo)q(dule)h(then)g Fm(X)28 b Fg(has)c(a)g(righ)o(t)f(zero)g Fn(0)1470 1514 y Fl(0)1507 1533 y Fi(2)j Fm(X)i Fg(and)83 1593 y Fi(8)p Fm(\030)15 b Fi(2)f Fm(X)239 1575 y Fj(N)287 1593 y Fn(:)g(0)p Fm(\030)i Fn(=)e(0)452 1575 y Fl(0)464 1593 y Fg(.)-6 1653 y(\(b\))24 b(If)d Fm(X)26 b Fg(is)21 b(a)h(non-empt)o(y)e Fi(L)p Fg(-bimo)q(dule)g(and)i(has)g(a)g(left)f (zero)g Fn(0)1281 1635 y Fl(0)1314 1653 y Fg(then)g Fn(0)1454 1635 y Fl(0)1488 1653 y Fg(is)g(also)h(the)83 1713 y(\(unique\))15 b(righ)o(t)h(zero)g(of)h Fm(X)k Fg(and)700 1811 y Fi(8)p Fm(\030)16 b Fi(2)e Fm(X)857 1791 y Fj(N)905 1811 y Fn(:)f(0)p Fm(\030)k Fn(=)d(0)1070 1791 y Fl(0)1082 1811 y Fm(;)696 1909 y Fi(8)p Fm(x)f Fi(2)h Fm(X)k Fn(:)13 b Fm(x)p Fn(0)950 1889 y Fj(N)998 1909 y Fn(=)h(0)1074 1889 y Fl(0)1086 1909 y Fm(:)-1 1991 y Fg(\(c\))24 b(If)16 b Fm(f)j Fn(:)13 b Fm(X)19 b Fi(\000)-9 b(!)14 b Fm(Y)27 b Fg(is)16 b(a)h(homomorphism)c (of)k(left)e Fi(L)p Fg(-mo)q(dules,)h(then)g Fm(f)5 b Fn(\(0\))14 b(=)g(0)p Fg(.)0 2051 y Fu(Pr)o(oof:)21 b Fn(\(a\))d(F)l(or)e(some)g Fm(\030)h Fi(2)e Fm(X)22 b Fn(de\014ne)16 b(0)797 2033 y Fl(0)824 2051 y Fn(:=)e(0)p Fm(\030)k Fi(2)d Fm(X)t Fn(.)23 b(Then)17 b Fm(l)q Fn(0)1250 2033 y Fl(0)p Fj(N)1307 2051 y Fn(=)e Fm(l)q Fn(\(0)p Fm(\030)r Fn(\))1461 2033 y Fj(N)1510 2051 y Fn(=)g Fm(l)q Fn(0)1603 2033 y Fj(N)1636 2051 y Fm(\030)j Fn(=)0 2112 y(0)p Fm(\030)f Fn(=)d(0)138 2094 y Fl(0)167 2112 y Fn(and)j (uniqueness)f(is)g(obtained)h(as)g(follo)o(ws.)22 b(Let)16 b(0)1109 2094 y Fl(00)1145 2112 y Fi(2)f Fm(X)20 b Fn(b)q(e)d(a)g(righ) o(t)f(zero.)22 b(Then)0 2172 y(0)24 2154 y Fl(00)59 2172 y Fn(=)14 b(0)e Fi(\001)f Fn(0)196 2154 y Fl(00)p Fj(N)263 2172 y Fn(=)i(0)f Fi(\001)f Fn(0)399 2154 y Fl(0)p Fj(N)456 2172 y Fn(=)j(0)532 2154 y Fl(0)544 2172 y Fn(.)0 2232 y(\(b\))i Fm(X)21 b Fn(has)16 b(a)h(righ)o(t)f(zero)f(0)513 2214 y Fl(00)551 2232 y Fn(b)o(y)h(part)g(\(a\))h(whic)o(h)e(is)h(a)g (zero)g(b)o(y)g(3.3)g(hence)g(b)o(y)f(uniqueness)0 2292 y(of)k(the)f(left)g(zero)g(w)o(e)g(get)g(0)518 2274 y Fl(0)548 2292 y Fn(=)g(0)628 2274 y Fl(00)667 2292 y Fn(=)g(0)p Fm(\030)r Fn(.)29 b(Th)o(us)18 b Fm(x)p Fn(0)990 2274 y Fj(N)1042 2292 y Fn(=)f Fm(x)p Fn(\(0)1168 2274 y Fj(N)1202 2292 y Fm(\025)p Fn(\))h(=)g(\()p Fm(x)p Fn(0)1394 2274 y Fj(N)1427 2292 y Fn(\))p Fm(\025)h Fn(implies)d(b)o(y) 0 2352 y(uniqueness)g Fm(x)p Fn(0)298 2334 y Fj(N)345 2352 y Fn(=)e(0)421 2334 y Fl(0)433 2352 y Fn(.)0 2413 y(\(c\))i Fm(f)5 b Fn(\(0\))14 b(=)g Fm(f)5 b Fn(\(0)p Fm(\030)r Fn(\))15 b(=)f(0)p Fm(f)467 2395 y Fj(N)502 2413 y Fn(\()p Fm(\030)r Fn(\))g(=)g(0.)p 1689 2386 V 1689 2411 2 25 v 1711 2411 V 1689 2413 24 2 v 0 2516 a(A)j(non-empt)o(y)e(righ)o(t)i Fi(L)p Fn(-mo)q(dule,)f(ho)o(w)o(ev)o (er,)g(will)g(in)h(general)g(not)g(ha)o(v)o(e)f(a)i(left)e(zero.)24 b(The)0 2576 y(elemen)o(t)14 b Fm(x)p Fn(0)233 2558 y Fj(N)282 2576 y Fi(2)h Fm(X)22 b Fn(satis\014es)17 b(\(ii\),)f(but)h (it)g(will)f(not)h(b)q(e)h(unique,)e(e.g.)23 b Fm(X)d Fn(=)15 b Fi(f)p Fn(0)p Fm(;)8 b Fn(0)1534 2558 y Fl(0)1546 2576 y Fi(g)17 b Fn(where)0 2636 y(b)q(oth)g(elemen)o(ts)d(satisfy)i (\(ii\).)p eop %%Page: 12 12 12 11 bop 233 265 a Fo(4.)52 b(\(Semi-\)Additive)22 b(Theories)i(and)e (Scalars)0 367 y Fn(The)15 b(elemen)o(ts)d(of)i Fi(L)h Fn(can)g(b)q(e)g(considered)f(as)i(op)q(erators)g(whic)o(h)e(pro)q (duce)h(allo)o(w)o(able)e(linear)0 428 y(com)o(binations)18 b(of)i(elemen)o(ts)d(in)i(a)h(left)e Fi(L)p Fn(-mo)q(dule.)31 b(This)19 b(is)h(the)f(essence)g(of)h(all)f(the)g(ex-)0 488 y(amples)14 b(of)i(con)o(v)o(exit)o(y)d(theories.)21 b(In)15 b(some)f(sense)i(the)f(m)o(ultiplic)o(ation)e(of)j(elemen)o(ts) d(in)i(the)0 548 y(mo)q(dule)c(b)o(y)h(certain)g(elemen)o(ts)e(\(whic)o (h)i(w)o(e)g(will)g(call)f(scalars)i(and)h(whic)o(h)d(will)h(b)q(e)h (discussed)0 608 y(later)i(on\))h(and)g(the)f(addition)g(in)g(the)g(mo) q(dule)f(are)i(hidden)f(among)g(these)g(op)q(erators.)22 b(Ob-)0 668 y(serv)o(e,)f(ho)o(w)o(ev)o(er,)f(that)i(the)f(addition)g (prop)q(er)h(is)e(v)o(ery)g(often)h(not)h(allo)o(w)o(able)e(in)h(con)o (v)o(ex)0 728 y(sets.)0 789 y(In)d(some)e(cases,)i(ho)o(w)o(ev)o(er,)f (there)g(will)g(b)q(e)h(a)g(structure)g(of)g(an)g(addition)g(on)h(the)e (mo)q(dule.)0 849 y(In)i(section)g(3)h(w)o(e)f(sa)o(w)h(that)f(a)h (zero)f(is)g(carried)g(o)o(v)o(er)f(from)h Fi(L)g Fn(to)h(mo)q(dules)f (o)o(v)o(er)f Fi(L)p Fn(.)31 b(W)l(e)0 909 y(will)14 b(study)h(no)o(w)h(ho)o(w)g(m)o(uc)o(h)d(of)i(an)h(additiv)o(e)e (structure)h(will)f(b)q(e)h(transferred)g(in)g(a)h(similar)0 969 y(w)o(a)o(y)l(.)k(Throughout)c(this)e(section)f(w)o(e)h(shall)g (assume)f(that)i Fi(L)f Fn(is)g(a)h(unital)e(left)g(linear)h(theory)0 1029 y(whic)o(h)i(has)h(a)f(zero:)21 b(0)14 b Fi(2)g(L)p Fn(.)0 1090 y(F)l(or)f Fm(i;)8 b(j)16 b Fi(2)e Fm(N)19 b Fn(with)13 b Fm(i)g Fi(6)p Fn(=)h Fm(j)i Fn(and)d(for)h(elemen)o(ts)c Fm(x;)e(y)14 b Fn(in)f(a)g(left)f Fi(L)p Fn(-mo)q(dule)h Fm(X)k Fn(let)12 b Fm(")p Fn(\()p Fm(x;)c(y)r Fi(j)p Fm(i;)g(j)s Fn(\))k Fi(2)0 1150 y Fm(X)44 1132 y Fj(N)94 1150 y Fn(b)q(e)17 b(giv)o(en)e(b)o(y)466 1266 y Fm(")p Fn(\()p Fm(x;)8 b(y)r Fi(j)p Fm(i;)g(j)s Fn(\)\()p Fm(k)r Fn(\))k(:=)822 1167 y Fh(8)822 1204 y(>)822 1217 y(<)822 1291 y(>)822 1304 y(:)879 1206 y Fn(0)p Fm(;)94 b Fn(if)16 b Fm(k)f Fi(6)p Fn(=)f Fm(i;)8 b(j;)879 1266 y(x;)90 b Fn(if)16 b Fm(k)f Fn(=)f Fm(i;)879 1326 y(y)r(;)92 b Fn(if)16 b Fm(k)f Fn(=)f Fm(j:)0 1447 y Fw(4.1)20 b(De\014nition:)26 b Fi(L)18 b Fn(is)g Fg(semi-additiv)o(e)p Fn(,)c(if)j(there)g(are)h Fm(i;)8 b(j)18 b Fi(2)f Fm(N)5 b Fn(,)17 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Fn(+)g Fm(y)r Fg(.)23 b(In)17 b(particular)g(the)f(addition)h(do)q(es)h(not)f(dep)q(end)0 2636 y(on)g(the)f(c)o(hoice)f(of)h Fm(i;)8 b(j)17 b Fi(2)d Fm(N)21 b Fg(with)16 b Fm(i)e Fi(6)p Fn(=)f Fm(j)s Fg(.)p eop %%Page: 13 13 13 12 bop 0 265 a Fu(Pr)o(oof:)20 b Fn(Since)296 375 y Fm(")p Fn(\()p Fm(\016)r Fn(\()p Fm(i)398 354 y Fl(0)409 375 y Fn(\))p Fm(;)8 b(\016)r Fn(\()p Fm(j)516 354 y Fl(0)527 375 y Fn(\))p Fi(j)p Fm(i;)g(j)s Fn(\))p Fm(")p Fn(\()p Fm(\016)r Fn(\()p Fm(i)743 354 y Fl(0)753 375 y Fn(\))p Fm(;)g(\016)r Fn(\()p Fm(j)860 354 y Fl(0)870 375 y Fn(\))p Fi(j)p Fm(i)920 354 y Fl(0)932 375 y Fm(;)g(j)977 354 y Fl(0)988 375 y Fn(\))14 b(=)g Fm(")p Fn(\()p Fm(\016)r Fn(\()p Fm(i)1175 354 y Fl(0)1185 375 y Fn(\))p Fm(;)8 b(\016)r Fn(\()p Fm(j)1292 354 y Fl(0)1303 375 y Fn(\))p Fi(j)p Fm(i;)g(j)s Fn(\))0 485 y(w)o(e)16 b(ha)o(v)o(e)83 586 y Fm(a)109 566 y Fl(0)120 586 y Fm(")p Fn(\()p Fm(\016)r Fn(\()p Fm(i)222 566 y Fl(0)232 586 y Fn(\))p Fm(;)8 b(\016)r Fn(\()p Fm(j)339 566 y 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Fi(j)p Fm(i;)g(j)s Fn(\))570 804 y(=)14 b Fm(a)648 784 y Fl(0)0 979 y Fn(and)138 1089 y Fm(a)164 1068 y Fl(0)176 1089 y Fm(")p Fn(\()p Fm(x;)8 b(y)r Fi(j)p Fm(i)325 1068 y Fl(0)335 1089 y Fm(;)g(j)380 1068 y Fl(0)391 1089 y Fn(\))14 b(=)g Fm(a")p Fn(\()p Fm(\016)r Fn(\()p Fm(i)604 1068 y Fl(0)614 1089 y Fn(\))p Fm(;)8 b(\016)r Fn(\()p Fm(j)721 1068 y Fl(0)732 1089 y Fn(\))p Fi(j)p Fm(i;)g(j)s Fn(\))p Fm(")p Fn(\()p Fm(x;)g(y)r Fi(j)p Fm(i)995 1068 y Fl(0)1004 1089 y Fm(;)g(j)1049 1068 y Fl(0)1060 1089 y Fn(\))14 b(=)g Fm(a")p Fn(\()p Fm(x;)8 b(y)r Fi(j)p Fm(i;)g(j)s Fn(\))k(=)h Fm(x)e Fn(+)g Fm(y)r(:)p 1689 1172 24 2 v 1689 1197 2 25 v 1711 1197 V 1689 1199 24 2 v 0 1284 a Fn(F)l(rom)k(no)o(w)i(on)f(addition)h(is)f(written)g(b) o(y)f(means)h(of)g(the)g(initially)e(c)o(hosen)i Fm(i)g Fn(and)h Fm(j)s Fn(.)0 1394 y Fw(4.4)h(De\014nition:)24 b Fn(An)16 b(addition)g Fm(a)g Fn(in)g Fi(L)g Fn(is)g(called)0 1454 y Fg(asso)q(ciativ)o(e)21 b Fn(if)g(there)f(are)i Fm(i;)8 b(j;)g(k)24 b Fi(2)e Fm(N)5 b Fn(,)22 b(m)o(utually)d (distinct,)i(suc)o(h)g(that)h Fm(\016)r Fn(\()p Fm(i)p Fn(\))13 b(+)i(\()p Fm(\016)r Fn(\()p Fm(j)s Fn(\))f(+)0 1514 y Fm(\016)r Fn(\()p Fm(k)r Fn(\)\))f(=)h(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))c(+)h Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))g(+)g Fm(\016)r Fn(\()p Fm(k)r Fn(\))k(holds,)0 1574 y Fg(comm)o(utativ)n(e)e Fn(if)j Fm(\016)r Fn(\()p Fm(i)p Fn(\))10 b(+)h Fm(\016)r Fn(\()p Fm(j)s Fn(\))i(=)h Fm(\016)r Fn(\()p Fm(j)s Fn(\))d(+)g Fm(\016)r Fn(\()p Fm(i)p Fn(\),)0 1635 y Fg(with)16 b(zero)g(elemen)o (t)d Fn(if)j Fm(\016)r Fn(\()p Fm(i)p Fn(\))10 b(+)h(0)j(=)g(0)e(+)f Fm(\016)r Fn(\()p Fm(i)p Fn(\))h(=)i Fm(\016)r Fn(\()p Fm(i)p Fn(\),)0 1695 y Fg(with)h(in)o(v)o(erses)f Fn(if)g(there)h (exists)g(\()p Fi(\000)p Fm(\016)r Fn(\()p Fm(i)p Fn(\)\))d Fi(2)i(L)i Fn(with)f Fm(\016)r Fn(\()p Fm(i)p Fn(\))8 b(+)h(\()p Fi(\000)p Fm(\016)r Fn(\()p Fm(i)p Fn(\)\))k(=)g(\()p Fi(\000)p Fm(\016)r Fn(\()p Fm(i)p Fn(\)\))8 b(+)h Fm(\016)r Fn(\()p Fm(i)p Fn(\))k(=)h(0.)0 1805 y Fw(4.5)21 b(Prop)r(osition:)57 b Fg(Let)19 b Fi(L)g Fg(b)q(e)g(a)h(unital)e(left)g(linear)g(theory)h (with)g(an)g(addition)g Fm(a)p Fg(.)29 b(If)0 1865 y Fm(a)17 b Fg(is)g(asso)q(ciativ)o(e,)g(comm)o(utativ)n(e,)d(with)j (zero)g(elemen)o(t,)d(or)j(with)g(in)o(v)o(erses)f(then)h(so)h(is)f (the)0 1925 y(addition)f(of)h(an)g Fi(L)p Fg(-mo)q(dule)e Fm(X)t Fg(.)0 2010 y Fu(Pr)o(oof:)20 b Fn(F)l(or)d Fm(x)c Fi(2)h Fm(X)21 b Fn(denote)16 b(b)o(y)g Fm(")p Fn(\()p Fm(x)p Fi(j)p Fm(i)p Fn(\))d Fi(2)h Fm(X)879 1992 y Fj(N)929 2010 y Fn(the)i(map)545 2151 y Fm(")p Fn(\()p Fm(x)p Fi(j)p Fm(i)p Fn(\)\()p Fm(t)p Fn(\))c(:=)798 2078 y Fh(\()853 2121 y Fn(0)p Fm(;)93 b Fn(if)16 b Fm(t)d Fi(6)p Fn(=)h Fm(i;)853 2181 y(x;)89 b Fn(if)16 b Fm(t)d Fn(=)h Fm(i:)0 2296 y Fn(F)l(or)j Fm(x;)8 b(y)r(;)g(z)15 b Fi(2)g Fm(X)21 b Fn(and)c(m)o(utually)e(distinct)g Fm(i;)8 b(j;)g(k)17 b Fi(2)d Fm(N)22 b Fn(denote)17 b(b)o(y)f Fm(")p Fn(\()p Fm(x;)8 b(y)r(;)g(z)r Fi(j)p Fm(i;)g(j;)g(k)r Fn(\))13 b Fi(2)i Fm(X)1679 2278 y Fj(N)0 2356 y Fn(the)h(map)405 2503 y Fm(")p Fn(\()p Fm(x;)8 b(y)r(;)g(z)r Fi(j)p Fm(i;)g(j;)g(k)r Fn(\)\()p Fm(t)p Fn(\))j(:=)844 2378 y Fh(8)844 2416 y(>)844 2428 y(>)844 2441 y(>)844 2453 y(<)844 2528 y(>)844 2540 y(>)844 2553 y(>)844 2565 y(:)901 2412 y Fn(0)p Fm(;)94 b Fn(if)16 b Fm(t)d Fi(6)p Fn(=)h Fm(i;)8 b(j;)g(k)r(;)901 2472 y(x;)90 b Fn(if)16 b Fm(t)d Fn(=)h Fm(i;)901 2532 y(y)r(;)92 b Fn(if)16 b Fm(t)d Fn(=)h Fm(j;)901 2593 y(z)r(;)93 b Fn(if)16 b Fm(t)d Fn(=)h Fm(k)r(:)p eop %%Page: 14 14 14 13 bop 0 265 a Fn(Then)6 366 y(\()p Fm(x)11 b Fn(+)g Fm(y)r Fn(\))f(+)h Fm(z)44 b Fn(=)14 b(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))c(+)h Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\()p Fm(x)f Fn(+)h Fm(y)r(;)d(z)r Fi(j)p Fm(i;)g(j)s Fn(\))284 439 y(=)14 b(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))c(+)h Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))e(+)i Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\()p Fm(x;)d(y)r(;)g(z)r Fi(j)p Fm(i;)g(j;)g(k)r Fn(\))p Fm(;)g(\016)r Fn(\()p Fm(k)r Fn(\))p Fm(")p Fn(\()o Fm(x;)g(y)r(;)f(z)r Fi(j)p Fm(i;)g(j;)h(k)r Fn(\))p Fi(j)p Fm(i;)g(j)s Fn(\))284 512 y(=)14 b(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))c(+)h Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))e(+)i Fm(\016)r Fn(\()p Fm(j)s Fn(\))p Fm(;)d(\016)r Fn(\()p Fm(k)r Fn(\))p Fi(j)p Fm(i;)g(j)s Fn(\))p Fm(")p Fn(\()p Fm(x;)g(y)r(;)f(z)r Fi(j)p Fm(i;)h(j;)g(k)r Fn(\))284 584 y(=)14 b(\(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))c(+)h Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))f(+)h Fm(\016)r Fn(\()p Fm(k)r Fn(\)\))p Fm(")p Fn(\()p Fm(x;)d(y)r(;)g(z)r Fi(j)p Fm(i;)g(j;)g(k)r Fn(\))0 759 y(and)14 b(similarly)d Fm(x)5 b Fn(+)g(\()p Fm(y)i Fn(+)e Fm(z)r Fn(\))13 b(=)h(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))5 b(+)g(\()p Fm(\016)r Fn(\()p Fm(j)s Fn(\))g(+)g Fm(\016)r Fn(\()p Fm(k)r Fn(\)\)\))p Fm(")p Fn(\()p Fm(x;)j(y)r(;)g(z)r Fi(j)p Fm(i)o(;)g(j;)g(k)r Fn(\).)18 b(Th)o(us)13 b(asso)q(ciativit)o(y)0 819 y(is)j(inherited)f (b)o(y)h(the)g(mo)q(dules.)0 879 y(F)l(urthermore)d Fm(x)7 b Fn(+)g Fm(y)16 b Fn(=)e(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))7 b(+)g Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\()p Fm(x;)h(y)r Fi(j)p Fm(i;)g(j)s Fn(\))j(and)16 b Fm(y)9 b Fn(+)e Fm(x)14 b Fn(=)f(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))7 b(+)g Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\()p Fm(y)r(;)h(x)p Fi(j)p Fm(i;)g(j)s Fn(\))k(=)0 939 y(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))6 b(+)g Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\()p Fm(\016)r Fn(\()p Fm(j)s Fn(\))p Fm(;)i(\016)q Fn(\()p Fm(i)p Fn(\))p Fi(j)p Fm(i)o(;)g(j)s Fn(\))p Fm(")p Fn(\()p Fm(x;)f(y)r Fi(j)p Fm(i;)g(j)s Fn(\))k(=)j(\()p Fm(\016)r Fn(\()p Fm(j)s Fn(\))6 b(+)g Fm(\016)r Fn(\()p Fm(i)p Fn(\)\))p Fm(")p Fn(\()p Fm(x;)i(y)r Fi(j)p Fm(i)o(;)g(j)s Fn(\))j(sho)o(w)j(that)g (comm)o(u-)0 999 y(tativit)o(y)h(is)h(inherited.)0 1060 y(F)l(or)26 b(the)g(other)g(t)o(w)o(o)g(la)o(ws)h(w)o(e)e(ha)o(v)o(e)h Fm(x)17 b Fn(+)h(0)31 b(=)g(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))16 b(+)i Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\()p Fm(x;)8 b Fn(0)p Fi(j)p Fm(i;)g(j)s Fn(\))29 b(=)i(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))17 b(+)0 1120 y Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))p Fm(;)8 b Fn(0)p Fi(j)p Fm(i;)g(j)s Fn(\))p Fm(")p Fn(\()p Fm(x;)g Fn(0)p Fi(j)p Fm(i;)f(j)s Fn(\))20 b(=)i(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))13 b(+)h(0\))p Fm(")p Fn(\()p Fm(x;)8 b Fn(0)p Fi(j)p Fm(i;)g(j)s Fn(\))21 b(and)g Fm(x)h Fn(=)g Fm(\016)r Fn(\()p Fm(i)p Fn(\))p Fm(")p Fn(\()p Fm(x;)8 b Fn(0)p Fi(j)p Fm(i;)g(j)s Fn(\))19 b(resp.)0 1180 y Fm(x)11 b Fn(+)g(\()p Fi(\000)p Fm(x)p Fn(\))i(=)h(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))d(+)g Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\()p Fm(x;)d Fi(\000)p Fm(x)p Fi(j)p Fm(i;)g(j)s Fn(\))j(=)j(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))c(+)h Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))p Fm(;)d Fi(\000)p Fm(\016)r Fn(\()p Fm(i)p Fn(\))p Fi(j)p Fm(i;)f(j)s Fn(\))p Fm(")p Fn(\()p Fm(x;)h Fn(0)p Fi(j)p Fm(i;)f(j)s Fn(\))12 b(=)0 1240 y(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))e(+)h(\()p Fi(\000)p Fm(\016)r Fn(\()p Fm(i)p Fn(\)\)\))p Fm(")p Fn(\()p Fm(x;)d Fn(0)p Fi(j)p Fm(i;)g(j)s Fn(\))14 b(and)j(0)d(=)g(0)p Fm(")p Fn(\()p Fm(x;)8 b Fn(0)p Fi(j)p Fm(i;)g(j)s Fn(\).)p 1689 1214 24 2 v 1689 1239 2 25 v 1711 1239 V 1689 1241 24 2 v 0 1350 a Fw(4.6)27 b(Example:)40 b Fn(Let)24 b Fi(L)h Fn(b)q(e)f(a)h(unital)e(sequen)o(tial)g(left)g(linear)h(theory)f(as)i (de\014ned)f(in)0 1410 y(example)9 b(1.4.a\))j(with)f(0)j(=)g(\(0)p Fm(;)8 b Fn(0)p Fm(;)g(:)g(:)g(:)o Fn(\))14 b Fi(2)g(L)p Fn(.)20 b(An)o(y)11 b(elemen)n(t)e(of)i(the)g(form)g(\()p Fm(a)1389 1417 y Fk(1)1408 1410 y Fm(;)d(a)1456 1417 y Fk(2)1475 1410 y Fm(;)g Fn(0)p Fm(;)g Fn(0)p Fm(;)g(:)g(:)g(:)p Fn(\))13 b Fi(2)0 1471 y(L)18 b Fn(de\014nes)f(an)g(addition.)25 b(The)17 b(addition)g(is)h(comm)o(utativ)n(e)c(if)j(and)g(only)g(if)g Fm(a)1454 1478 y Fk(1)1489 1471 y Fn(=)e Fm(a)1568 1478 y Fk(2)1588 1471 y Fn(.)24 b(It)17 b(is)0 1531 y(asso)q(ciativ)o(e)h (if)g(and)i(only)e(if)g Fm(a)572 1513 y Fk(2)572 1543 y Fj(i)609 1531 y Fn(=)g Fm(a)691 1538 y Fj(i)723 1531 y Fn(and)h Fm(a)846 1538 y Fk(1)866 1531 y Fm(a)892 1538 y Fk(2)929 1531 y Fn(=)f Fm(a)1011 1538 y Fk(2)1030 1531 y Fm(a)1056 1538 y Fk(1)1075 1531 y Fn(.)29 b(The)18 b(addition)h(is)f(an)h(addition)0 1591 y(with)e(zero)h(elemen)o(t)c(if) j(and)i(only)e(if)g Fm(a)721 1598 y Fk(1)757 1591 y Fn(=)f Fm(a)837 1598 y Fk(2)872 1591 y Fn(=)g(1.)26 b(If)17 b Fi(L)h Fn(is)g(de\014ned)f(o)o(v)o(er)g(a)h Ff(ring)g Fm(R)g Fn(then)0 1651 y(the)e(addition)g(is)g(an)h(addition)g(with)f (in)o(v)o(erses)e(if)i(and)h(only)f(if)g(\()p Fi(\000)p Fn(1)p Fm(;)8 b Fn(0)p Fm(;)g Fn(0)p Fm(;)g(:)g(:)g(:)o Fn(\))14 b Fi(2)g(L)p Fn(.)0 1761 y(No)o(w)h(w)o(e)g(study)h(somewhat)f (more)g(in)g(detail)g(sequences)f(in)h Fi(L)1165 1743 y Fj(N)1215 1761 y Fn(with)g(en)o(tries)g Fm(\016)r Fn(\()p Fm(i)p Fn(\),)f(where)0 1821 y Fm(\016)k Fn(is)e(the)g(unit)g(in)g Fi(L)p Fn(.)0 1931 y Fw(4.7)22 b(De\014nition:)30 b Fn(Let)19 b Fm(\033)h Fn(:)e Fm(N)24 b Fi(\000)-8 b(!)18 b Fm(N)25 b Fn(b)q(e)19 b(a)g(map.)29 b(Then)20 b(w)o(e)e(de\014ne)h Fm(\016)1401 1913 y Fj(\033)1443 1931 y Fn(:=)f Fm(\016)c Fi(\016)f Fm(\033)r Fn(,)19 b(i.e.)0 1992 y Fm(\016)24 1974 y Fj(\033)47 1992 y Fn(\()p Fm(i)p Fn(\))f(=)g Fm(\016)r Fn(\()p Fm(\033)r Fn(\()p Fm(i)p Fn(\)\).)27 b(A)19 b(map)f Fm(\033)h Fn(:)f Fm(N)24 b Fi(\000)-9 b(!)18 b Fm(N)24 b Fn(is)19 b(called)f(an)h(\()p Fm(i;)8 b(j)s Fn(\))p Fg(-map)18 b Fn(if)g Fm(\033)1401 1974 y Fl(\000)p Fk(1)1447 1992 y Fn(\()p Fi(f)p Fm(j)s Fi(g)p Fn(\))g(=)h Fi(f)p Fm(i)p Fi(g)p Fn(.)0 2052 y(W)l(e)d(call)f Fm(b)f Fi(2)g(L)j Fn(an)f Fm(i)p Fn(-)p Fg(scalar)h Fn(if)470 2162 y Fi(8)p Fm(j)f Fi(2)e Fm(N)21 b Fi(8)p Fn(\()p Fm(i;)8 b(j)s Fn(\)-maps)14 b Fm(\033)o(;)8 b(\034)19 b Fn(:)14 b Fm(b\016)1075 2141 y Fj(\033)1111 2162 y Fn(=)g Fm(b\016)1208 2141 y Fj(\034)1229 2162 y Fm(:)0 2322 y Fn(A)e(little)f(calculation)h(for)g (a)h(sequen)o(tial)e(left)h(linear)g(theory)g Fi(L)h Fn(with)f(zero)h(o)o(v)o(er)e(the)h(semiring)0 2382 y Fm(R)17 b Fn(sho)o(ws)g(that)g(the)f Fm(i)p Fn(-scalars)g(are)g(of)h (the)f(form)f(\(0)p Fm(;)8 b(:)g(:)g(:)g(;)g Fn(0)p Fm(;)g(r)1131 2389 y Fj(i)1145 2382 y Fm(;)g Fn(0)p Fm(;)g(:)g(:)g(:)o Fn(\).)0 2492 y Fw(4.8)18 b(Lemma:)40 b Fg(\(a\))24 b(F)l(or)17 b Fm(\033)o(;)8 b(\034)19 b Fn(:)13 b Fm(N)19 b Fi(\000)-8 b(!)14 b Fm(N)21 b Fg(w)o(e)16 b(ha)o(v)o(e)f Fm(\016)1040 2474 y Fj(\033)q(\034)1097 2492 y Fn(=)e Fm(\016)1172 2474 y Fj(\034)1193 2492 y Fm(\016)1217 2474 y Fj(\033)1254 2492 y Fi(2)h(L)p Fg(.)-6 2552 y(\(b\))24 b(If)16 b Fm(\033)i Fg(is)e(an)g Fn(\()p Fm(i;)8 b(j)s Fn(\))p Fg(-map)15 b(and)i Fm(\034)22 b Fg(is)16 b(a)h Fn(\()p Fm(j;)8 b(k)r Fn(\))p Fg(-map)16 b(then)g Fm(\034)6 b(\033)17 b Fg(is)f(an)h Fn(\()p Fm(i;)8 b(k)r Fn(\))p Fg(-map.)-1 2612 y(\(c\))24 b Fm(\016)r Fn(\()p Fm(i)p Fn(\))15 b Fg(is)h(an)h Fm(i)p Fg(-scalar)f(for)h(all)f Fm(i)d Fi(2)h Fm(N)5 b Fg(.)p eop %%Page: 15 15 15 14 bop -6 265 a Fg(\(d\))24 b(Let)16 b Fm(l)f Fi(2)f(L)p Fg(.)21 b(Then)c Fm(l)q(\016)r Fn(\()p Fm(i)p Fn(\))539 247 y Fj(N)587 265 y Fg(is)f(an)h Fm(i)p Fg(-scalar.)-1 325 y(\(e\))24 b(If)16 b Fm(b)d Fi(2)h(L)j Fg(is)f(an)h Fm(i)p Fg(-scalar)f(and)h Fm(\033)h Fg(is)e(an)h Fn(\()p Fm(i;)8 b(j)s Fn(\))p Fg(-map)15 b(then)h Fm(b\016)1190 307 y Fj(\033)1229 325 y Fg(is)g(a)g Fm(j)s Fg(-scalar.)2 385 y(\(f)s(\))25 b(If)15 b Fm(b)h Fg(is)g(an)g Fm(i)p Fg(-scalar)g(and)h Fm(\030)r(;)8 b(\021)16 b Fi(2)e Fm(X)726 367 y Fj(N)776 385 y Fg(with)i Fm(\030)r Fn(\()p Fm(i)p Fn(\))e(=)g Fm(\021)r Fn(\()p Fm(i)p Fn(\))p Fg(,)h(then)h Fm(b\030)g Fn(=)e Fm(b\021)r Fg(.)20 b(In)c(particular)83 454 y Fm(b\030)j Fg(dep)q(ends)d(only)g(on)h(the)f Fm(i)608 436 y Fg(th)672 454 y(comp)q(onen)o(t)f(of)i Fm(\030)r Fg(.)0 539 y Fu(Pr)o(oof:)j Fn(Straigh)o(tforw)o(ard)c(substitutions)f (of)h(the)f(de\014nitions)g(and)h(simple)c(calculations.)p 1689 573 24 2 v 1689 598 2 25 v 1711 598 V 1689 600 24 2 v 0 710 a Fw(4.9)18 b(De\014nition:)24 b Fn(F)l(or)16 b Fm(x)e Fi(2)g Fm(X)20 b Fn(and)d(an)g Fm(i)p Fn(-scalar)f Fm(b)e Fi(2)g(L)i Fn(w)o(e)g(de\014ne)g Fm(bx)d Fn(:=)h Fm(b")p Fn(\()p Fm(x)p Fi(j)p Fm(i)p Fn(\).)0 820 y(No)o(w)19 b(w)o(e)h(w)o(an)o(t)f(to)h(study)g(elemen)o(ts)d(in)i Fm(X)820 802 y Fj(N)852 790 y Fd(a)893 820 y Fn(and)h(in)f Fm(X)1095 802 y Fj(N)1127 790 y Fd(b)1165 820 y Fn(for)g(a)i(set)e Fm(X)t Fn(,)h(where)g Fm(a)f Fn(and)0 880 y Fm(b)i Fn(are)g (non-negativ)o(e)f(in)o(tegers.)35 b(Let)21 b Fm(\037)g Fn(:)g Fm(N)856 862 y Fj(a)899 880 y Fi(\000)-8 b(!)21 b Fm(N)1045 862 y Fj(b)1084 880 y Fn(b)q(e)g(a)g(map.)34 b(F)l(or)21 b(\004)g Fi(2)h Fm(X)1586 862 y Fj(N)1618 850 y Fd(b)1657 880 y Fn(w)o(e)0 940 y(de\014ne)c(\004)176 922 y Fj(\037)218 940 y Fn(:=)f(\004)12 b Fi(\016)h Fm(\037)18 b Fi(2)g Fm(X)514 922 y Fj(N)546 910 y Fd(a)567 940 y Fn(.)28 b(Then)19 b(w)o(e)f(ha)o(v)o(e)g(\004)961 922 y Fj(#\037)1024 940 y Fn(=)f(\(\004)1131 922 y Fj(#)1154 940 y Fn(\))1173 922 y Fj(\037)1216 940 y Fn(for)i Fm(\037)e Fn(:)h Fm(N)1417 922 y Fj(a)1456 940 y Fi(\000)-9 b(!)18 b Fm(N)1598 922 y Fj(b)1634 940 y Fn(and)0 1000 y Fm(#)c Fn(:)f Fm(N)114 982 y Fj(b)145 1000 y Fi(\000)-8 b(!)13 b Fm(N)283 982 y Fj(c)301 1000 y Fn(.)0 1110 y(No)o(w)20 b(let)f(a)i(map)e Fi(L)14 b(\002)f Fm(X)490 1092 y Fj(N)544 1110 y Fi(3)21 b Fn(\()p Fm(l)q(;)8 b(\030)r Fn(\))20 b Fi(7!)g Fm(l)q(\030)j Fi(2)d Fm(Z)k Fn(b)q(e)d(giv)o(en.)31 b(W)l(e)20 b(observ)o(e)f(that)i Fm(X)1604 1092 y Fj(N)1636 1080 y Fd(b)1674 1096 y Fi(\030)1674 1112 y Fn(=)0 1170 y(\()p Fm(X)63 1152 y Fj(N)97 1170 y Fn(\))116 1152 y Fj(N)148 1141 y Fd(b)p Fe(\000)p Fc(1)221 1170 y Fn(and)c(de\014ne)457 1286 y Fi(L)491 1266 y Fj(N)523 1254 y Fd(a)555 1286 y Fi(\002)11 b Fm(X)649 1266 y Fj(N)681 1254 y Fd(b)712 1286 y Fi(3)j Fn(\(\003)p Fm(;)8 b Fn(\004\))13 b Fi(7!)h Fn(\003\004)f Fi(2)h Fm(Z)1127 1266 y Fj(N)1159 1254 y Fd(a)p Fc(+)p Fd(b)p Fe(\000)p Fc(1)0 1396 y Fn(b)o(y)252 1457 y(\(\003\004\)\()p Fm(i)393 1464 y Fk(1)412 1457 y Fm(;)8 b(:)g(:)g(:)f(;)h(i)538 1464 y Fj(a)p Fk(+)p Fj(b)p Fl(\000)p Fk(1)646 1457 y Fn(\))14 b(:=)f(\003\()p Fm(i)814 1464 y Fk(1)833 1457 y Fm(;)8 b(:)g(:)g(:)g(;)g(i)960 1464 y Fj(a)980 1457 y Fn(\)\004\()p Fi(\000)p Fm(;)g(i)1129 1464 y Fj(a)p Fk(+1)1194 1457 y Fm(;)g(:)g(:)g(:)f(;)h(i)1320 1464 y Fj(a)p Fk(+)p Fj(b)p Fl(\000)p Fk(1)1428 1457 y Fn(\))p Fm(:)0 1594 y Fn(F)l(or)k(a)h(left)e Fi(L)p Fn(-mo)q(dule)g Fm(X)16 b Fn(one)d(c)o(hec)o(ks)d(\004)740 1576 y Fj(#\037)798 1594 y Fn(=)k Fm(\016)874 1576 y Fj(\037)898 1594 y Fn(\004)931 1576 y Fj(#)953 1594 y Fn(,)f(whic)o(h)e(is)h(a)g(generalization)f(of)i(Lemma)0 1654 y(4.8)k(\(a\).)0 1764 y Fw(4.10)i(Lemm)o(a:)45 b Fg(Let)17 b Fm(X)k Fg(b)q(e)16 b(a)h(left)f Fi(L)p Fg(-mo)q(dule.)22 b(If)16 b Fn(\003)e Fi(2)g(L)1125 1746 y Fj(N)1157 1734 y Fd(a)1178 1764 y Fg(,)i(M)f Fi(2)f(L)1349 1746 y Fj(N)1381 1734 y Fd(b)1399 1764 y Fg(,)i(and)h Fn(\004)d Fi(2)h Fm(X)1663 1746 y Fj(N)1695 1734 y Fd(c)0 1824 y Fg(then)668 1884 y Fn(\(\003)p Fg(M)o Fn(\)\004)f(=)f(\003\()p Fg(M)p Fn(\004\))p Fm(:)0 1996 y Fu(Pr)o(oof:)124 2098 y Fn(\(\003\(M\004\)\)) 41 b(\()p Fm(i)389 2105 y Fk(1)408 2098 y Fm(;)8 b(:)g(:)g(:)f(;)h(i) 534 2105 y Fj(a)p Fk(+)p Fj(b)p Fk(+)p Fj(c)p Fl(\000)p Fk(2)685 2098 y Fn(\))353 2171 y(=)13 b(\003\()p Fm(i)474 2178 y Fk(1)494 2171 y Fm(;)8 b(:)g(:)g(:)f(;)h(i)620 2178 y Fj(a)640 2171 y Fn(\)\(M\004\)\()p Fi(\000)p Fm(;)g(i)872 2178 y Fj(a)p Fk(+1)937 2171 y Fm(;)g(:)g(:)g(:)f(;)h(i)1063 2178 y Fj(a)p Fk(+)p Fj(b)p Fk(+)p Fj(c)p Fl(\000)p Fk(2)1214 2171 y Fn(\))353 2243 y(=)13 b(\003\()p Fm(i)474 2250 y Fk(1)494 2243 y Fm(;)8 b(:)g(:)g(:)f(;)h(i)620 2250 y Fj(a)640 2243 y Fn(\)\(M\()p Fi(\000)p Fm(;)g(i)820 2250 y Fj(a)p Fk(+1)885 2243 y Fm(;)g(:)g(:)g(:)f(;)h(i)1011 2250 y Fj(a)p Fk(+)p Fj(b)p Fl(\000)p Fk(1)1119 2243 y Fn(\)\004\()p Fi(\000)p Fm(;)g(i)1268 2250 y Fj(a)p Fk(+)p Fj(b)1330 2243 y Fm(;)g(:)g(:)g(:)g(;)g(i)1457 2250 y Fj(a)p Fk(+)p Fj(b)p Fk(+)p Fj(c)p Fl(\000)p Fk(2)1607 2243 y Fn(\)\))353 2316 y(=)13 b(\(\003\()p Fm(i)493 2323 y Fk(1)513 2316 y Fm(;)8 b(:)g(:)g(:)f(;)h(i)639 2323 y Fj(a)659 2316 y Fn(\)M\()p Fi(\000)p Fm(;)g(i)820 2323 y Fj(a)p Fk(+1)885 2316 y Fm(;)g(:)g(:)g(:)f(;)h(i)1011 2323 y Fj(a)p Fk(+)p Fj(b)p Fl(\000)p Fk(1)1119 2316 y Fn(\)\)\004\()p Fi(\000)p Fm(;)g(i)1287 2323 y Fj(a)p Fk(+)p Fj(b)1349 2316 y Fm(;)g(:)g(:)g(:)g(;)g(i)1476 2323 y Fj(a)p Fk(+)p Fj(b)p Fk(+)p Fj(c)p Fl(\000)p Fk(2)1626 2316 y Fn(\))353 2389 y(=)13 b(\(\003M\()p Fm(i)538 2396 y Fk(1)557 2389 y Fm(;)8 b(:)g(:)g(:)g(;)g(i)684 2396 y Fj(a)p Fk(+)p Fj(b)p Fl(\000)p Fk(1)792 2389 y Fn(\)\004\()p Fi(\000)p Fm(;)g(i)941 2396 y Fj(a)p Fk(+)p Fj(b)1003 2389 y Fm(;)g(:)g(:)g(:)f(;)h(i)1129 2396 y Fj(a)p Fk(+)p Fj(b)p Fk(+)p Fj(c)p Fl(\000)p Fk(2)1280 2389 y Fn(\))353 2461 y(=)13 b(\(\(\003M\)\004\)\()p Fm(i)628 2468 y Fk(1)647 2461 y Fm(;)8 b(:)g(:)g(:)f(;)h(i)773 2468 y Fj(a)p Fk(+)p Fj(b)p Fk(+)p Fj(c)p Fl(\000)p Fk(2)924 2461 y Fn(\))p Fm(:)p 1689 2536 V 1689 2561 2 25 v 1711 2561 V 1689 2563 24 2 v eop %%Page: 16 16 16 15 bop 0 265 a Fn(W)l(e)17 b(\014x)g(maps)g Fm(\033)h Fn(:)d Fm(N)21 b Fi(\000)-8 b(!)15 b Fm(N)i Fi(\002)12 b Fm(N)23 b Fn(and)18 b Fm(\034)j Fn(:)16 b Fm(N)h Fi(\002)12 b Fm(N)21 b Fi(\000)-9 b(!)16 b Fm(N)23 b Fn(with)17 b Fm(\034)6 b(\033)17 b Fn(=)f(id)1455 272 y Fj(N)1507 265 y Fn(and)i Fm(\033)r(\034)j Fn(=)0 325 y(id)41 332 y Fj(N)t Fl(\002)p Fj(N)133 325 y Fn(.)g(F)l(urthermore)15 b(let)g Fm(\031)h Fn(:)d Fm(N)j Fi(\002)11 b Fm(N)19 b Fi(\000)-8 b(!)13 b Fm(N)k Fi(\002)11 b Fm(N)21 b Fn(b)q(e)16 b(the)g(map)g(with)g Fm(\031)r Fn(\()p Fm(i;)8 b(j)s Fn(\))13 b(=)g(\()p Fm(j;)8 b(i)p Fn(\).)0 435 y Fw(4.11)20 b(Corollary:)52 b Fg(Let)18 b Fm(l)q(;)8 b(m)16 b Fi(2)h(L)p Fg(.)26 b(If)17 b Fm(l)q Fn(\()p Fm(m\016)881 417 y Fj(\034)901 435 y Fn(\))g(=)f Fm(m)p Fn(\()p Fm(l)q(\016)1093 417 y Fj(\034)t(\031)1135 435 y Fn(\))i Fg(then)g Fm(l)q Fn(\()p Fm(m)p Fn(\004\))d(=)h Fm(m)p Fn(\()p Fm(l)q Fn(\004)1595 417 y Fj(\031)1618 435 y Fn(\))h Fg(for)0 495 y(all)f Fn(\004)d Fi(2)h Fm(X)205 477 y Fj(N)t Fl(\002)p Fj(N)298 495 y Fg(.)0 580 y Fu(Pr)o(oof:)0 640 y Fm(l)q Fn(\()p Fm(m)p Fn(\004\))19 b(=)h Fm(l)q Fn(\()p Fm(m)p Fn(\004)318 622 y Fj(\033)q(\034)360 640 y Fn(\))g(=)g Fm(l)q Fn(\()p Fm(m)p Fn(\()p Fm(\016)578 622 y Fj(\034)598 640 y Fn(\004)631 622 y Fj(\033)654 640 y Fn(\)\))h(=)f Fm(l)q Fn(\(\()p Fm(m\016)892 622 y Fj(\034)912 640 y Fn(\)\004)964 622 y Fj(\033)987 640 y Fn(\))g(=)g(\()p Fm(l)q Fn(\()p Fm(m\016)1205 622 y Fj(\034)1226 640 y Fn(\)\)\004)1297 622 y Fj(\033)1340 640 y Fn(=)g(\()p Fm(m)p Fn(\()p Fm(l)q(\016)1519 622 y Fj(\034)t(\031)1561 640 y Fn(\)\)\004)1632 622 y Fj(\033)1675 640 y Fn(=)0 701 y Fm(m)p Fn(\(\()p Fm(l)q(\016)121 683 y Fj(\034)t(\031)163 701 y Fn(\)\004)215 683 y Fj(\033)238 701 y Fn(\))13 b(=)h Fm(m)p Fn(\()p Fm(l)q Fn(\()p Fm(\016)443 683 y Fj(\034)t(\031)485 701 y Fn(\004)518 683 y Fj(\033)541 701 y Fn(\)\))g(=)f Fm(m)p Fn(\()p Fm(l)q Fn(\004)755 683 y Fj(\033)q(\034)t(\031)819 701 y Fn(\))g(=)h Fm(m)p Fn(\()p Fm(l)q Fn(\004)1014 683 y Fj(\031)1036 701 y Fn(\).)p 1689 674 24 2 v 1689 699 2 25 v 1711 699 V 1689 701 24 2 v 0 811 a Fw(4.12)k(Lemm)o(a:)43 b Fg(If)16 b Fm(b)d Fi(2)h(L)j Fg(is)e(a)h Fm(k)r Fg(-scalar)h(with)e Fm(b)p Fn(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))9 b(+)i Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))i(=)h Fm(b\016)r Fn(\()p Fm(i)p Fn(\))9 b(+)h Fm(b\016)r Fn(\()p Fm(j)s Fn(\))15 b Fg(then)g(for)0 871 y(all)h Fi(L)p Fg(-mo)q(dules)g Fm(X)k Fg(and)d(all)f Fm(x;)8 b(y)15 b Fi(2)f Fm(X)21 b Fg(w)o(e)15 b(ha)o(v)o(e)h Fm(b)p Fn(\()p Fm(x)10 b Fn(+)h Fm(y)r Fn(\))j(=)f Fm(bx)e Fn(+)g Fm(by)r Fg(.)0 956 y Fu(Pr)o(oof:)19 b Fn(This)c(is)f(a)h(sp)q(ecial)f(case)g(of)h (Lemma)d(4.11)j(but)f(can)h(also)g(b)q(e)f(obtained)h(b)o(y)f(simply)0 1016 y(m)o(ultiplyi)o(ng)g(the)i(equation)83 1118 y Fm(b")p Fn(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))9 b(+)i Fm(\016)r Fn(\()p Fm(j)s Fn(\))p Fi(j)p Fm(k)r Fn(\))83 b(=)13 b Fm(b)p Fn(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))d(+)h Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))511 1190 y(=)i Fm(b\016)r Fn(\()p Fm(i)p Fn(\))d(+)h Fm(b\016)r Fn(\()p Fm(j)s Fn(\))511 1263 y(=)i(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))d(+)h Fm(\016)r Fn(\()p Fm(j)s Fn(\)\))p Fm(")p Fn(\()p Fm(b")p Fn(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))p Fi(j)p Fm(k)r Fn(\))p Fm(;)d(b")p Fn(\()p Fm(\016)r Fn(\()p Fm(j)s Fn(\))p Fi(j)p Fm(k)r Fn(\))p Fi(j)o Fm(i;)g(j)s Fn(\))0 1437 y(from)15 b(the)h(righ)o(t)g(b)o(y)g Fm(")p Fn(\()p Fm(x;)8 b(y)r Fi(j)p Fm(i;)g(j)s Fn(\).)p 1689 1411 V 1689 1436 2 25 v 1711 1436 V 1689 1438 24 2 v 0 1547 a(In)16 b(view)f(of)i(this)f(result)g(Lemma)e(4.11)j(can)f(b)q(e)h (seen)f(as)h(a)f(general)g(distributiv)o(e)f(la)o(w.)0 1657 y(The)f(to)q(ols)i(dev)o(elop)q(ed)d(b)o(y)h(no)o(w)g(let)g(us)h (construct)f(some)f(in)o(teresting)g(examples)g(of)h(certain)0 1718 y(righ)o(t)23 b(mo)q(dules)f(and)h(bimo)q(dules.)41 b(F)l(or)23 b(this)g(purp)q(ose)h(let)e(\001)j(=)g Fi(f)p Fm(\016)r Fn(\()p Fm(i)p Fn(\))p Fi(j)p Fm(i)f Fi(2)i Fm(N)5 b Fi(g)26 b(\022)f(L)p Fn(.)0 1778 y(Then)e(b)o(y)f(the)g (unitary)h(la)o(w)f(\()p Fi(L)624 1785 y Fk(1)644 1778 y Fn(\))h Fm(\016)r Fn(\()p Fm(i)p Fn(\))p Fm(\025)h Fn(=)g Fm(\025)p Fn(\()p Fm(i)p Fn(\))f(it)f(is)h(clear)f(that)h(\001)f (is)g(a)h(unital)g(left)0 1838 y(linear)c(subtheory)h(of)g Fi(L)p Fn(.)32 b(By)19 b(Lemma)f(1.3.\(c\))h(there)g(is)g(a)i (bijection)d(b)q(et)o(w)o(een)h(\001)1581 1820 y Fj(N)1634 1838 y Fn(and)0 1898 y Fi(D)d Fn(=)d(Map)q(\()p Fm(N)r(;)8 b(N)d Fn(\).)21 b(W)l(e)13 b(observ)o(e)g(that)i Fi(D)g Fn(is)f(a)g(monoid)f(under)h(the)f(comp)q(osition)h(of)g(maps.)0 1958 y(W)l(e)j(will)e(consider)i(sets)g Fm(M)5 b Fn(,)17 b(whic)o(h)f(are)h(\(left\))f Fi(D)q Fn(-sets,)i(i.e.)k(on)17 b(whic)o(h)g Fi(D)h Fn(op)q(erates,)g(suc)o(h)0 2019 y(that)i(the)f(unital)h(and)g(asso)q(ciativ)o(e)f(la)o(ws)h(hold:)28 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Fj(\033)0 325 y Fn(i\013)g Fm(l)q(\033)104 307 y Fj(N)137 325 y Fn(\()p Fm(\026)p Fn(\))e(=)g Fm(\033)r Fn(\()p Fm(l)q(\026)p Fn(\).)p 1689 298 24 2 v 1689 323 2 25 v 1711 323 V 1689 325 24 2 v 0 435 a Fw(4.14)k(Theorem:)44 b Fg(Let)16 b Fi(L)f Fn(=)e Fm(R)623 417 y Fk(\()p Fj(N)t Fk(\))701 435 y Fg(b)q(e)j(the)g(semiring.)k(Then)c(the)g(functor)g (from)f Fm(R)p Fg(-Mo)q(d)0 495 y(to)i Fi(L)p Fg(-Mo)q(d)g(\(as)g (de\014ned)f(in)g(example)e(2.8.b\))i(is)g(an)h(equiv)m(alence)d(of)j (categories.)0 580 y Fu(Pr)o(oof:)23 b Fn(W)l(e)18 b(describ)q(e)f(the) h(in)o(v)o(erse)e(functor.)26 b(Let)18 b Fm(M)23 b Fn(b)q(e)18 b(a)g(left)f Fi(L)p Fn(-mo)q(dule.)25 b(Since)17 b(the)0 640 y(addition)i(in)f Fi(L)h Fn(is)f(describ)q(ed)g(b)o(y)g Fm(a)f Fn(=)g(\(1)p Fm(;)8 b Fn(1)p Fm(;)g Fn(0)p Fm(;)g Fn(0)p Fm(;)g(:)g(:)g(:)q Fn(\))18 b(and)h(the)f(addition)h(is)f(asso)q (ciativ)o(e,)0 701 y(comm)o(utativ)n(e,)h(with)h(zero)h(elemen)o(t)c (and)22 b(with)e(in)o(v)o(erses,)g Fm(M)26 b Fn(is)21 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2463 y(X)630 2473 y(X)671 2483 y(X)713 2494 y(X)754 2504 y(X)796 2514 y(X)837 2525 y(X)879 2535 y(X)920 2546 y(X)962 2556 y(X)1003 2566 y(X)1045 2577 y(X)1086 2587 y(X)1128 2597 y(X)1166 2607 y(X)-42 b(z)1145 2418 y Fl(L)1171 2411 y Fm(F)1203 2418 y Fl(K)1232 2411 y Fn(\()p Fm(X)15 b Fi(\002)c Fm(Y)1395 2393 y Fj(N)1429 2411 y Fn(\))p Fm(=)p Fn(Rel)p 1022 2398 109 2 v 1089 2397 a Fb(-)1066 2388 y Fj(\027)1021 2529 y(f)1042 2517 y Fe(0)925 2452 y Fb(H)966 2473 y(H)1008 2494 y(H)1049 2514 y(H)1091 2535 y(H)1132 2556 y(H)1174 2577 y(H)1215 2597 y(H)1234 2607 y(H)-42 b(j)1325 2658 y Fm(Z)p 1343 2607 2 176 v 1344 2607 a Fb(?)1363 2523 y Fj(g)p eop %%Page: 22 22 22 21 bop 0 265 a Fn(pro)o(vides)18 b(a)g(univ)o(ersal)g(solution)g (for)h(the)f(univ)o(ersal)f(problem)g(of)i(tensor)f(pro)q(ducts.)29 b(The)0 325 y(details)16 b(are)g(straigh)o(tforw)o(ard)h(to)g(c)o(hec)o (k.)p 1689 298 24 2 v 1689 323 2 25 v 1711 323 V 1689 325 24 2 v 0 435 a Fw(6.3)h(Corollary:)46 b Fg(The)16 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Fn(\))789 2477 y Fi(\030)790 2492 y Fn(=)842 2490 y Fi(L)p Fg(-Mo)q(d-)q Fi(M)p Fn(\()p Fm(X)q(;)g Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)g(Z)t Fn(\)\))p Fm(:)p eop %%Page: 23 23 23 22 bop 0 265 a Fu(Pr)o(oof:)21 b Fn(W)l(e)c(de\014ne)f Fm(\033)g Fn(:)e Fi(L)p Fn(-Mo)q(d-)r Fi(K)q Fn(\()p Fm(X)h Fi(\012)809 272 y Fl(M)864 265 y Fm(Y)s(;)8 b(Z)t Fn(\))15 b Fi(\000)-9 b(!)14 b(L)p Fn(-Mo)q(d-)r Fi(M)p Fn(\()p Fm(X)q(;)8 b Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)g(Z)t Fn(\)\))17 b(b)o(y)0 325 y Fm(\033)r Fn(\()p Fm(f)5 b Fn(\)\()p Fm(x)p Fn(\)\()p Fm(\021)r Fn(\))13 b(:=)g Fm(f)5 b Fn(\()p Fm(x)12 b Fi(\012)432 332 y Fl(M)486 325 y Fm(\021)r Fn(\).)21 b(Then)218 433 y Fm(\033)r Fn(\()p Fm(f)5 b Fn(\)\()p Fm(x)p Fn(\)\()p Fm(\021)r(\024)p Fn(\))41 b(=)g Fm(f)5 b Fn(\()p Fm(x)11 b Fi(\012)719 440 y Fl(M)774 433 y Fn(\()p Fm(\021)r(\024)p Fn(\)\))j(=)g Fm(f)5 b Fn(\()p Fm(x)11 b Fi(\012)1077 440 y Fl(M)1132 433 y Fm(\021)r Fn(\))p Fm(\024)i Fn(=)h Fm(\033)r Fn(\()p Fm(f)5 b Fn(\)\()p Fm(x)p Fn(\)\()p Fm(\021)r Fn(\))p Fm(\024;)179 494 y Fn(\()p Fm(\033)r Fn(\()p Fm(f)g Fn(\)\()p Fm(x)p Fn(\))p Fm(\026)p Fn(\)\()p Fm(\021)r Fn(\))41 b(=)g Fm(\033)r Fn(\()p Fm(f)5 b Fn(\)\()p Fm(x)p Fn(\)\()p Fm(\026\021)r Fn(\))14 b(=)f Fm(f)5 b Fn(\()p Fm(x)12 b Fi(\012)1041 501 y Fl(M)1095 494 y Fm(\026\021)r Fn(\))i(=)g Fm(f)5 b Fn(\()p Fm(x\026)12 b Fi(\012)1391 501 y Fl(M)1446 494 y Fm(\021)r Fn(\))514 554 y(=)41 b Fm(\033)r Fn(\()p Fm(f)5 b Fn(\)\()p Fm(x\026)p Fn(\)\()p Fm(\021)r Fn(\))p Fm(;)163 614 y Fn(\()p Fm(l)q(\033)r Fn(\()p Fm(f)g Fn(\))295 596 y Fj(N)328 614 y Fn(\()p Fm(\030)r Fn(\)\)\()p Fm(\021)r Fn(\))42 b(=)f Fm(l)q Fn(\()p Fm(f)657 596 y Fj(N)t Fl(\000)718 614 y Fn(\()p Fm(\030)14 b Fi(\012)811 621 y Fl(M)866 614 y Fm(\021)r Fn(\)\))g(=)f Fm(f)5 b Fn(\()p Fm(l)q(\030)14 b Fi(\012)1133 621 y Fl(M)1188 614 y Fm(\021)r Fn(\))g(=)f Fm(\033)r Fn(\()p Fm(f)5 b Fn(\)\()p Fm(l)q(\030)r Fn(\)\()p Fm(\021)r Fn(\))p Fm(:)0 721 y Fn(Th)o(us)17 b Fm(\033)g Fn(is)f(w)o(ell-de\014ned.)k(The)d(in)o(v)o(erse)d(map)h(is)h (de\014ned)g(b)o(y)222 831 y Fm(\033)252 810 y Fl(\000)p Fk(1)313 831 y Fn(:)d Fi(L)p Fn(-Mo)q(d-)r Fi(M)p Fn(\()p Fm(X)q(;)8 b Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)g(Z)t Fn(\)\))14 b Fi(\000)-8 b(!)13 b(L)p Fn(-Mo)q(d-)r Fi(K)q Fn(\()p Fm(X)i Fi(\012)1327 838 y Fl(M)1382 831 y Fm(Y)s(;)8 b(Z)t Fn(\))0 941 y(b)o(y)16 b Fm(\033)98 923 y Fl(\000)p Fk(1)145 941 y Fn(\()p Fm(g)r Fn(\)\()p Fm(x)11 b Fi(\012)305 948 y Fl(M)360 941 y Fm(\021)r Fn(\))j(:=)f Fm(g)r Fn(\()p Fm(x)p Fn(\)\()p Fm(\021)r Fn(\).)22 b(Since)15 b Fm(g)r Fn(\()p Fm(x\026)p Fn(\)\()p Fm(\021)r Fn(\))g(=)f Fm(g)r Fn(\()p Fm(x)p Fn(\)\()p Fm(\026\021)r Fn(\),)i Fm(g)r Fn(\()p Fm(l)q(\030)r Fn(\)\()p Fm(\021)r Fn(\))e(=)g Fm(l)q(g)1540 923 y Fj(N)1574 941 y Fn(\()p Fm(\030)r Fn(\)\()p Fm(\021)r Fn(\),)0 1001 y(and)19 b Fm(g)r Fn(\()p Fm(x)p Fn(\)\()p Fm(\021)r(\024)p Fn(\))d(=)h Fm(g)r Fn(\()p Fm(x)p Fn(\)\()p Fm(\021)r Fn(\))p Fm(\024)p Fn(,)h(the)f(map)h Fm(\033)792 983 y Fl(\000)p Fk(1)838 1001 y Fn(\()p Fm(g)r Fn(\))g(is)g(a)h(w)o(ell-de\014ned)d(bimo)q(dule) h(homomor-)0 1061 y(phism)e(on)h Fm(X)f Fi(\012)305 1068 y Fl(M)360 1061 y Fm(Y)c Fn(.)21 b(Ob)o(viously)15 b Fm(\033)693 1043 y Fl(\000)p Fk(1)756 1061 y Fn(is)h(the)f(in)o(v)o (erse)g(map)g(to)h Fm(\033)r Fn(.)21 b(It)15 b(is)h(easily)f(seen)h (that)0 1121 y(these)g(maps)g(are)g(natural)h(transformations.)p 1689 1095 24 2 v 1689 1120 2 25 v 1711 1120 V 1689 1122 24 2 v 0 1231 a Fw(6.7)h(Prop)r(osition:)46 b Fg(Let)524 1238 y Fl(L)550 1231 y Fm(X)590 1238 y Fl(M)635 1231 y Fg(,)665 1238 y Fl(M)709 1231 y Fm(Y)737 1238 y Fl(K)766 1231 y Fg(,)796 1238 y Fl(K)825 1231 y Fm(Z)858 1238 y Fl(I)898 1231 y Fg(b)q(e)17 b(bimo)q(dules.)j(Then)c(the)g(map)55 1341 y Fm(\013)f Fn(:)e(\()p Fm(X)i Fi(\012)241 1348 y Fl(M)296 1341 y Fm(Y)d Fn(\))f Fi(\012)405 1348 y Fl(K)444 1341 y Fm(Z)18 b Fi(3)c Fn(\()p Fm(x)d Fi(\012)639 1348 y Fl(M)694 1341 y Fm(\021)r Fn(\))g Fi(\012)789 1348 y Fl(K)829 1341 y Fm(\020)18 b Fi(7!)13 b Fm(x)e Fi(\012)1009 1348 y Fl(M)1064 1341 y Fn(\()p Fm(\021)i Fi(\012)1159 1348 y Fl(K)1199 1341 y Fm(\020)t Fn(\))h Fi(2)g Fm(X)h Fi(\012)1398 1348 y Fl(M)1453 1341 y Fn(\()p Fm(Y)22 b Fi(\012)1561 1348 y Fl(K)1601 1341 y Fm(Z)t Fn(\))0 1451 y Fg(is)d(a)g(bimo)q(dule)e(homomorphism,)f(whic)o(h)i(is)h (coheren)o(t)f(in)g(the)h(sense)g(of)g(monoidal)f(cate-)0 1511 y(gories.)0 1597 y Fu(Pr)o(oof:)j Fn(W)l(e)c(\014rst)g(sho)o(w)g (that)g(the)f(map)g(in)g(the)h(prop)q(osition)g(is)g(w)o(ell)e (de\014ned.)22 b(F)l(or)17 b(that)0 1657 y(purp)q(ose)g(w)o(e)f (consider)g(the)g(map)394 1767 y Fm(\014)g Fn(:)e Fm(X)h Fi(\012)560 1774 y Fl(M)615 1767 y Fm(Y)25 b Fi(\000)-8 b(!)13 b(H)p Fm(om)p Fn(\()p Fm(Z)q(;)8 b(X)16 b Fi(\012)1040 1774 y Fl(M)1095 1767 y Fn(\()p Fm(Y)22 b Fi(\012)1203 1774 y Fl(K)1243 1767 y Fm(Z)t Fn(\)\))0 1877 y(with)16 b Fm(\014)s Fn(\()p Fm(x)10 b Fi(\012)238 1884 y Fl(M)293 1877 y Fm(\021)r Fn(\)\()p Fm(\020)t Fn(\))k(:=)f Fm(x)e Fi(\012)558 1884 y Fl(M)613 1877 y Fn(\()p Fm(\021)i Fi(\012)708 1884 y Fl(K)747 1877 y Fm(\020)t Fn(\).)22 b(The)16 b(map)f Fm(\014)k Fn(is)d(w)o(ell-de\014ned)f(since)419 1985 y(\()p Fm(l)q(\030)r Fn(\))d Fi(\012)547 1992 y Fl(M)602 1985 y Fn(\()p Fm(\021)h Fi(\012)697 1992 y Fl(K)736 1985 y Fm(\020)t Fn(\))42 b(=)f Fm(l)q Fn(\()p Fm(\030)14 b Fi(\012)1010 1992 y Fl(M)1065 1985 y Fn(\()p Fm(\021)f Fi(\012)1160 1992 y Fl(K)1200 1985 y Fm(\020)t Fn(\)\))p Fm(;)440 2045 y(x\026)e Fi(\012)547 2052 y Fl(M)602 2045 y Fn(\()p Fm(\021)i Fi(\012)697 2052 y Fl(K)736 2045 y Fm(\020)t Fn(\))42 b(=)f Fm(x)11 b Fi(\012)979 2052 y Fl(M)1034 2045 y Fn(\()p Fm(\026\021)i Fi(\012)1158 2052 y Fl(K)1198 2045 y Fm(\020)t Fn(\))p Fm(;)441 2106 y(x)e Fi(\012)519 2113 y Fl(M)574 2106 y Fn(\()p Fm(\021)r(\024)g Fi(\012)697 2113 y Fl(K)736 2106 y Fm(\020)t Fn(\))42 b(=)f Fm(x)11 b Fi(\012)979 2113 y Fl(M)1034 2106 y Fn(\()p Fm(\021)i Fi(\012)1129 2113 y Fl(K)1169 2106 y Fm(\024\020)t Fn(\))p Fm(;)440 2166 y(x)e Fi(\012)518 2173 y Fl(M)573 2166 y Fn(\()p Fm(\021)i Fi(\012)668 2173 y Fl(K)708 2166 y Fm(\020)t(#)p Fn(\))41 b(=)g(\()p Fm(x)11 b Fi(\012)998 2173 y Fl(M)1053 2166 y Fn(\()p Fm(\021)i Fi(\012)1148 2173 y Fl(K)1188 2166 y Fm(\020)t Fn(\)\))p Fm(#:)0 2275 y Fn(The)e(adjoin)o(t)g(map)f(is)h Fm(\013)k Fn(:)e(\()p Fm(X)t Fi(\012)577 2282 y Fl(M)622 2275 y Fm(Y)e Fn(\))q Fi(\012)720 2282 y Fl(K)749 2275 y Fm(Z)18 b Fi(\000)-9 b(!)14 b Fm(X)t Fi(\012)977 2282 y Fl(M)1022 2275 y Fn(\()p Fm(Y)e Fi(\012)1120 2282 y Fl(K)1149 2275 y Fm(Z)t Fn(\))f(with)g Fm(\013)p Fn(\(\()p Fm(x)q Fi(\012)1459 2282 y Fl(M)1503 2275 y Fm(\021)r Fn(\))q Fi(\012)1588 2282 y Fl(K)1617 2275 y Fm(\020)t Fn(\))j(=)0 2335 y Fm(x)d Fi(\012)78 2342 y Fl(M)133 2335 y Fn(\()p Fm(\021)i Fi(\012)228 2342 y Fl(K)267 2335 y Fm(\020)t Fn(\))p Fm(:)0 2396 y Fn(The)j(coherence)f(diagram)h(is)63 2473 y(\(\()p Fm(U)g Fi(\012)11 b Fm(X)t Fn(\))g Fi(\012)g Fm(Y)g Fn(\))g Fi(\012)g Fm(Z)172 b Fn(\()p Fm(U)16 b Fi(\012)11 b Fn(\()p Fm(X)k Fi(\012)c Fm(Y)g Fn(\)\))g Fi(\012)g Fm(Z)p 495 2462 139 2 v 592 2461 a Fb(-)530 2448 y Fj(\013)p Fl(\012)p Fk(1)1233 2473 y Fm(U)16 b Fi(\012)11 b Fn(\(\()p Fm(X)k Fi(\012)c Fm(Y)g Fn(\))g Fi(\012)g Fm(Z)t Fn(\))p 1080 2462 V 1177 2461 a Fb(-)1137 2451 y Fj(\013)p 271 2573 2 78 v 271 2573 a Fb(?)229 2541 y Fj(\013)p 1440 2573 V 1441 2573 a Fb(?)1461 2543 y Fj(\013)p Fl(\012)p Fk(1)63 2619 y Fn(\()p Fm(U)16 b Fi(\012)11 b Fm(X)t Fn(\))g Fi(\012)g Fn(\()p Fm(Y)22 b Fi(\012)11 b Fm(Z)t Fn(\))753 b Fm(U)16 b Fi(\012)11 b Fn(\()p Fm(X)k Fi(\012)c Fn(\()p Fm(Y)22 b Fi(\012)11 b Fm(Z)t Fn(\)\))p Fm(:)p 495 2608 711 2 v 1164 2607 a Fb(-)838 2597 y Fj(\013)p eop %%Page: 24 24 24 23 bop 0 265 a Fn(By)16 b(elemen)o(t)n(wise)d(computations)j(it)g (is)g(easy)g(to)h(see)e(that)i(this)f(diagram)g(comm)o(utes.)p 1689 238 24 2 v 1689 263 2 25 v 1711 263 V 1689 265 24 2 v 0 375 a(W)l(e)g(will)f(see)h(in)g(Cor.)22 b(6.17)17 b(that)f Fm(\013)h Fn(is)f(surjectiv)o(e.)0 485 y Fw(6.8)i(Remark:)23 b Fn(W)l(e)16 b(cannot)h(sho)o(w)g(that)g Fm(\013)g Fn(is)f(an)h (isomorphism.)i(In)d(fact)h(w)o(e)e(conjecture)0 545 y(that)e(it)f(is)g(not)h(in)g(general.)19 b(Ho)o(w)o(ev)o(er,)11 b(ho)o(w)i(close)f(it)g(is)h(to)f(b)q(eing)h(bijectiv)o(e,)e(can)h(b)q (e)h(seen)f(b)o(y)0 605 y(prop)q(ositions)i(6.16)g(and)g(6.17.)21 b(Nev)o(ertheless)11 b Fi(L)p Fn(-Mo)q(d-)q Fi(L)i Fn(b)q(eha)o(v)o(es) g(lik)o(e)e(a)j(closed)e(monoidal)0 665 y(category)l(.)27 b(The)18 b(ab)q(o)o(v)o(e)g(p)q(en)o(tagon)h(diagram)e(su\016ces)h(to)h (generate)e(coherence.)26 b(And)18 b(the)0 726 y(inner)13 b(hom-functors)h(ha)o(v)o(e)f(the)g(usual)h(prop)q(erties.)21 b(Before)13 b(w)o(e)g(pro)o(v)o(e)g(this,)g(w)o(e)h(sho)o(w)g(that)0 786 y(there)i(is)g(a)g(t)o(w)o(o-sided)g(unit)g(for)h(the)f(tensor)g (pro)q(duct.)0 896 y Fw(6.9)i(Prop)r(osition:)46 b Fg(There)15 b(are)i(natural)f(isomorphisms)348 1006 y Fm(s)d Fn(:)h Fi(L)d(\012)496 1013 y Fl(L)533 1006 y Fm(X)592 992 y Fi(\030)592 1008 y Fn(=)644 1006 y Fm(X)102 b Fg(and)114 b Fm(d)15 b Fn(:)e Fm(X)j Fi(\012)1140 1013 y Fl(M)1194 1006 y Fi(M)1268 992 y(\030)1268 1008 y Fn(=)1321 1006 y Fm(X)0 1116 y Fg(whic)o(h)g(satisfy)g(the)g(coherence)f(diagrams)h (for)g(monoidal)g(categories)92 1224 y Fn(\()p Fm(X)g Fi(\012)206 1231 y Fl(L)243 1224 y Fi(L)p Fn(\))11 b Fi(\012)346 1231 y Fl(L)384 1224 y Fm(Y)538 1198 y Fj(\013)509 1224 y Fi(\000)-8 b(!)86 b Fm(X)16 b Fi(\012)771 1231 y Fl(L)808 1224 y Fn(\()p Fi(L)c(\012)912 1231 y Fl(L)949 1224 y Fm(Y)f Fn(\))180 b Fi(L)12 b(\012)1272 1231 y Fl(L)1309 1224 y Fi(L)42 b Fn(=)f Fi(L)12 b(\012)1549 1231 y Fl(L)1586 1224 y Fi(L)262 1285 y Fj(d)p Fl(\012)307 1291 y Fe(L)331 1285 y Fj(Y)22 b Fi(&)253 b(.)14 b Fj(X)s Fl(\012)799 1291 y Fe(L)823 1285 y Fj(s)422 b(d)14 b Fi(&)121 b(.)14 b Fj(s)464 1345 y Fm(X)i Fi(\012)559 1352 y Fl(L)596 1345 y Fm(Y)763 b Fi(L)114 b Fm(:)0 1482 y Fu(Pr)o(oof:)19 b Fn(W)l(e)13 b(de\014ne)f Fm(s)433 1464 y Fl(0)459 1482 y Fn(:)h Fi(L)t(\002)t Fm(X)611 1464 y Fj(N)660 1482 y Fi(\000)-9 b(!)14 b Fm(X)j Fn(b)o(y)c Fm(s)899 1464 y Fl(0)911 1482 y Fn(\()p Fm(l)q(;)8 b(\030)r Fn(\))14 b(:=)f Fm(l)q(\030)r Fn(.)20 b(Ob)o(viously)12 b Fm(s)1411 1464 y Fl(0)1436 1482 y Fn(is)h(bilinear,)f(so)0 1543 y(it)k(factors)i(uniquely)d(through)j(a)g(bimo)q(dule)d (homomorphism)f Fm(s)j Fn(on)g(the)g(tensor)g(pro)q(duct.)0 1603 y(W)l(e)23 b(de\014ne)f Fm(s)261 1585 y Fl(\000)p Fk(1)334 1603 y Fn(:)i Fm(X)30 b Fi(\000)-8 b(!)25 b(L)16 b(\012)637 1610 y Fl(L)678 1603 y Fm(X)28 b Fn(b)o(y)22 b Fm(s)843 1585 y Fl(\000)p Fk(1)890 1603 y Fn(\()p Fm(x)p Fn(\))j(:=)g Fm(\016)r Fn(\(1\))15 b Fi(\012)1198 1610 y Fl(L)1240 1603 y Fm(x)1268 1585 y Fj(N)1301 1603 y Fn(.)42 b(Then)23 b Fm(ss)1537 1585 y Fl(\000)p Fk(1)1584 1603 y Fn(\()p Fm(x)p Fn(\))i(=)0 1663 y Fm(s)p Fn(\()p Fm(\016)r Fn(\(1\))13 b Fi(\012)180 1670 y Fl(L)220 1663 y Fm(x)248 1645 y Fj(N)281 1663 y Fn(\))20 b(=)g Fm(\016)r Fn(\(1\))p Fm(x)492 1645 y Fj(N)545 1663 y Fn(=)g Fm(x)g Fn(hence)f Fm(ss)836 1645 y Fl(\000)p Fk(1)903 1663 y Fn(=)h(id)o Fi(j)1015 1670 y Fj(X)1069 1663 y Fn(and)g Fm(s)1190 1645 y Fl(\000)p Fk(1)1237 1663 y Fm(s)p Fn(\()p Fm(l)14 b Fi(\012)1347 1670 y Fl(L)1387 1663 y Fm(\030)r Fn(\))20 b(=)g Fm(s)1530 1645 y Fl(\000)p Fk(1)1577 1663 y Fn(\()p Fm(l)q(\030)r Fn(\))h(=)0 1723 y Fm(\016)r Fn(\(1\))11 b Fi(\012)136 1730 y Fl(L)173 1723 y Fn(\()p Fm(l)q(\030)r Fn(\))250 1705 y Fj(N)298 1723 y Fn(=)j Fm(\016)r Fn(\(1\))d Fi(\012)486 1730 y Fl(L)523 1723 y Fm(l)539 1705 y Fj(N)573 1723 y Fm(\030)16 b Fn(=)e Fm(\016)r Fn(\(1\))p Fm(l)764 1705 y Fj(N)808 1723 y Fi(\012)847 1730 y Fl(L)885 1723 y Fm(\030)i Fn(=)e Fm(l)e Fi(\012)1040 1730 y Fl(L)1078 1723 y Fm(\030)r Fn(.)22 b(Ob)o(viously)15 b(it)h(is)g(su\016cien)o(t)f(to)0 1783 y(sho)o(w)f(that)h Fm(s)245 1765 y Fl(\000)p Fk(1)292 1783 y Fm(s)e Fn(and)i(id)e(agree)h(on)g(elemen)o(ts)d(of)j(the)g(form) f Fm(l)7 b Fi(\012)1171 1790 y Fl(L)1202 1783 y Fm(\030)r Fn(,)15 b(if)e(w)o(e)g(kno)o(w)h(that)h Fm(d)1619 1765 y Fl(\000)p Fk(1)1680 1783 y Fn(is)0 1844 y(an)f Fi(L)p Fn(-bimo)q(dule)e(homomorphism.)17 b(W)l(e)c(ha)o(v)o(e)g Fm(s)901 1826 y Fl(\000)p Fk(1)948 1844 y Fn(\()p Fm(l)q(\030)r Fn(\))h(=)g Fm(\016)r Fn(\(1\))5 b Fi(\012)1221 1851 y Fl(L)1251 1844 y Fn(\()p Fm(l)q(\030)r Fn(\))1328 1826 y Fj(N)1376 1844 y Fn(=)13 b Fm(\016)r Fn(\(1\))5 b Fi(\012)1557 1851 y Fl(L)1588 1844 y Fm(l)1604 1826 y Fj(N)1637 1844 y Fm(\030)17 b Fn(=)0 1904 y Fm(l)12 b Fi(\012)66 1911 y Fl(L)103 1904 y Fm(\030)k Fn(=)e Fm(l)q(\016)e Fi(\012)281 1911 y Fl(L)318 1904 y Fm(\030)17 b Fn(=)d Fm(l)q Fn(\()p Fm(\016)r Fn(\(1\))c Fi(\012)578 1911 y Fl(L)615 1904 y Fm(x)643 1886 y Fj(N)643 1916 y(i)677 1904 y Fn(\))k(=)f Fm(l)q Fn(\()p Fm(s)819 1886 y Fl(\000)p Fk(1)866 1904 y Fn(\))885 1886 y Fj(N)919 1904 y Fn(\()p Fm(\030)r Fn(\),)j(where)g(w)o(e)g(used)240 2014 y Fm(\016)r Fn(\()p Fm(i)p Fn(\))10 b Fi(\012)368 2021 y Fl(L)405 2014 y Fm(\030)17 b Fn(=)d Fm(\016)r Fn(\()p Fm(j)s Fn(\))p Fm(\016)r Fn(\()p Fm(i)p Fn(\))659 1993 y Fj(N)702 2014 y Fi(\012)741 2021 y Fl(L)778 2014 y Fm(\030)i Fn(=)e Fm(\016)r Fn(\()p Fm(j)s Fn(\))d Fi(\012)1002 2021 y Fl(L)1039 2014 y Fm(\016)r Fn(\()p Fm(i)p Fn(\))1118 1993 y Fj(N)1150 2014 y Fm(\030)17 b Fn(=)d Fm(\016)r Fn(\()p Fm(j)s Fn(\))c Fi(\012)1374 2021 y Fl(L)1411 2014 y Fm(x)1439 1993 y Fj(N)1439 2026 y(i)0 2124 y Fn(for)22 b(all)e(c)o(hoices)h(of)g Fm(i;)8 b(j)25 b Fi(2)d Fm(N)5 b Fn(.)37 b(F)l(urthermore)20 b(w)o(e)g(ha)o(v)o(e)h Fm(s)1121 2106 y Fl(\000)p Fk(1)1168 2124 y Fn(\()p Fm(x\025)p Fn(\))h(=)h Fm(\016)r Fn(\(1\))14 b Fi(\012)1484 2131 y Fl(L)1525 2124 y Fn(\()p Fm(x\025)p Fn(\))1619 2106 y Fj(N)1675 2124 y Fn(=)0 2184 y Fm(\016)r Fn(\(1\))c Fi(\012)135 2191 y Fl(L)171 2184 y Fm(x)199 2166 y Fj(N)232 2184 y Fm(\025)15 b Fn(=)e Fm(s)349 2166 y Fl(\000)p Fk(1)397 2184 y Fn(\()p Fm(x)p Fn(\))p Fm(\025)p Fn(.)21 b(Consequen)o(tly)15 b Fm(s)g Fn(and)i Fm(s)984 2166 y Fl(\000)p Fk(1)1047 2184 y Fn(are)e(in)o(v)o(erse)f(to)i(eac)o(h)g (other.)21 b(They)0 2244 y(are)16 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2604 V 1689 2629 2 25 v 1711 2629 V 1689 2631 24 2 v eop %%Page: 25 25 25 24 bop 0 265 a Fw(6.10)27 b(De\014nition:)41 b Fm(E)29 b Fi(2)e(L)p Fn(-Mo)q(d-)r Fi(L)d Fn(together)g(with)f Fi(L)p Fn(-bimo)q(dule)g(homomorphisms)0 325 y Fi(r)13 b Fn(:)h Fm(E)g Fi(\012)172 332 y Fl(L)209 325 y Fm(E)j Fi(\000)-9 b(!)14 b Fm(E)19 b Fn(and)e Fm(\021)f Fn(:)d Fi(L)h(\000)-8 b(!)13 b Fm(E)20 b Fn(is)c(called)f(a)i Ff(monoid)p Fn(,)e(if)h(the)g(follo)o(wing)g(diagrams)0 385 y(comm)o(ute)326 453 y(\()p Fm(E)e Fi(\012)434 460 y Fl(L)471 453 y Fm(E)s Fn(\))d Fi(\012)579 460 y Fl(L)616 453 y Fm(E)487 b(E)14 b Fi(\012)1228 460 y Fl(L)1266 453 y Fm(E)p 670 441 456 2 v 1084 440 a Fb(-)848 425 y Fl(r\012)904 431 y Fe(L)929 425 y Fk(1)326 575 y Fm(E)g Fi(\012)415 582 y Fl(L)452 575 y Fn(\()p Fm(E)g Fi(\012)560 582 y Fl(L)597 575 y Fm(E)s Fn(\))p 490 528 2 54 v 491 528 a Fb(?)449 509 y Fj(\013)p 1221 650 2 176 v 1222 650 a Fb(?)1241 574 y Fl(r)p 490 650 2 54 v 491 650 a Fb(?)373 632 y Fk(1)p Fl(\012)418 638 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Fm(\025)p Fn(\))g Fi(\016)g Fm(\017)1075 1536 y Fl(0)1086 1557 y Fn(\))j(=)g Fm(m)p Fn(\()p Fm(e;)8 b(\025\017)1326 1536 y Fl(0)1337 1557 y Fn(\))p Fm(;)166 1630 y(m)p Fn(\()p Fm(l)q(\017;)g(\017)306 1609 y Fl(0)316 1630 y Fn(\))14 b(=)f(\()p Fm(\021)r Fn(\()p Fm(l)q Fn(\))e Fi(\016)g Fm(\017)p Fn(\))g Fi(\016)g Fm(\017)652 1609 y Fl(0)663 1630 y Fn(\))j(=)f Fm(\021)r Fn(\()p Fm(l)q Fn(\))e Fi(\016)g Fn(\()p Fm(\017)g Fi(\016)g Fm(\017)980 1609 y Fl(0)991 1630 y Fn(\))j(=)g Fm(l)q(m)1135 1609 y Fj(N)1167 1630 y Fn(\()p Fm(\017;)8 b(\017)1248 1609 y Fl(0)1259 1630 y Fn(\))p Fm(;)166 1702 y(m)p Fn(\()p Fm(e;)g(\017)293 1682 y Fl(0)303 1702 y Fm(\025)p Fn(\))14 b(=)g Fm(e)d Fi(\016)g Fn(\()p Fm(\017)525 1682 y Fl(0)547 1702 y Fi(\016)g Fm(\021)609 1682 y Fj(N)643 1702 y Fn(\()p Fm(\025)p Fn(\)\))j(=)g(\()p Fm(e)c Fi(\016)h Fm(\017)902 1682 y Fl(0)914 1702 y Fn(\))g Fi(\016)g Fm(\021)1006 1682 y Fj(N)1039 1702 y Fn(\()p Fm(\025)p Fn(\))j(=)g Fm(m)p Fn(\()p Fm(e;)8 b(\017)1298 1682 y Fl(0)1308 1702 y Fn(\))p Fm(\025:)0 1804 y Fn(The)14 b(induced)g(map)g Fi(r)g Fn(is)g(asso)q(ciativ)o(e,)g(since)g(the)g(m)o(ultipli)o(cation) e(is)i(asso)q(ciativ)o(e.)20 b(W)l(e)15 b(use)0 1864 y(the)h(same)f(sym)o(b)q(ol)g(for)h(the)g(units)g(in)g Fi(L)g Fn(and)h(in)e Fm(E)s Fn(,)h(hence)f Fm(\021)r Fn(\()p Fm(\016)r Fn(\))e(=)h Fm(\016)r Fn(.)21 b(The)16 b(unit)g(prop)q(ert)o(y)0 1924 y(for)j(a)g(monoid)f(follo)o(ws)h(from)e Fm(\016)r Fn(\(1\))c Fi(\016)f Fm(x)740 1906 y Fj(N)792 1924 y Fn(=)18 b Fm(\016)r Fn(\(1\))p Fm(x)962 1906 y Fj(N)1013 1924 y Fn(=)g Fm(x)g Fn(and)i Fm(x)12 b Fi(\016)h Fm(\016)19 b Fn(=)f Fm(x\016)i Fn(=)e Fm(x)p Fn(.)28 b(Hence)0 1985 y(\()p Fm(E)s(;)8 b Fi(r)p Fm(;)g(\021)r Fn(\))15 b(is)h(a)h(monoid.)0 2070 y(Con)o(v)o(ersely)e(let)g Fm(E)20 b Fn(b)q(e)c(a)h(monoid.)j(Then)c(there)g(is)g(a)h(m)o(ultipli) o(cation)518 2187 y Fm(m)d Fn(:)f Fm(E)h Fi(\002)d Fm(E)741 2166 y Fj(N)815 2160 y Fl(\012)788 2187 y Fi(\000)-8 b(!)13 b Fm(E)h Fi(\012)971 2194 y Fl(L)1009 2187 y Fm(E)1087 2160 y Fl(r)1061 2187 y Fi(\000)-8 b(!)14 b Fm(E)0 2297 y Fn(whic)o(h)d(ob)o(viously)g(is)g(asso)q(ciativ)o(e.)19 b(F)l(rom)11 b(the)g(unit)g(prop)q(ert)o(y)h(of)f(a)h(monoid)f(w)o(e)g (get)h Fm(\021)r Fn(\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\)\))q Fi(\016)0 2357 y Fm(\017)i Fn(=)f Fi(r)p Fn(\()p Fm(\021)d Fi(\012)219 2364 y Fl(L)254 2357 y Fn(1\)\()p Fm(\016)r Fn(\()p Fm(i)p Fn(\))e Fi(\012)442 2364 y Fl(L)477 2357 y Fm(\017)p Fn(\))13 b(=)h Fi(r)p Fn(\()p Fm(\021)c Fi(\012)715 2364 y Fl(L)750 2357 y Fn(1\)\()p Fm(\016)r Fn(\(1\))f Fi(\012)946 2364 y Fl(L)980 2357 y Fm(e)1003 2339 y Fj(N)1003 2369 y(i)1037 2357 y Fn(\))14 b(=)f Fm(e)1144 2364 y Fj(i)1172 2357 y Fn(=)h Fm(\016)r Fn(\()p Fm(i)p Fn(\))p Fm(\017)p Fn(,)f(using)j(an)f(equalit)o(y)0 2417 y(from)g(the)g(pro)q (of)i(of)f(the)g(previous)f(prop)q(osition)i(and)g Fm(e)9 b Fi(\016)h Fm(\021)1101 2399 y Fj(N)1135 2417 y Fn(\()p Fm(\016)r Fn(\))j(=)h Fi(r)p Fn(\()p Fm(E)e Fi(\012)1410 2424 y Fl(L)1447 2417 y Fm(\021)r Fn(\)\()p Fm(e)d Fi(\012)1582 2424 y Fl(L)1618 2417 y Fm(\016)r Fn(\))14 b(=)0 2477 y Fm(e)22 b Fn(=)g Fm(e\016)r Fn(,)f(where)g(w)o(e)g(abbreviated)f Fm(\021)707 2459 y Fj(N)741 2477 y Fn(\()p Fm(\016)r Fn(\))h(=:)h Fm(\016)r Fn(.)36 b(Then)21 b Fm(\016)h Fn(is)f(the)g(unit)g(of)h Fm(E)i Fn(and)e Fm(E)i Fn(is)0 2537 y(a)d(left)f(linear)g(theory)l(.)35 b(The)20 b(map)g Fm(\021)j Fn(preserv)o(es)d(the)g(unit)h(and)g(induces)f(the)h(bimo)q (dules)0 2598 y(structure)c(of)g Fm(E)s Fn(,)g(since)f Fm(\021)r Fn(\()p Fm(l)q Fn(\))f(=)g Fm(\021)r Fn(\()p Fm(l)q(\016)r Fn(\))f(=)h Fm(l)q(\021)r Fn(\()p Fm(\016)r Fn(\))f(=)i Fm(l)q(\016)r Fn(,)f(hence)i Fm(\021)r Fn(\()p Fm(l)q Fn(\))11 b Fi(\016)g Fm(\017)k Fn(=)g(\()p Fm(l)q(\016)r Fn(\))c Fi(\016)h Fm(\017)i Fn(=)i Fm(l)q(\017)g Fn(and)p eop %%Page: 26 26 26 25 bop 0 265 a Fm(e)13 b Fi(\016)f Fm(\021)99 247 y Fj(N)133 265 y Fn(\()p Fm(\025)p Fn(\))19 b(=)f Fm(e)13 b Fi(\016)f Fn(\()p Fm(\025\016)r Fn(\))19 b(=)f Fm(e\025)p Fn(.)30 b(Finally)17 b Fm(\021)k Fn(is)e(a)g(homomorphism)d(of)k(left)e (linear)g(theories)0 325 y(since)d Fm(\021)r Fn(\()p Fm(l)q(\025)p Fn(\))f(=)g Fm(\021)r Fn(\()p Fm(l)q(\016)r(\025)p Fn(\))f(=)h Fm(l)q(\016)r(\025)f Fn(=)h(\()p Fm(l)q(\016)r(\025)p Fn(\))d Fi(\016)g Fm(\016)k Fn(=)f(\()p Fm(l)q(\016)r Fn(\))c Fi(\016)h Fn(\()p Fm(\025\016)r Fn(\))i(=)h Fm(\021)r Fn(\()p Fm(l)q Fn(\))d Fi(\016)g Fm(\021)1297 307 y Fj(N)1330 325 y Fn(\()p Fm(\025)p Fn(\).)p 1689 298 24 2 v 1689 323 2 25 v 1711 323 V 1689 325 24 2 v 0 435 a Fw(6.12)18 b(Remark:)23 b Fn(Let)16 b Fm(X)21 b Fn(and)16 b Fm(Y)28 b Fn(b)q(e)16 b Fi(L)p Fn(-bimo)q(dules.)21 b(The)16 b(ev)m(aluation)548 545 y Fm(ev)f Fn(:)e Fi(H)p Fm(om)p Fn(\()p Fm(X)q(;)8 b(Y)k Fn(\))f Fi(\012)936 552 y Fl(L)973 545 y Fm(X)19 b Fi(\000)-9 b(!)14 b Fm(Y)0 655 y Fn(with)h Fm(ev)r Fn(\()p Fm(f)d Fi(\012)253 662 y Fl(L)288 655 y Fm(\030)r Fn(\))i(:=)g Fm(f)5 b Fn(\()p Fm(\030)r Fn(\))16 b(is)e(the)h(counit)g(for)g(the)g(pair)g(of)g(adjoin)o(t)g(functors)g Fi(\000)8 b(\012)1540 662 y Fl(L)1575 655 y Fm(X)19 b Fn(and)0 715 y Fi(H)p Fm(om)p Fn(\()p Fm(X)q(;)8 b Fi(\000)p Fn(\).)0 825 y Fw(6.13)27 b(Corollary:)80 b Fg(Let)549 832 y Fl(L)576 825 y Fm(X)616 832 y Fl(M)660 825 y Fg(,)700 832 y Fl(M)744 825 y Fm(Y)772 832 y Fl(K)801 825 y Fg(,)841 832 y Fl(L)867 825 y Fm(Z)900 832 y Fl(K)954 825 y Fg(b)q(e)24 b(bimo)q(dules.)44 b(F)l(or)24 b(eac)o(h)g(bimo)q(dule)0 885 y(homomorphism)16 b Fm(f)23 b Fn(:)17 b Fm(X)g Fi(\012)521 892 y Fl(M)577 885 y Fm(Y)29 b Fi(\000)-8 b(!)17 b Fm(Z)23 b Fg(there)17 b(is)i(a)g(unique)e(bimo)q(dule)g(homomorphism)0 946 y Fm(g)f Fn(:)d Fm(X)19 b Fi(\000)-9 b(!)14 b(H)p Fm(om)p Fn(\()p Fm(Y)s(;)8 b(Z)t Fn(\))16 b Fg(suc)o(h)g(that)h(the)f (diagram)g(comm)o(ute)o(s:)470 1027 y Fm(X)f Fi(\012)564 1034 y Fl(M)619 1027 y Fm(Y)374 1172 y Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)8 b(Z)t Fn(\))j Fi(\012)660 1179 y Fl(M)715 1172 y Fm(Y)p 563 1125 2 78 v 564 1125 a Fb(?)433 1095 y Fj(g)q Fl(\012)478 1101 y Fe(M)516 1095 y Fj(Y)402 b(f)700 1058 y Fb(X)742 1068 y(X)783 1079 y(X)825 1089 y(X)866 1099 y(X)908 1110 y(X)949 1120 y(X)971 1125 y(X)-42 b(z)1132 1172 y Fm(Z)q(:)p 769 1161 336 2 v 1063 1160 a Fb(-)919 1150 y Fj(ev)0 1281 y Fu(Pr)o(oof:)23 b Fn(This)17 b(is)g(a)h(standard)g(consequence)f(of)g(the)g(fact)h(that)f Fi(\000)12 b(\012)1316 1288 y Fl(M)1372 1281 y Fm(Y)28 b Fn(is)17 b(left-adjoin)o(t)0 1342 y(to)g Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)8 b Fi(\000)p Fn(\).)p 1689 1315 24 2 v 1689 1340 2 25 v 1711 1340 V 1689 1342 24 2 v 0 1452 a Fw(6.14)27 b(Corollary:)77 b Fg(The)24 b(ev)m(aluation)g Fm(ev)j Fn(:)g Fi(H)p Fm(om)p Fn(\()p Fm(X)q(;)8 b(Y)k Fn(\))k Fi(\012)1220 1459 y Fl(M)1280 1452 y Fm(X)31 b Fi(\000)-9 b(!)27 b Fm(Y)35 b Fg(de\014nes)23 b(a)0 1512 y(comp)q(osition)16 b(of)g(maps)104 1622 y Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)8 b(Z)t Fn(\))j Fi(\012)390 1629 y Fl(M)445 1622 y Fi(H)p Fm(om)p Fn(\()p Fm(X)q(;)d(Y)k Fn(\))i Fi(3)g Fm(f)j Fi(\012)835 1629 y Fl(M)890 1622 y Fm( )e Fi(7!)f Fn(\()p Fm(x)f Fi(7!)h Fm(f)5 b Fn(\()p Fm( )r Fn(\()p Fm(\030)r Fn(\)\))14 b Fi(2)g(H)p Fm(om)p Fn(\()p Fm(X)q(;)8 b(Z)t Fn(\))p Fm(;)0 1732 y Fg(whic)o(h)16 b(is)g(asso)q(ciativ)o(e)g(in)g(the)g(sense)g(that)g(the)g(diagram)72 1809 y Fn(\()p Fi(H)p Fm(om)p Fn(\()p Fm(Z)q(;)8 b(U)d Fn(\))12 b Fi(\012)f(H)p Fm(om)p Fn(\()p Fm(Y)s(;)d(Z)t Fn(\)\))j Fi(\012)g(H)p Fm(om)p Fn(\()p Fm(X)q(;)d(Y)k Fn(\))112 b Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)8 b(U)d Fn(\))12 b Fi(\012)e(H)p Fm(om)p Fn(\()p Fm(X)q(;)e(Y)13 b Fn(\))p 973 1798 83 2 v 1014 1797 a Fb(-)72 1931 y Fi(H)p Fm(om)p Fn(\()p Fm(Z)q(;)8 b(U)d Fn(\))12 b Fi(\012)f Fn(\()p Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)d(Z)t Fn(\))j Fi(\012)g(H)p Fm(om)p Fn(\()p Fm(X)q(;)d(Y)k Fn(\)\))p 514 1885 2 54 v 515 1885 a Fb(?)p 1343 2006 2 176 v 1344 2006 a(?)p 514 2006 2 54 v -871 w(?)241 2053 y Fi(H)p Fm(om)p Fn(\()p Fm(Z)q(;)c(U)d Fn(\))12 b Fi(\012)e(H)p Fm(om)p Fn(\()p Fm(X)q(;)e(Z)t Fn(\))432 b Fi(H)p Fm(om)p Fn(\()p Fm(X)q(;)8 b(U)d Fn(\))p 804 2042 402 2 v 1164 2041 a Fb(-)0 2137 y Fg(comm)o(utes.)0 2223 y Fu(Pr)o(oof:)26 b Fn(The)19 b(map)f Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)8 b(Z)t Fn(\))13 b Fi(\012)701 2230 y Fl(M)758 2223 y Fi(H)p Fm(om)p Fn(\()p Fm(X)q(;)8 b(Y)k Fn(\))19 b Fi(\000)-9 b(!)18 b(H)p Fm(om)p Fn(\()p Fm(X)q(;)8 b(Z)t Fn(\))20 b(induced)e(b)o(y)h(the)0 2283 y(ev)m(aluation)d(is)g(uniquely)f(determined)f(b)o(y)i(the)g (follo)o(wing)g(diagram)24 2360 y(\()p Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)8 b(Z)t Fn(\))j Fi(\012)g(H)p Fm(om)p Fn(\()p Fm(X)q(;)d(Y)13 b Fn(\)\))e Fi(\012)f Fm(X)290 b Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)8 b(Z)t Fn(\))j Fi(\012)g Fn(\()p Fi(H)p Fm(om)p Fn(\()p Fm(X)q(;)d(Y)k Fn(\))f Fi(\012)g Fm(X)t Fn(\))p 728 2349 257 2 v 943 2348 a Fb(-)p 368 2557 2 176 v 369 2557 a(?)286 2477 y Fj(g)q Fl(\012)p Fk(1)1176 2482 y Fi(H)p Fm(om)p Fn(\()p Fm(Y)s(;)d(Z)t Fn(\))j Fi(\012)g Fm(Y)p 1343 2436 2 54 v 1344 2436 a Fb(?)1363 2418 y Fk(1)p Fl(\012)p Fj(ev)p 1343 2557 V 1344 2557 a Fb(?)1363 2538 y Fj(ev)193 2604 y Fi(H)p Fm(om)p Fn(\()p Fm(X)q(;)d(Z)t Fn(\))k Fi(\012)f Fm(X)786 b(Z)q(:)p 559 2593 740 2 v 1257 2592 a Fb(-)p eop %%Page: 27 27 27 26 bop 0 265 a Fn(It)16 b(is)g(easy)g(to)h(see)f(that)g(the)g(comp)q (osition)g(is)g(describ)q(ed)g(as)h(giv)o(en)e(in)h(the)g(Corollary)l (.)p 1689 238 24 2 v 1689 263 2 25 v 1711 263 V 1689 265 24 2 v 0 375 a Fw(6.15)32 b(Corollary:)98 b Fg(F)l(or)28 b(a)h(bimo)q(dule)854 382 y Fl(L)880 375 y Fm(X)920 382 y Fl(M)993 375 y Fg(the)f(set)g(of)h(inner)e(endomorphisms)0 435 y Fi(E)t Fm(nd)p Fn(\()p Fm(X)t Fn(\))15 b(:=)f Fi(H)p Fm(om)p Fn(\()p Fm(X)q(;)8 b(X)t Fn(\))17 b Fg(is)g(a)f(left)g(linear)f (theory)l(.)0 520 y Fu(Pr)o(oof:)24 b Fn(By)17 b(Corollary)h(6.14)g Fi(E)t Fm(nd)p Fn(\()p Fm(X)t Fn(\))h(is)f(a)g(monoid)f(in)h(the)f (quasi-monoidal)g(category)0 580 y Fi(L)p Fn(-Mo)q(d-)q Fi(L)p Fn(.)22 b(The)15 b(unitary)g(la)o(w)g(is)g(giv)o(en)f(b)o(y)h (the)g(map)f Fi(L)g(3)g Fm(l)h Fi(7!)f Fn(\()p Fm(\030)i Fi(7!)e Fm(l)q(\030)r Fn(\))g Fi(2)g(E)t Fm(nd)p Fn(\()p Fm(X)t Fn(\).)22 b(By)0 640 y(Prop)q(osition)17 b(6.11)g(it)f(is)g(a)h (left)e(linear)h(theory)l(.)p 1689 614 V 1689 639 2 25 v 1711 639 V 1689 641 24 2 v 0 750 a Fw(6.16)g(Lemm)o(a:)42 b Fg(Let)14 b Fm(N)19 b Fg(b)q(e)14 b(an)h(in\014nite)e(set.)20 b(Let)15 b Fn(\()p Fm(x)1019 757 y Fj(i)1039 750 y Fi(\012)1078 757 y Fl(M)1129 750 y Fm(\021)1153 757 y Fj(i)1167 750 y Fn(\))1186 757 y Fj(i)p Fl(2)p Fj(N)1269 750 y Fi(2)f Fn(\()p Fm(X)d Fi(\012)1425 757 y Fl(M)1475 750 y Fm(Y)g Fn(\))1533 732 y Fj(N)1567 750 y Fg(.)21 b(Then)0 811 y(there)16 b(are)g(elemen)o(ts)d Fm(\030)428 793 y Fl(0)455 811 y Fi(2)h Fm(X)546 793 y Fj(N)580 811 y Fg(,)h Fm(\021)635 793 y Fl(0)661 811 y Fi(2)f Fm(Y)747 793 y Fj(N)797 811 y Fg(suc)o(h)i(that)501 921 y Fn(\()p Fm(x)548 928 y Fj(i)573 921 y Fi(\012)612 928 y Fl(M)667 921 y Fm(\021)691 928 y Fj(i)705 921 y Fn(\))724 928 y Fj(i)p Fl(2)p Fj(N)807 921 y Fn(=)e(\()p Fm(\030)901 900 y Fl(0)913 921 y Fn(\()p Fm(i)p Fn(\))d Fi(\012)1018 928 y Fl(M)1072 921 y Fm(\021)1098 900 y Fl(0)1110 921 y Fn(\))1129 928 y Fj(i)p Fl(2)p Fj(N)1198 921 y Fm(:)0 1056 y Fu(Pr)o(oof:)19 b Fn(Since)12 b Fm(N)19 b Fn(is)13 b(in\014nite)f(there)h(is)g(a)g(bijection)f Fm(\034)20 b Fn(:)13 b Fm(N)e Fi(\002)5 b Fm(N)18 b Fi(\000)-8 b(!)13 b Fm(N)19 b Fn(with)13 b(in)o(v)o(erse)e(map)0 1116 y Fm(\033)r Fn(.)19 b(W)l(e)12 b(de\014ne)g(\001)i(:=)f Fm(\016)r(\034)k Fn(suc)o(h)12 b(that)h(\001\()p Fm(i;)8 b Fi(\000)p Fn(\))13 b Fi(2)h(M)947 1098 y Fj(N)980 1116 y Fn(,)f Fm(\021)1033 1098 y Fl(0)1058 1116 y Fn(:=)g Fm(\021)r(\033)r Fn(,)f(and)h Fm(\030)1319 1098 y Fl(0)1331 1116 y Fn(\()p Fm(i)p Fn(\))h(:=)f Fm(\030)r Fn(\()p Fm(i)p Fn(\)\001\()p Fm(i;)8 b Fi(\000)p Fn(\).)0 1176 y(Then)16 b(w)o(e)e(ha)o(v)o(e)h Fm(\030)332 1158 y Fl(0)344 1176 y Fn(\()p Fm(i)p Fn(\))9 b Fi(\012)447 1183 y Fl(M)501 1176 y Fm(\021)527 1158 y Fl(0)552 1176 y Fn(=)14 b Fm(\030)r Fn(\()p Fm(i)p Fn(\)\001\()p Fm(i;)8 b Fi(\000)p Fn(\))g Fi(\012)886 1183 y Fl(M)939 1176 y Fm(\021)965 1158 y Fl(0)991 1176 y Fn(=)13 b Fm(\030)r Fn(\()p Fm(i)p Fn(\))d Fi(\012)1169 1183 y Fl(M)1222 1176 y Fn(\001\()p Fm(i;)e Fi(\000)p Fn(\))p Fm(\021)1405 1158 y Fl(0)1429 1176 y Fn(=)14 b Fm(\030)r Fn(\()p Fm(i)p Fn(\))c Fi(\012)1608 1183 y Fl(M)1661 1176 y Fm(\021)1685 1183 y Fj(i)1699 1176 y Fn(,)0 1236 y(since)15 b(\001\()p Fm(i;)8 b(j)s Fn(\))p Fm(\021)15 b Fn(=)f Fm(\016)r(\034)6 b Fn(\()p Fm(i;)i(j)s Fn(\))p Fm(\021)14 b Fn(=)g Fm(\021)618 1218 y Fl(0)630 1236 y Fm(\034)6 b Fn(\()p Fm(i;)i(j)s Fn(\))13 b(=)g Fm(\021)845 1243 y Fj(i)860 1236 y Fn(\()p Fm(j)s Fn(\).)p 1689 1209 V 1689 1234 2 25 v 1711 1234 V 1689 1236 24 2 v 0 1346 a Fw(6.17)18 b(Theorem:)44 b Fg(The)17 b(asso)q(ciativit)o(y)e(map)h Fm(\013)g Fg(is)g(surjectiv)o(e.)0 1431 y Fu(Pr)o(oof:)k Fn(By)c(6.16)h(the)f(generators)h Fm(x)11 b Fi(\012)771 1438 y Fl(M)826 1431 y Fn(\()p Fm(\021)r Fn(\()p Fm(i)p Fn(\))f Fi(\012)975 1438 y Fl(K)1015 1431 y Fm(\020)1036 1438 y Fj(i)1050 1431 y Fn(\))1069 1438 y Fj(i)p Fl(2)p Fj(N)1155 1431 y Fn(are)16 b(in)g(the)g(image)f (of)h Fm(\013)p Fn(.)p 1689 1405 V 1689 1429 2 25 v 1711 1429 V 1689 1431 24 2 v 0 1591 a(Bibliograph)o(y)90 1798 y([1])59 b(Bos,)17 b(W.,)h(and)g(W)l(ol\013,)g(G.:)24 b(A\016ne)16 b(R\177)-24 b(aume)17 b(I.)f(Mitt.Math.Sem.Gie)o(\031en,)f Fw(129)p Fn(,)201 1858 y(1978,)i(1-115.)90 1929 y([2])59 b(|,)15 b(|:)21 b(A\016ne)16 b(R\177)-24 b(aume)15 b(I)q(I.)g (Mitt.Math.Sem.)o(Gie\031en,)e Fw(130)p Fn(,)j(1978,)h(0-83.)90 2000 y([3])59 b(Gudder,)15 b(S.P)l(.,)g(and)h(Sc)o(hro)q(ec)o(k,)e(F.:) 21 b(Generalized)14 b(Con)o(v)o(exit)o(y)l(.)19 b(SIAM)14 b(J.Math.)201 2060 y(Anal.)21 b Fw(11)p Fn(,)15 b(1980,)j(984-1001.)90 2131 y([4])59 b(Kneser,)15 b(H.:)21 b(Kon)o(v)o(exe)14 b(Ra)q(\177)-26 b(ume.)21 b(Arc)o(h.d.Math.)f Fw(3)p Fn(,)c(1952,)h(198-206.)90 2202 y([5])59 b(Ostermann,)13 b(F.,)g(and)h(Sc)o(hmidt,)e(J.:)20 b(Der)13 b(baryzen)o(trisc)o(he)f (Kalk)q(\177)-26 b(ul)14 b(als)g(axioma-)201 2262 y(tisc)o(he)20 b(Grundlage)i(der)e(a\016nen)i(Geometrie.)33 b(J.Reine)20 b(u.Angew.Math.)p Fw(224)p Fn(,)201 2322 y(1966,)d(44-57.)90 2393 y([6])59 b(P)o(areigis,)19 b(B.:)26 b(Categories)19 b(and)h(F)l(unctors.)30 b(Academic)16 b(Press,)k(New)e(Y)l(ork)h(|)201 2453 y(London,)e(1970.)90 2524 y([7])59 b(|:)20 b(Non-additiv)o(e)14 b(ring)h(and)h(mo)q(dule)e(theory)h(I.)f(General)g(theory)h(of)h (monoids.)201 2584 y(Publ.Math.)21 b(\(Debrecen\))15 b Fw(24)p Fn(,)h(1977,)h(189-204.)p eop %%Page: 28 28 28 27 bop 90 265 a Fn([8])59 b(Pumpl)q(\177)-26 b(un,)13 b(D.:)20 b(Regularly)13 b(Ordered)h(Banac)o(h)f(Spaces)h(and)h(P)o (ositiv)o(ely)c(Con)o(v)o(ex)201 325 y(Spaces.)21 b(Results)16 b(in)g(Math.)21 b Fw(7)p Fn(,)16 b(1984,)i(85-112.)90 396 y([9])59 b(Pumpl)q(\177)-26 b(un,)12 b(D.,)h(and)g(R\177)-24 b(ohrl,)13 b(H.:)18 b(Banac)o(h)13 b(Spaces)f(and)h(T)l(otally)g(Con)o (v)o(ex)e(Spaces)201 456 y(I.)k(Comm.in)f(Alg.)20 b Fw(12)p Fn(,)c(1984,)h(953-1019.)66 527 y([10])59 b(|,)13 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Fw(29)p Fn(,)c(1949,)i(25-30.) 66 1182 y([15])59 b(Wic)o(k)o(enh\177)-24 b(auser,)15 b(A.:)23 b(P)o(ositiv)o(ely)15 b(Con)o(v)o(ex)i(Spaces.)24 b(Diplomarb)q(eit)16 b(FU)g(Hagen)201 1243 y(1987.)16 1478 y(Bo)q(do)h(P)o(areigis)16 1538 y(Math.)22 b(Inst.,)15 b(Univ.)20 b(M)q(\177)-26 b(unc)o(hen)16 1598 y(8000)18 b(M)q(\177)-26 b(unc)o(hen)16 b(2)16 1658 y(German)o(y)883 1478 y(Helm)o(ut)d(R\177)-24 b(ohrl)883 1538 y(9322)18 b(La)f(Jolla)f(F)l(arms)f(Rd.)883 1598 y(La)i(Jolla,)f(CA)g(92037)883 1658 y(USA)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF