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Fq(b)28 b Fp(2)g Fq(V)1848 5507 y Fo(G)1907 5492 y Fu(,)33 b(and)f Fq(')2220 5507 y Fo(s)2257 5492 y Fu([)p Fq(b)p Fu(])d(=)e(id)32 b(for)g(all)e Fq(b)f Fp(2)f Fq(V)3103 5507 y Fo(G)3162 5492 y Fu(.)0 5651 y(\(1\))34 b(Let)h Fq(')400 5666 y Fo(g)472 5651 y Fu(:)c Fq(G)607 5666 y Fn(1)678 5651 y Fp(\000)-59 b(!)31 b Fq(F)48 b Fu(b)s(e)35 b(the)g(iden)m(tit)m(y)g(map)f(on)h(the)g(v)m (ertices.)51 b(Let)35 b Fp(F)44 b Fu(b)s(e)35 b(de\014ned)i(with)d(the) h(w)m(ord)0 5767 y Fq(\013)29 b Fu(=)f(\()p Fq(abcd)p Fu(\),)33 b(and)g(let)f Fp(G)39 b Fu(b)s(e)33 b(de\014ned)h(with)e(the) i(w)m(ord)f Fq(\014)h Fu(=)28 b(\()p Fq(abcd)p Fu(\).)44 b(If)33 b(w)m(e)h(require)f(that)f(the)i(follo)m(wing)p eop %%Page: 7 7 7 6 bop 1235 251 a Fr(SEQUENTIAL)32 b(D)n(YNAMICAL)g(SYSTEMS)1197 b(7)0 450 y Fu(diagrams)31 b(comm)m(ute)1509 609 y Fq(k)1563 573 y Fo(n)2289 609 y Fq(k)2343 573 y Fo(n)p 1639 576 623 4 v 2178 574 a Fh(-)1898 531 y Fo(f)1932 539 y Fj(\013)1972 555 y(i)1500 999 y Fq(k)1554 963 y Fo(m)2280 999 y Fq(k)2334 963 y Fo(m)p 1649 966 603 4 v 2168 964 a Fh(-)1901 918 y Fo(g)1935 930 y Fj(\014)1970 946 y(i)p 1558 896 4 254 v 1560 896 a Fh(?)1439 786 y Fo(')1485 763 y Fg(\003)p 2338 896 V 2340 896 a Fh(?)2379 786 y Fo(')2425 763 y Fg(\003)0 1108 y Fu(then)i(the)g(diagram)1150 1255 y Fq(k)1204 1219 y Fo(n)1540 1255 y Fq(k)1594 1219 y Fo(n)p 1280 1222 233 4 v 1429 1220 a Fh(-)1360 1188 y Fo(f)1394 1196 y Fj(a)1140 1645 y Fq(k)1194 1609 y Fo(m)1530 1645 y Fq(k)1584 1609 y Fo(m)p 1289 1612 213 4 v 1419 1610 a Fh(-)1360 1578 y Fo(g)1394 1586 y Fj(a)p 1199 1542 4 254 v 1201 1542 a Fh(?)1080 1432 y Fo(')1126 1409 y Fg(\003)p 1589 1542 V 1591 1542 a Fh(?)1470 1432 y Fo(')1516 1409 y Fg(\003)1540 1255 y Fq(k)1594 1219 y Fo(n)1930 1255 y Fq(k)1984 1219 y Fo(n)p 1670 1222 233 4 v 1819 1220 a Fh(-)1753 1188 y Fo(f)1787 1200 y Fj(b)1530 1645 y Fq(k)1584 1609 y Fo(m)1920 1645 y Fq(k)1974 1609 y Fo(m)p 1679 1612 213 4 v 1809 1610 a Fh(-)1753 1578 y Fo(g)1787 1590 y Fj(b)p 1979 1542 4 254 v 1981 1542 a Fh(?)1860 1432 y Fo(')1906 1409 y Fg(\003)p 2369 1542 V 2371 1542 a Fh(?)2250 1432 y Fo(')2296 1409 y Fg(\003)1930 1255 y Fq(k)1984 1219 y Fo(n)2320 1255 y Fq(k)2374 1219 y Fo(n)p 2060 1222 233 4 v 2209 1220 a Fh(-)2143 1188 y Fo(f)2177 1196 y Fj(c)1920 1645 y Fq(k)1974 1609 y Fo(m)2310 1645 y Fq(k)2364 1609 y Fo(m)p 2069 1612 213 4 v 2199 1610 a Fh(-)2143 1578 y Fo(g)2177 1586 y Fj(c)p 2759 1542 4 254 v 2761 1542 a Fh(?)2640 1432 y Fo(')2686 1409 y Fg(\003)2320 1255 y Fq(k)2374 1219 y Fo(n)2710 1255 y Fq(k)2764 1219 y Fo(n)p 2450 1222 233 4 v 2599 1220 a Fh(-)2531 1188 y Fo(f)2565 1200 y Fj(d)2310 1645 y Fq(k)2364 1609 y Fo(m)2700 1645 y Fq(k)2754 1609 y Fo(m)p 2459 1612 213 4 v 2589 1610 a Fh(-)2531 1578 y Fo(g)2565 1590 y Fj(d)0 1754 y Fu(comm)m(utes,)g(and)f(th)m(us)i(w)m(e) f(ha)m(v)m(e)h(a)f(comm)m(utativ)m(e)e(diagram)g(for)h(the)h(global)d (up)s(date)j(functions.)0 1960 y(\(2\))g(Consider)i(the)f(graph)g (morphism)e Fq(')1535 1975 y Fo(g)1605 1960 y Fu(:)e Fq(G)1739 1975 y Fn(2)1808 1960 y Fp(\000)-59 b(!)29 b Fq(F)14 b Fu(,)34 b(giv)m(en)g(b)m(y)h(the)f(inclusion)f(on)h(the)g (v)m(ertices.)49 b(W)-8 b(e)0 2077 y(use)37 b(the)f(w)m(ords)h Fq(\013)d Fu(=)g(\()p Fq(abcd)p Fu(\))i(and)g Fq(\014)j Fu(=)33 b(\()p Fq(abc)p Fu(\).)54 b(Then)37 b(w)m(e)g(are)f(forced)h (to)e(require)i(that)e(the)i(follo)m(wing)0 2193 y(diagram)30 b(comm)m(utes.)1150 2340 y Fq(k)1204 2304 y Fo(n)1540 2340 y Fq(k)1594 2304 y Fo(n)p 1280 2307 233 4 v 1429 2305 a Fh(-)1360 2273 y Fo(f)1394 2281 y Fj(a)1140 2730 y Fq(k)1194 2694 y Fo(m)1530 2730 y Fq(k)1584 2694 y Fo(m)p 1289 2697 213 4 v 1419 2695 a Fh(-)1360 2663 y Fo(g)1394 2671 y Fj(a)p 1199 2627 4 254 v 1201 2627 a Fh(?)1080 2517 y Fo(')1126 2494 y Fg(\003)p 1589 2627 V 1591 2627 a Fh(?)1470 2517 y Fo(')1516 2494 y Fg(\003)1540 2340 y Fq(k)1594 2304 y Fo(n)1930 2340 y Fq(k)1984 2304 y Fo(n)p 1670 2307 233 4 v 1819 2305 a Fh(-)1753 2273 y Fo(f)1787 2285 y Fj(b)1530 2730 y Fq(k)1584 2694 y Fo(m)1920 2730 y Fq(k)1974 2694 y Fo(m)p 1679 2697 213 4 v 1809 2695 a Fh(-)1753 2663 y Fo(g)1787 2675 y Fj(b)p 1979 2627 4 254 v 1981 2627 a Fh(?)1860 2517 y Fo(')1906 2494 y Fg(\003)p 2369 2627 V 2371 2627 a Fh(?)2250 2517 y Fo(')2296 2494 y Fg(\003)1930 2340 y Fq(k)1984 2304 y Fo(n)2320 2340 y Fq(k)2374 2304 y Fo(n)p 2060 2307 233 4 v 2209 2305 a Fh(-)2143 2273 y Fo(f)2177 2281 y Fj(c)1920 2730 y Fq(k)1974 2694 y Fo(m)2310 2730 y Fq(k)2364 2694 y Fo(m)p 2069 2697 213 4 v 2199 2695 a Fh(-)2143 2663 y Fo(g)2177 2671 y Fj(c)p 2759 2627 4 254 v 2761 2627 a Fh(?)2640 2517 y Fo(')2686 2494 y Fg(\003)2320 2340 y Fq(k)2374 2304 y Fo(n)2710 2340 y Fq(k)2764 2304 y Fo(n)p 2450 2307 233 4 v 2599 2305 a Fh(-)2531 2273 y Fo(f)2565 2285 y Fj(d)2310 2730 y Fq(k)2364 2694 y Fo(m)2700 2730 y Fq(k)2754 2694 y Fo(m)p 2459 2697 213 4 v 2589 2695 a Fh(-)2536 2677 y Fn(id)0 2839 y Fu(So)38 b(w)m(e)i(ha)m(v)m(e)f(to)f(assume)h(that)g(the)f(partial)e(diagram)h (concerning)h Fq(f)2627 2854 y Fo(d)2706 2839 y Fu(comm)m(utes)g(with)g (the)h(iden)m(tit)m(y)0 2955 y(as)h(lo)m(w)m(er)g(arro)m(w,)h(since)f Fq(')1008 2919 y Fk(\000)p Fn(1)1008 2980 y Fo(g)1102 2955 y Fu(\()p Fq(d)p Fu(\))g(=)f Fp(;)p Fu(.)64 b(Again)39 b(w)m(e)i(ha)m(v)m(e)f(that)g(the)g(diagram)d(for)i(the)h(global)e(up)s (date)0 3071 y(functions)33 b(comm)m(utes.)0 3278 y(\(3\))45 b(Use)g Fq(')433 3293 y Fo(g)522 3278 y Fu(:)k Fq(G)675 3293 y Fn(3)763 3278 y Fp(\000)-59 b(!)48 b Fq(F)14 b Fu(,)48 b(with)c Fq(')1379 3293 y Fo(g)1419 3278 y Fu(\()p Fq(a)p Fu(\))49 b(=)g Fq(a)c Fu(and)g Fq(')2082 3293 y Fo(g)2122 3278 y Fu(\()p Fq(b)p Fu(\))k(=)g Fq(')2477 3293 y Fo(g)2517 3278 y Fu(\()p Fq(c)p Fu(\))f(=)h Fq(b)p Fu(,)g(and)c Fq(\013)k Fu(=)g(\()p Fq(abcd)p Fu(\),)f(resp.)0 3394 y Fq(\014)33 b Fu(=)28 b(\()p Fq(abc)p Fu(\).)44 b(Then)34 b(w)m(e)f(ha)m(v)m(e)h(to)e(consider)h(the)g(follo)m(wing)d (diagram:)1150 3588 y Fq(k)1204 3551 y Fo(n)1540 3588 y Fq(k)1594 3551 y Fo(n)p 1280 3555 233 4 v 1429 3553 a Fh(-)1360 3521 y Fo(f)1394 3529 y Fj(a)1140 3978 y Fq(k)1194 3941 y Fo(m)1530 3978 y Fq(k)1584 3941 y Fo(m)p 1289 3945 213 4 v 1419 3943 a Fh(-)1360 3911 y Fo(g)1394 3919 y Fj(a)p 1199 3875 4 254 v 1201 3875 a Fh(?)1080 3765 y Fo(')1126 3741 y Fg(\003)p 1589 3875 V 1591 3875 a Fh(?)1470 3765 y Fo(')1516 3741 y Fg(\003)1540 3588 y Fq(k)1594 3551 y Fo(n)1930 3588 y Fq(k)1984 3551 y Fo(n)p 1670 3555 233 4 v 1819 3553 a Fh(-)1753 3521 y Fo(f)1787 3533 y Fj(b)1530 3978 y Fq(k)1584 3941 y Fo(m)1920 3978 y Fq(k)1974 3941 y Fo(m)p 1679 3945 213 4 v 1809 3943 a Fh(-)1720 3911 y Fo(g)1754 3919 y Fj(c)1786 3911 y Fo(g)1820 3923 y Fj(b)p 1979 3875 4 254 v 1981 3875 a Fh(?)1860 3765 y Fo(')1906 3741 y Fg(\003)p 2369 3875 V 2371 3875 a Fh(?)2250 3765 y Fo(')2296 3741 y Fg(\003)1930 3588 y Fq(k)1984 3551 y Fo(n)2320 3588 y Fq(k)2374 3551 y Fo(n)p 2060 3555 233 4 v 2209 3553 a Fh(-)2143 3521 y Fo(f)2177 3529 y Fj(c)1920 3978 y Fq(k)1974 3941 y Fo(m)2310 3978 y Fq(k)2364 3941 y Fo(m)p 2069 3945 213 4 v 2199 3943 a Fh(-)2146 3924 y Fn(id)p 2759 3875 4 254 v 2761 3875 a Fh(?)2640 3765 y Fo(')2686 3741 y Fg(\003)2320 3588 y Fq(k)2374 3551 y Fo(n)2710 3588 y Fq(k)2764 3551 y Fo(n)p 2450 3555 233 4 v 2599 3553 a Fh(-)2531 3521 y Fo(f)2565 3533 y Fj(d)2310 3978 y Fq(k)2364 3941 y Fo(m)2700 3978 y Fq(k)2754 3941 y Fo(m)p 2459 3945 213 4 v 2589 3943 a Fh(-)2536 3924 y Fn(id)0 4108 y Fu(Observ)m(e)i(that)f (there)g(are)f(no)h(v)m(ertices)g(b)s(eing)f(mapp)s(ed)g(in)m(to)g Fq(c;)17 b(d)27 b Fp(2)h Fq(V)21 b Fu(\()p Fq(F)14 b Fu(\),)31 b(but)g(that)f(the)h(t)m(w)m(o)g(v)m(ertices)0 4224 y Fq(b)38 b Fu(and)f Fq(c)h Fu(in)e Fq(G)548 4239 y Fn(3)625 4224 y Fu(are)h(mapp)s(ed)g(to)h Fq(b)f Fu(in)g Fq(F)14 b Fu(.)58 b(Ob)m(viously)37 b(the)h(order)f(of)g Fq(g)2710 4239 y Fo(b)2782 4224 y Fu(and)g Fq(g)3023 4239 y Fo(c)3095 4224 y Fu(is)g(imp)s(ortan)m(t.)56 b(So)37 b(a)0 4340 y(map)32 b(b)s(et)m(w)m(een)j(the)e(w)m(ords)g(of)f(the)i(t) m(w)m(o)f(SDS)f(should)h(not)f(only)g(b)s(e)h(compatible)e(with)h(the)h (graph)g(map)0 4456 y(but)g(it)e(should)i(also)e(b)s(e)i(order)g (preserving)g(in)f(some)h(sense.)0 4663 y(\(4\))i(No)m(w)i(w)m(e)f (consider)h Fq(')980 4678 y Fo(g)1053 4663 y Fu(:)c Fq(G)1190 4678 y Fn(2)1262 4663 y Fp(\000)-59 b(!)33 b Fq(F)14 b Fu(,)36 b(resp.)54 b Fq(')1871 4678 y Fo(g)1944 4663 y Fu(:)33 b Fq(G)2081 4678 y Fn(3)2154 4663 y Fp(\000)-60 b(!)33 b Fq(F)14 b Fu(,)36 b(sending)g Fq(a)g Fu(to)f Fq(a)p Fu(,)i Fq(b)f Fu(to)g Fq(b)p Fu(,)h(and)f Fq(c)f Fu(to)h Fq(a)p Fu(,)0 4779 y(and)d Fq(\013)28 b Fu(=)g(\()p Fq(abcd)p Fu(\),)k(resp.)45 b Fq(\014)33 b Fu(=)28 b(\()p Fq(abc)p Fu(\).)44 b(Then)33 b(w)m(e)h(ha)m(v)m(e)g(to)e(consider)h (the)g(follo)m(wing)d(diagram:)1150 4973 y Fq(k)1204 4936 y Fo(n)1540 4973 y Fq(k)1594 4936 y Fo(n)p 1280 4940 233 4 v 1429 4938 a Fh(-)1360 4906 y Fo(f)1394 4914 y Fj(a)1140 5363 y Fq(k)1194 5326 y Fo(m)1530 5363 y Fq(k)1584 5326 y Fo(m)p 1289 5330 213 4 v 1419 5328 a Fh(-)1327 5296 y Fo(g)1361 5304 y Fj(c)1393 5296 y Fo(g)1427 5304 y Fj(a)p 1199 5260 4 254 v 1201 5260 a Fh(?)1080 5150 y Fo(')1126 5126 y Fg(\003)p 1589 5260 V 1591 5260 a Fh(?)1470 5150 y Fo(')1516 5126 y Fg(\003)1540 4973 y Fq(k)1594 4936 y Fo(n)1930 4973 y Fq(k)1984 4936 y Fo(n)p 1670 4940 233 4 v 1819 4938 a Fh(-)1753 4906 y Fo(f)1787 4918 y Fj(b)1530 5363 y Fq(k)1584 5326 y Fo(m)1920 5363 y Fq(k)1974 5326 y Fo(m)p 1679 5330 213 4 v 1809 5328 a Fh(-)1753 5296 y Fo(g)1787 5308 y Fj(b)p 1979 5260 4 254 v 1981 5260 a Fh(?)1860 5150 y Fo(')1906 5126 y Fg(\003)p 2369 5260 V 2371 5260 a Fh(?)2250 5150 y Fo(')2296 5126 y Fg(\003)1930 4973 y Fq(k)1984 4936 y Fo(n)2320 4973 y Fq(k)2374 4936 y Fo(n)p 2060 4940 233 4 v 2209 4938 a Fh(-)2143 4906 y Fo(f)2177 4914 y Fj(c)1920 5363 y Fq(k)1974 5326 y Fo(m)2310 5363 y Fq(k)2364 5326 y Fo(m)p 2069 5330 213 4 v 2199 5328 a Fh(-)2146 5309 y Fn(id)p 2759 5260 4 254 v 2761 5260 a Fh(?)2640 5150 y Fo(')2686 5126 y Fg(\003)2320 4973 y Fq(k)2374 4936 y Fo(n)2710 4973 y Fq(k)2764 4936 y Fo(n)p 2450 4940 233 4 v 2599 4938 a Fh(-)2531 4906 y Fo(f)2565 4918 y Fj(d)2310 5363 y Fq(k)2364 5326 y Fo(m)2700 5363 y Fq(k)2754 5326 y Fo(m)p 2459 5330 213 4 v 2589 5328 a Fh(-)2536 5309 y Fn(id)0 5535 y Fu(Observ)m(e)c(that)f(the)g(comp)s(osition)d(in) i(the)h(lo)m(w)m(er)f(ro)m(w)h(giv)m(es)g Fq(g)2236 5550 y Fo(b)2270 5535 y Fq(g)2317 5550 y Fo(c)2352 5535 y Fq(g)2399 5550 y Fo(a)2440 5535 y Fu(.)41 b(In)25 b(case)g(of)f(the)h (graph)f Fq(G)3428 5550 y Fn(2)3492 5535 y Fu(this)g(is)h(the)0 5651 y(global)g(up)s(date)j(function)g Fq(g)j Fu(=)c Fq(g)1211 5666 y Fo(c)1246 5651 y Fq(g)1293 5666 y Fo(b)1327 5651 y Fq(g)1374 5666 y Fo(a)1415 5651 y Fu(,)i(since)f Fq(b)g Fu(and)g Fq(c)g Fu(are)g(in)f(di\013eren)m(t)h(connected)h(comp) s(onen)m(ts)f(of)g(the)0 5767 y(graph,)j(so)g(that)f Fq(g)676 5782 y Fo(b)741 5767 y Fu(and)h Fq(g)976 5782 y Fo(c)1041 5767 y Fu(comm)m(ute)f(under)i(comp)s(osition.)40 b(In)31 b(the)g(case)h(of)e(the)h(graph)g Fq(G)3434 5782 y Fn(3)3473 5767 y Fu(,)g(ho)m(w)m(ev)m(er,)p eop %%Page: 8 8 8 7 bop 0 251 a Fr(8)932 b(REINHARD)23 b(LA)n(UBENBA)n(CHER)h(AND)h (BODO)g(P)-6 b(AREIGIS)0 450 y Fu(these)28 b(t)m(w)m(o)g(lo)s(cal)d(up) s(date)i(functions)g(do)g(not)g(comm)m(ute)g(in)f(general,)i(so)f(that) g(this)g(construction)g(should)0 566 y(not)g(giv)m(e)h(a)f(morphism)f (in)h(our)h(category)-8 b(,)29 b(and)e(w)m(e)i(ha)m(v)m(e)g(to)e (exclude)i(it.)41 b(This)28 b(can)g(b)s(e)f(done)i(b)m(y)f(lo)s(oking)0 683 y(at)33 b(order)g(preserving)h(maps)f(from)e(the)j(sub)m(w)m(ords)h (on)e(the)h(connected)h(comp)s(onen)m(ts)e(of)g Fq(G)3411 698 y Fn(2)3450 683 y Fu(,)h(resp.)46 b Fq(G)3834 698 y Fn(3)3873 683 y Fu(,)0 799 y(to)32 b(the)h(w)m(ord)g Fq(\013)h Fu(on)e(the)h(graph)f Fq(F)14 b Fu(.)0 996 y(No)m(w)33 b(w)m(e)h(\014x)f(some)f(notation)g(that)g(w)m(e)i(will)c (use)j(in)f(the)h(follo)m(wing)d(de\014nition.)0 1155 y(Let)h Fq(\014)228 1170 y Fn(\()p Fo(l)q Fn(\))340 1155 y Fu(denote)g(the)h(sub)m(w)m(ord)g(of)f Fq(\014)36 b Fu(whose)c(letters)f(b)s(elong)f(to)g(the)i(connected)g(comp)s(onen)m (t)f Fq(G)3607 1170 y Fn(\()p Fo(l)q Fn(\))3688 1155 y Fu(.)43 b(Let)0 1271 y Fp(j)p Fq(\014)6 b Fp(j)37 b Fu(denote)i(the)g(ordered)g(set)g(of)f(indices)g Fp(f)p Fu(1)p Fq(;)17 b(:)g(:)g(:)f(;)49 b Fu(length)32 b(of)g Fq(\014)6 b Fp(g)38 b Fu(and)g(let)g Fp(j)p Fq(\014)2940 1287 y Fn(\()p Fo(l)q Fn(\))3020 1271 y Fp(j)g Fu(denote)h(the)g (ordered)0 1387 y(subset)34 b(of)e(indices)h(of)f Fq(\014)897 1403 y Fn(\()p Fo(l)q Fn(\))1010 1387 y Fu(\(in)g Fp(j)p Fq(\014)6 b Fp(j)p Fu(\).)0 1547 y(Observ)m(e)42 b(that)e(the)h (\(unordered\))g(set)g Fp(j)p Fq(\014)6 b Fp(j)39 b Fu(decomp)s(oses)i (in)m(to)e(a)h(disjoin)m(t)f(union)h(of)f(the)i(\(unordered\))0 1663 y(sets)34 b Fp(j)p Fq(\014)274 1678 y Fn(\()p Fo(l)q Fn(\))354 1663 y Fp(j)p Fu(.)0 1826 y Fy(De\014nition)27 b(2.4.)37 b Fu(Let)25 b Fp(F)37 b Fu(=)1105 1745 y Fl(\000)1150 1826 y Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(a)p Fu(]\))p Fq(;)g Fu(\()p Fq(f)1625 1841 y Fo(i)1681 1826 y Fu(:)28 b Fq(k)1790 1790 y Fo(n)1864 1826 y Fp(\000)-59 b(!)27 b Fq(k)2063 1790 y Fo(n)2110 1826 y Fu(\))p Fq(;)17 b(\013)2255 1745 y Fl(\001)2325 1826 y Fu(\(with)25 b Fq(V)2635 1841 y Fo(F)2721 1826 y Fu(=)i Fp(f)p Fq(a)2925 1841 y Fn(1)2965 1826 y Fq(;)17 b(:)g(:)g(:)f(;)h(a)3235 1841 y Fo(n)3281 1826 y Fp(g)25 b Fu(and)h Fq(f)3587 1841 y Fo(i)3642 1826 y Fu(=)i Fq(f)3794 1841 y Fo(a)3831 1851 y Fj(i)3862 1826 y Fu(\))0 1954 y(and)33 b Fp(G)h Fu(=)386 1873 y Fl(\000)432 1954 y Fq(G;)17 b Fu(\()p Fq(k)s Fu([)p Fq(b)p Fu(]\))p Fq(;)g Fu(\()p Fq(g)907 1969 y Fo(j)971 1954 y Fu(:)27 b Fq(k)1079 1918 y Fo(m)1174 1954 y Fp(\000)-60 b(!)27 b Fq(k)1372 1918 y Fo(m)1439 1954 y Fu(\))p Fq(;)17 b(\014)1582 1873 y Fl(\001)1659 1954 y Fu(\(with)33 b Fq(V)1977 1969 y Fo(G)2063 1954 y Fu(=)28 b Fp(f)p Fq(b)2258 1969 y Fn(1)2298 1954 y Fq(;)17 b(:)g(:)g(:)e(;)i(b)2557 1969 y Fo(m)2624 1954 y Fp(g)p Fu(\))32 b(b)s(e)h(SDS.)0 2114 y(A)50 b Ft(\()p Fq(Z)7 b Ft(-\)morphism)48 b(of)j(se)-5 b(quential)50 b(dynamic)-5 b(al)49 b(systems)h Fq(')56 b Fu(:)h(\()p Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(a)p Fu(]\))p Fq(;)g Fu(\()p Fq(f)2943 2129 y Fo(i)3028 2114 y Fu(:)56 b Fq(k)3165 2078 y Fo(n)3269 2114 y Fp(\000)-59 b(!)56 b Fq(k)3497 2078 y Fo(n)3544 2114 y Fu(\))p Fq(;)17 b(\013)q Fu(\))56 b Fp(\000)-60 b(!)0 2231 y Fu(\()p Fq(G;)17 b Fu(\()p Fq(k)s Fu([)p Fq(b)p Fu(]\))p Fq(;)g Fu(\()p Fq(g)513 2246 y Fo(j)577 2231 y Fu(:)28 b Fq(k)686 2195 y Fo(m)780 2231 y Fp(\000)-60 b(!)28 b Fq(k)979 2195 y Fo(m)1045 2231 y Fu(\))p Fq(;)17 b(\014)6 b Fu(\))32 b(consists)h(of)241 2428 y(-)41 b(a)32 b(graph)g(morphism)f Fq(')1192 2443 y Fo(g)1260 2428 y Fu(:)c Fq(G)h Fp(\000)-59 b(!)27 b Fq(F)46 b Fu(\(rev)m(erse)35 b(direction!\),)241 2544 y(-)41 b(a)32 b(family)e(of)i(maps)g(\()p Fq(')1162 2559 y Fo(s)1199 2544 y Fu([)p Fq(b)p Fu(])c(:)g Fq(k)s Fu([)p Fq(')1522 2559 y Fo(g)1562 2544 y Fu(\()p Fq(b)p Fu(\)])g Fp(\000)-59 b(!)27 b Fq(k)s Fu([)p Fq(b)p Fu(])33 b Fp(j)g Fq(b)28 b Fp(2)g Fq(V)2342 2559 y Fo(G)2401 2544 y Fq(;)17 b(')2509 2559 y Fo(s)2545 2544 y Fu([)p Fq(b)p Fu(])28 b Fp(2)h Fq(Z)7 b Fu(\),)241 2660 y(-)41 b(and)32 b(a)h(family)d(of)i(order)g(preserving)i(maps)1608 2852 y Fl(e)-67 b Fq(')1660 2868 y Fn(\()p Fo(l)q Fn(\))1768 2852 y Fu(:)28 b Fp(j)p Fq(\014)1906 2868 y Fn(\()p Fo(l)q Fn(\))1986 2852 y Fp(j)g(\000)-60 b(!)27 b(j)p Fq(\013)q Fp(j)315 3044 y Fu(for)32 b(eac)m(h)h(connected)h(comp)s(onen)m(t)f Fq(G)1714 3060 y Fn(\()p Fo(l)q Fn(\))1827 3044 y Fu(of)f Fq(G)0 3241 y Fu(suc)m(h)i(that)241 3439 y(-)41 b Fp(8)p Fq(l)100 b Fp(8)p Fq(j)45 b Fp(2)39 b(j)p Fq(\014)827 3454 y Fn(\()p Fo(l)q Fn(\))908 3439 y Fp(j)f Fu(:)h Fq(')1104 3454 y Fo(g)1144 3439 y Fu(\()p Fq(\014)1237 3454 y Fo(j)1273 3439 y Fu(\))g(=)g Fq(\013)1538 3454 y Fn(~)-46 b Fo(')1573 3468 y Fi(\()p Fj(l)p Fi(\))1645 3454 y Fn(\()p Fo(j)t Fn(\))1736 3439 y Fu(,)41 b(i.e.)63 b(all)49 b Fl(e)-67 b Fq(')2198 3454 y Fn(\()p Fo(l)q Fn(\))2317 3439 y Fu(are)40 b(compatible)d(with)h(the)i(giv)m(en)f (graph)315 3558 y(morphism)30 b Fq(')834 3573 y Fo(g)874 3558 y Fu(.)241 3679 y(-)41 b(if)36 b Fq(i)h Fp(2)f(j)p Fq(\013)q Fp(j)h Fu(and)p 931 3624 64 4 v 37 w Fq(')995 3703 y Fn(\()p Fo(l)q Fn(\))1076 3679 y Fu(\()p Fq(i)p Fu(\))g(:=)f(\()p Fq(\014)1454 3694 y Fo(j)1490 3595 y Fl(\014)1490 3654 y(\014)1536 3679 y(e)-67 b Fq(')1588 3695 y Fn(\()p Fo(l)q Fn(\))1668 3679 y Fu(\()p Fq(j)6 b Fu(\))36 b(=)h Fq(i)p Fu(\))g(is)g(the)h(sub)m(w)m(ord)i(of)d Fq(\014)2885 3695 y Fn(\()p Fo(l)q Fn(\))3003 3679 y Fu(mapp)s(ed)h(in)m(to)f Fq(\013)3645 3694 y Fo(i)3710 3679 y Fu(then)315 3797 y(the)c(diagram)1509 3984 y Fq(k)1563 3948 y Fo(n)2289 3984 y Fq(k)2343 3948 y Fo(n)p 1639 3951 623 4 v 2178 3949 a Fh(-)1898 3906 y Fo(f)1932 3914 y Fj(\013)1972 3930 y(i)1500 4374 y Fq(k)1554 4338 y Fo(m)2280 4374 y Fq(k)2334 4338 y Fo(m)p 1649 4341 603 4 v 2168 4339 a Fh(-)1714 4218 y Ff(Q)1780 4288 y Fj(l)1816 4218 y Ff(Q)p 1883 4259 41 3 v 70 x Fj(')1923 4308 y Fi(\()p Fj(l)p Fi(\))1995 4288 y(\()p Fj(i)p Fi(\))2081 4267 y Fo(g)2115 4279 y Fj(\014)2150 4295 y(j)p 1558 4271 4 254 v 1560 4271 a Fh(?)1439 4161 y Fo(')1485 4138 y Fg(\003)p 2338 4271 V 2340 4271 a Fh(?)2379 4161 y Fo(')2425 4138 y Fg(\003)315 4512 y Fu(comm)m(utes,)39 b(where)g(the)f(pro)s(duct)1638 4437 y Fl(Q)p 1733 4503 46 3 v 1733 4541 a Fo(')1779 4558 y Fi(\()p Fj(l)p Fi(\))1851 4541 y Fn(\()p Fo(i)p Fn(\))1950 4512 y Fq(g)1997 4527 y Fo(\014)2037 4537 y Fj(j)2111 4512 y Fu(is)g(tak)m(en)h(in)e(the)h (order)g(of)g(the)g(en)m(tries)g(in)g(the)315 4654 y(sub)m(w)m(ord)p 699 4599 64 4 v 33 w Fq(')763 4677 y Fn(\()p Fo(l)q Fn(\))844 4654 y Fu(\()p Fq(i)p Fu(\).)43 b(\(If)p 1158 4599 V 32 w Fq(')1222 4677 y Fn(\()p Fo(l)q Fn(\))1303 4654 y Fu(\()p Fq(i)p Fu(\))31 b(is)h(the)g(empt)m(y)g(w)m(ord,)h(then)f (the)g(pro)s(duct)g(is)g(assumed)g(to)g(b)s(e)g(the)315 4777 y(iden)m(tit)m(y)g(map.\))0 4974 y(In)48 b(order)g(to)f(further)h (motiv)-5 b(ate)46 b(this)h(de\014nition)g(w)m(e)i(giv)m(e)e(a)h(few)g (examples)g(that)f(sho)m(w)i(that)f(the)0 5090 y(conditions)32 b(are)g(necessary)j(for)d(our)h(further)f(studies.)0 5244 y Fy(Example)d(2.5.)37 b Fu(\(1\))25 b(This)i(example)e(sho)m(ws)j (that)d(t)m(w)m(o)i(v)m(ery)h(simple)c(SDS)i(that)g(should)g(b)s(e)g (isomorphic,)0 5361 y(are)k(indeed)h(isomorphic.)41 b(Here)31 b(w)m(e)h(use)f Fq(k)s Fu([)p Fq(a)p Fu(])d(=)g Fq(k)s Fu([)p Fq(b)p Fu(])g(=)g Fq(k)33 b Fu(and)e(w)m(e)g(need)h(the)e(set)h (maps)g Fq(')3378 5376 y Fo(s)3414 5361 y Fu([)p Fq(b)p Fu(])e(:)e Fq(k)k Fp(\000)-59 b(!)27 b Fq(k)s Fu(.)0 5520 y(Let)37 b Fq(F)50 b Fu(=)35 b Fp(f)p Fq(a)p Fp(g)j Fu(b)s(e)f(the)h(one)f(v)m(ertex)i(graph.)58 b(F)-8 b(or)36 b Fq(i)g Fp(2)g Fq(k)41 b Fu(de\014ne)d(the)g(set)g(map)f Fq(p)3043 5535 y Fo(i)3106 5520 y Fu(:)f Fq(k)j Fp(\000)-60 b(!)36 b Fq(k)k Fu(to)d(b)s(e)h(the)0 5638 y(pro)5 b(jection)39 b(of)h Fq(k)i Fu(on)m(to)e(the)g(elemen)m(t)g Fq(i)g Fp(2)g Fq(k)s Fu(.)65 b(Let)39 b Fq(i;)17 b(j)46 b Fp(2)40 b Fq(k)j Fu(with)c Fq(i)h Fp(6)p Fu(=)g Fq(j)6 b Fu(.)65 b(Let)39 b Fp(F)50 b Fu(=)3295 5557 y Fl(\000)3341 5638 y Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu(\))p Fq(;)g Fu(\()p Fq(p)3712 5653 y Fo(i)3779 5638 y Fu(:)40 b Fq(k)0 5765 y Fp(\000)-60 b(!)28 b Fq(k)s Fu(\))p Fq(;)17 b Fu(\()p Fq(a)p Fu(\))408 5685 y Fl(\001)486 5765 y Fu(and)32 b Fp(G)i Fu(=)872 5685 y Fl(\000)917 5765 y Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu(\))p Fq(;)g Fu(\()p Fq(p)1288 5780 y Fo(j)1352 5765 y Fu(:)28 b Fq(k)i Fp(\000)-59 b(!)27 b Fq(k)s Fu(\))p Fq(;)17 b Fu(\()p Fq(a)p Fu(\))1896 5685 y Fl(\001)1941 5765 y Fu(.)44 b(Then)34 b Fp(F)2376 5738 y(\030)2377 5770 y Fu(=)2481 5765 y Fp(G)6 b Fu(.)p eop %%Page: 9 9 9 8 bop 1235 251 a Fr(SEQUENTIAL)32 b(D)n(YNAMICAL)g(SYSTEMS)1197 b(9)0 450 y Fu(W)-8 b(e)33 b(de\014ne)i(the)e(isomorphism)d Fq(')f Fu(:)f Fp(F)38 b(\000)-60 b(!)28 b(G)39 b Fu(b)m(y)34 b Fq(')1890 465 y Fo(g)1958 450 y Fu(=)29 b(id)e(:)i Fp(f)p Fq(a)p Fp(g)f(\000)-60 b(!)28 b(f)p Fq(a)p Fp(g)p Fu(,)33 b Fq(')2827 465 y Fo(s)2864 450 y Fu([)p Fq(a)p Fu(])28 b(=)h Fq(\031)j Fu(:)c Fq(k)k Fp(\000)-60 b(!)28 b Fq(k)36 b Fu(where)e Fq(\031)0 566 y Fu(is)27 b(some)h(bijectiv)m(e)g (map)f(from)f Fq(k)31 b Fu(to)c Fq(k)k Fu(with)d Fq(\031)t Fu(\()p Fq(i)p Fu(\))f(=)h Fq(j)6 b Fu(.)42 b(Finally)25 b(let)39 b Fl(e)-67 b Fq(')2590 582 y Fn(\(1\))2712 566 y Fu(=)27 b(id.)41 b(Then)29 b Fq(')f Fu(is)f(a)h(morphism)0 686 y(of)41 b(SDS)h(with)f(in)m(v)m(erse)i Fq( )k Fu(=)c(\(id)o Fq(;)17 b(\031)1359 650 y Fk(\000)p Fn(1)1453 686 y Fq(;)g Fu(id)o(\).)71 b(In)41 b(fact)h(w)m(e)h(ha)m(v)m(e)g Fq(')2497 701 y Fo(g)2537 686 y Fu(\()p Fq(\014)2630 701 y Fn(1)2669 686 y Fu(\))g(=)g(id)o(\()p Fq(a)p Fu(\))g(=)g Fq(\013)3310 701 y Ff(e)-48 b Fo(')3347 715 y Fi(\(1\))3430 701 y Fn(\(1\))3566 686 y Fu(and)42 b(the)0 806 y(diagram)1533 938 y Fq(k)729 b(k)p 1616 905 669 4 v 2202 903 a Fh(-)1919 871 y Fo(p)1955 881 y Fj(i)1533 1327 y Fq(k)g(k)p 1616 1295 V 2202 1293 a Fh(-)1916 1254 y Fo(p)1952 1264 y Fj(j)p 1558 1225 4 254 v 1560 1225 a Fh(?)1341 1115 y Fo(')1387 1091 y Fg(\003)1423 1115 y Fn(=)p Fo(\031)p 2338 1225 V 2340 1225 a Fh(?)2379 1115 y Fo(')2425 1091 y Fg(\003)2461 1115 y Fn(=)p Fo(\031)0 1439 y Fu(comm)m(utes.)0 1598 y(\(2\))48 b(Consider)h(the)g(follo)m(wing)d(t)m(w)m(o)j(SDS)f Fp(F)64 b Fu(=)1906 1517 y Fl(\000)1952 1598 y Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu(\))p Fq(;)g Fu(\()p Fq(f)2322 1613 y Fo(i)2349 1598 y Fu(\))p Fq(;)g(\013)2494 1517 y Fl(\001)2588 1598 y Fu(and)48 b Fp(G)61 b Fu(=)3044 1517 y Fl(\000)3089 1598 y Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu(\))p Fq(;)g Fu(\()p Fq(g)3458 1613 y Fo(i)3485 1598 y Fu(\))p Fq(;)g(\013)3630 1517 y Fl(\001)3724 1598 y Fu(o)m(v)m(er)0 1726 y Fq(k)31 b Fu(=)c Fp(f)p Fu(0)p Fq(;)17 b Fu(1)p Fp(g)32 b Fu(where)i Fq(V)798 1741 y Fo(F)884 1726 y Fu(=)27 b Fp(f)p Fq(a;)17 b(b)p Fp(g)33 b Fu(and)f Fq(E)1517 1741 y Fo(F)1604 1726 y Fu(=)1707 1645 y Fl(\010)1766 1726 y Fp(f)p Fq(a;)17 b(b)p Fp(g)2002 1645 y Fl(\011)2060 1726 y Fu(.)43 b(Let)989 1900 y Fq(f)1037 1915 y Fo(a)1079 1900 y Fu(\()p Fq(x)1172 1915 y Fn(1)1212 1900 y Fq(;)17 b(x)1311 1915 y Fn(2)1351 1900 y Fu(\))27 b(=)h(\()p Fq(x)1613 1915 y Fn(2)1652 1900 y Fq(;)17 b(x)1751 1915 y Fn(2)1791 1900 y Fu(\))p Fq(;)195 b(f)2099 1915 y Fo(b)2134 1900 y Fu(\()p Fq(x)2227 1915 y Fn(1)2266 1900 y Fq(;)17 b(x)2365 1915 y Fn(2)2405 1900 y Fu(\))28 b(=)f(\()p Fq(x)2667 1915 y Fn(1)2707 1900 y Fq(;)p 2751 1845 95 4 v 17 w(x)2806 1915 y Fn(2)2846 1900 y Fu(\))p Fq(;)989 2016 y(g)1036 2031 y Fo(a)1078 2016 y Fu(\()p Fq(x)1171 2031 y Fn(1)1211 2016 y Fq(;)17 b(x)1310 2031 y Fn(2)1349 2016 y Fu(\))28 b(=)f(\()p 1556 1961 V Fq(x)1611 2031 y Fn(2)1651 2016 y Fq(;)17 b(x)1750 2031 y Fn(2)1790 2016 y Fu(\))p Fq(;)196 b(g)2098 2031 y Fo(b)2132 2016 y Fu(\()p Fq(x)2225 2031 y Fn(1)2265 2016 y Fq(;)17 b(x)2364 2031 y Fn(2)2404 2016 y Fu(\))27 b(=)h(\()p Fq(x)2666 2031 y Fn(1)2706 2016 y Fq(;)p 2750 1961 V 17 w(x)2805 2031 y Fn(2)2844 2016 y Fu(\))p Fq(:)0 2178 y Fu(The)j(up)s(date)g(sc)m(hedule)h Fq(\013)f Fu(is)f(arbitrary) -8 b(.)41 b(It)31 b(is)f(not)g(clear)g(if)f(these)j(SDS)e(are)g (isomorphic,)f(in)h(particular)0 2295 y(if)37 b(they)h(giv)m(e)g (isomorphic)e(state)i(spaces)i(as)e(describ)s(ed)g(in)f(section)h(3.)59 b(If)38 b(w)m(e)h(de\014ne)g Fq(')e Fu(:)f Fp(F)46 b(\000)-59 b(!)36 b(G)44 b Fu(b)m(y)0 2411 y Fq(')64 2426 y Fo(g)132 2411 y Fu(=)27 b(id,)32 b Fq(')440 2426 y Fo(s)477 2411 y Fu([)p Fq(a)p Fu(])c(=)f Fq(\034)11 b Fu(,)34 b Fq(')891 2426 y Fo(s)927 2411 y Fu([)p Fq(b)p Fu(])29 b(=)e(id)o(,)33 b(and)45 b Fl(e)-67 b Fq(')1549 2426 y Fn(\(1\))1671 2411 y Fu(=)27 b(id)32 b(then)h(this)f(is)g(an)h(isomorphism)d(since)j (the)g(diagrams)1088 2619 y Fq(k)1142 2583 y Fn(2)1478 2619 y Fq(k)1532 2583 y Fn(2)p 1210 2581 240 4 v 1367 2579 a Fh(-)1294 2547 y Fo(f)1328 2555 y Fj(a)1088 3009 y Fq(k)1142 2972 y Fn(2)1478 3009 y Fq(k)1532 2972 y Fn(2)p 1210 2971 V 1367 2969 a Fh(-)1294 2937 y Fo(g)1328 2945 y Fj(a)p 1133 2901 4 254 v 1135 2901 a Fh(?)806 2791 y Fo(')852 2768 y Fg(\003)888 2791 y Fn(=)p Fo(\034)8 b Fk(\002)p Fn(id)p 1523 2901 V 1525 2901 a Fh(?)1564 2791 y Fo(')1610 2768 y Fg(\003)1646 2791 y Fn(=)p Fo(\034)g Fk(\002)p Fn(id)2328 2619 y Fq(k)2382 2583 y Fn(2)2718 2619 y Fq(k)2772 2583 y Fn(2)p 2450 2581 240 4 v 2607 2579 a Fh(-)2537 2547 y Fo(f)2571 2559 y Fj(b)2328 3009 y Fq(k)2382 2972 y Fn(2)2718 3009 y Fq(k)2772 2972 y Fn(2)p 2450 2971 V 2607 2969 a Fh(-)2538 2937 y Fo(g)2572 2949 y Fj(b)p 2373 2901 4 254 v 2375 2901 a Fh(?)2047 2791 y Fo(')2093 2768 y Fg(\003)2129 2791 y Fn(=)p Fo(\034)g Fk(\002)p Fn(id)p 2763 2901 V 2765 2901 a Fh(?)2804 2791 y Fo(')2850 2768 y Fg(\003)2886 2791 y Fn(=)p Fo(\034)g Fk(\002)p Fn(id)0 3135 y Fu(comm)m(ute.)0 3294 y(\(3\))32 b(W)-8 b(e)32 b(determine)g(the)h(set)g(of)f(all)e(morphisms)h(in)g (some)h(v)m(ery)i(simple)d(cases.)44 b(Let)33 b Fq(k)s Fu([)p Fq(a)p Fu(])28 b(=)f Fq(k)s Fu([)p Fq(b)p Fu(])i(=)e Fq(k)k Fu(=)0 3411 y Fp(f)p Fu(0)p Fq(;)17 b Fu(1)p Fp(g)p Fu(.)41 b(W)-8 b(e)30 b(consider)g(4)f(SDS)g(all)f(de\014ned)i(on)g (the)g(trivial)c(one)k(p)s(oin)m(t)f(graph)g Fp(f)p Fq(a)p Fp(g)g Fu(and)h(with)f(the)h(trivial)0 3527 y(one)j(letter)f(up)s(date) h(sc)m(hedule)h Fq(a)p Fu(.)44 b(Let)1450 3686 y Fp(I)35 b Fu(:=)28 b(\()p Fq(a;)17 b(k)s(;)g Fu(id)26 b(:)i Fq(k)j Fp(\000)-60 b(!)27 b Fq(k)s(;)17 b(a)p Fu(\))p Fq(;)1460 3802 y Fp(T)53 b Fu(:=)27 b(\()p Fq(a;)17 b(k)s(;)g(\034)39 b Fu(:)28 b Fq(k)j Fp(\000)-60 b(!)27 b Fq(k)s(;)17 b(a)p Fu(\))p Fq(;)1395 3918 y Fp(P)1464 3933 y Fn(0)1532 3918 y Fu(:=)28 b(\()p Fq(a;)17 b(k)s(;)g(p)1943 3933 y Fn(0)2009 3918 y Fu(:)28 b Fq(k)j Fp(\000)-60 b(!)27 b Fq(k)s(;)17 b(a)p Fu(\))p Fq(;)1395 4034 y Fp(P)1464 4049 y Fn(1)1532 4034 y Fu(:=)28 b(\()p Fq(a;)17 b(k)s(;)g(p)1943 4049 y Fn(1)2009 4034 y Fu(:)28 b Fq(k)j Fp(\000)-60 b(!)27 b Fq(k)s(;)17 b(a)p Fu(\))p Fq(;)0 4197 y Fu(where)38 b Fq(\034)49 b Fu(is)36 b(the)h(in)m(terc)m(hange)h(map)e(sending)i(1)e (to)h(0)g(and)g(0)f(to)h(1,)h(and)f(the)g Fq(p)2982 4212 y Fo(i)3047 4197 y Fu(are)g(the)h(maps)e(in)h(the)0 4313 y(preceding)c(example.)43 b(In)33 b(order)g(to)f(\014nd)h(isomorphic)e (SDS,)h(w)m(e)i(ha)m(v)m(e)g(to)e(\014nd)h(out)f(if)g(the)h(diagram) 1533 4503 y Fq(k)729 b(k)p 1616 4470 669 4 v 2202 4468 a Fh(-)1929 4436 y Fo(f)1533 4893 y Fq(k)g(k)p 1616 4860 V 2202 4858 a Fh(-)1932 4826 y Fo(g)p 1558 4790 4 254 v 1560 4790 a Fh(?)1439 4680 y Fo(')1485 4657 y Fg(\003)p 2338 4790 V 2340 4790 a Fh(?)2379 4680 y Fo(')2425 4657 y Fg(\003)0 5025 y Fu(comm)m(utes.)58 b(If)38 b Fq(f)47 b Fp(6)p Fu(=)36 b Fq(g)41 b Fu(this)c(is)g(only)g(the)h(case)h(for)e Fq(f)47 b Fu(=)36 b Fq(p)2221 5040 y Fn(0)2298 5025 y Fu(and)i Fq(g)h Fu(=)d Fq(p)2740 5040 y Fn(1)2817 5025 y Fu(\(or)h(con)m(v)m(ersely\).)61 b(Th)m(us)39 b(w)m(e)0 5142 y(ha)m(v)m(e)33 b(that)f(three)h(of)e(the)h(four)g(SDS)g(are)g (nonisomorphic.)41 b(More)33 b(generally)e(w)m(e)i(ha)m(v)m(e)g(for)e (the)i(n)m(um)m(b)s(er)0 5258 y(of)f(morphisms)354 5417 y Fp(j)p Fu(Mor\()p Fp(I)7 b Fq(;)17 b Fp(I)7 b Fu(\))p Fp(j)28 b Fu(=)f(4)145 b Fp(j)p Fu(Mor\()p Fp(T)25 b Fq(;)17 b Fp(T)25 b Fu(\))p Fp(j)j Fu(=)f(2)109 b Fp(j)p Fu(Mor)o(\()p Fp(P)2262 5432 y Fn(0)2302 5417 y Fq(;)17 b Fp(P)2415 5432 y Fn(0)2455 5417 y Fu(\))p Fp(j)27 b Fu(=)g(2)98 b Fp(j)p Fu(Mor)o(\()p Fp(P)3108 5432 y Fn(1)3148 5417 y Fq(;)17 b Fp(P)3261 5432 y Fn(1)3301 5417 y Fu(\))p Fp(j)27 b Fu(=)g(2)354 5533 y Fp(j)p Fu(Mor\()p Fp(I)7 b Fq(;)17 b Fp(T)25 b Fu(\))p Fp(j)j Fu(=)f(0)127 b Fp(j)p Fu(Mor\()p Fp(T)25 b Fq(;)17 b Fp(I)7 b Fu(\))p Fp(j)28 b Fu(=)f(2)127 b Fp(j)p Fu(Mor)o(\()p Fp(P)2262 5548 y Fn(0)2302 5533 y Fq(;)17 b Fp(P)2415 5548 y Fn(1)2455 5533 y Fu(\))p Fp(j)27 b Fu(=)g(2)98 b Fp(j)p Fu(Mor)o(\()p Fp(P)3108 5548 y Fn(1)3148 5533 y Fq(;)17 b Fp(P)3261 5548 y Fn(0)3301 5533 y Fu(\))p Fp(j)27 b Fu(=)g(2)354 5649 y Fp(j)p Fu(Mor\()p Fp(I)7 b Fq(;)17 b Fp(P)770 5664 y Fn(0)810 5649 y Fu(\))p Fp(j)27 b Fu(=)g(1)98 b Fp(j)p Fu(Mor\()p Fp(P)1464 5664 y Fn(0)1503 5649 y Fq(;)17 b Fp(I)7 b Fu(\))p Fp(j)28 b Fu(=)f(2)98 b Fp(j)p Fu(Mor)o(\()p Fp(T)26 b Fq(;)17 b Fp(P)2386 5664 y Fn(0)2425 5649 y Fu(\))p Fp(j)28 b Fu(=)f(1)127 b Fp(j)p Fu(Mor)o(\()p Fp(P)3108 5664 y Fn(0)3148 5649 y Fq(;)17 b Fp(T)25 b Fu(\))p Fp(j)j Fu(=)f(0)354 5765 y Fp(j)p Fu(Mor\()p Fp(I)7 b Fq(;)17 b Fp(P)770 5780 y Fn(1)810 5765 y Fu(\))p Fp(j)27 b Fu(=)g(1)98 b Fp(j)p Fu(Mor\()p Fp(P)1464 5780 y Fn(1)1503 5765 y Fq(;)17 b Fp(I)7 b Fu(\))p Fp(j)28 b Fu(=)f(2)98 b Fp(j)p Fu(Mor)o(\()p Fp(T)26 b Fq(;)17 b Fp(P)2386 5780 y Fn(1)2425 5765 y Fu(\))p Fp(j)28 b Fu(=)f(1)127 b Fp(j)p Fu(Mor)o(\()p Fp(P)3108 5780 y Fn(1)3148 5765 y Fq(;)17 b Fp(T)25 b Fu(\))p Fp(j)j Fu(=)f(0)p Fq(:)p eop %%Page: 10 10 10 9 bop 0 251 a Fr(10)894 b(REINHARD)23 b(LA)n(UBENBA)n(CHER)h(AND)h (BODO)g(P)-6 b(AREIGIS)0 450 y Fu(\(4\))27 b(W)-8 b(e)28 b(consider)g(no)m(w)g(some)g(morphisms)e(that)h(only)g(dep)s(end)i(on)e (the)h(c)m(hoice)g(of)f(the)h(order)g(preserving)0 566 y(maps)45 b Fl(e)-68 b Fq(')319 582 y Fn(\()p Fo(l)q Fn(\))433 566 y Fu(but)32 b(that)h(are)f(de\014ned)i(on)f(the)g(same)f (morphism)f Fq(')2389 581 y Fo(g)2462 566 y Fu(of)h(graphs.)0 725 y(Let)27 b Fq(F)40 b Fu(b)s(e)26 b(de\014ned)i(b)m(y)f Fq(V)914 740 y Fo(F)1001 725 y Fu(=)g Fp(f)p Fq(a)1205 740 y Fn(1)1245 725 y Fq(;)17 b(a)1340 740 y Fn(2)1379 725 y Fp(g)27 b Fu(=)h Fp(f)p Fq(u;)17 b(v)t Fp(g)25 b Fu(\(with)h Fq(u)h Fu(=)g Fq(a)2327 740 y Fn(1)2367 725 y Fu(,)h Fq(v)j Fu(=)d Fq(a)2655 740 y Fn(2)2694 725 y Fu(\))f(and)f Fq(E)3014 740 y Fo(F)3101 725 y Fu(=)h Fp(ff)p Fq(u;)17 b(v)t Fp(gg)p Fu(,)26 b(and)h(let)0 842 y Fq(\013)32 b Fu(=)f(\()p Fq(u;)17 b(v)t(;)g(u)p Fu(\).)47 b(Let)35 b Fq(k)s Fu([)p Fq(u)p Fu(])c(=)g Fq(k)s Fu([)p Fq(v)t Fu(])g(=)g Fq(k)s Fu(.)49 b(A)m(t)35 b(this)g(time)e(w)m(e)i(do)g(not)f(\014x)i(the)f(lo)s(cal)d(functions)i Fq(f)3539 857 y Fo(u)3585 842 y Fq(;)17 b(f)3677 857 y Fo(v)3748 842 y Fu(:)31 b Fq(k)3860 806 y Fn(2)0 958 y Fp(\000)-60 b(!)28 b Fq(k)199 922 y Fn(2)238 958 y Fu(.)43 b(F)-8 b(urthermore)31 b(let)f Fq(G)h Fu(b)s(e)g(de\014ned)i(b) m(y)f Fq(V)1776 973 y Fo(G)1862 958 y Fu(=)c Fp(f)p Fq(b)2057 973 y Fn(1)2097 958 y Fp(g)f Fu(=)h Fp(f)p Fq(w)s Fp(g)i Fu(and)h Fq(E)2741 973 y Fo(G)2828 958 y Fu(=)c Fp(;)p Fu(,)32 b(let)e Fq(k)s Fu([)p Fq(w)s Fu(])d(=)h Fq(k)s Fu(,)j(and)h(let)0 1074 y Fq(\014)h Fu(=)28 b(\()p Fq(w)s(;)17 b(w)s Fu(\).)39 b(F)-8 b(or)25 b Fq(G)g Fu(w)m(e)i(also)d(do)i(not)f (\014x)h(the)g(lo)s(cal)d(function)i Fq(g)2351 1089 y Fo(w)2435 1074 y Fu(:)j Fq(k)i Fp(\000)-59 b(!)27 b Fq(k)s Fu(.)41 b(De\014ne)26 b(a)f(graph)h(morphism)0 1190 y Fq(')64 1205 y Fo(g)132 1190 y Fu(:)j Fq(G)f Fp(\000)-60 b(!)28 b Fq(F)47 b Fu(b)m(y)34 b Fq(')748 1205 y Fo(g)788 1190 y Fu(\()p Fq(w)s Fu(\))27 b(=)i Fq(u)p Fu(.)44 b(W)-8 b(e)33 b(also)f(\014x)i Fq(')1763 1205 y Fo(s)1799 1190 y Fu([)p Fq(w)s Fu(])28 b(:)h Fq(k)i Fp(\000)-60 b(!)28 b Fq(k)36 b Fu(to)d(b)s(e)g(the)g(iden)m(tit)m(y)-8 b(.)45 b(Then)34 b Fq(')3456 1154 y Fk(\003)3523 1190 y Fu(:)29 b Fq(k)3633 1154 y Fn(2)3701 1190 y Fp(\000)-60 b(!)28 b Fq(k)0 1307 y Fu(is)34 b(the)h(pro)5 b(jection)34 b(on)m(to)g(the)h (\014rst)g(comp)s(onen)m(t)f(pr)1922 1330 y Fn(1)1961 1307 y Fu(,)h(since)g Fq(')2328 1270 y Fk(\003)2367 1307 y Fu(\()p Fq(x)p Fu([)p Fq(u)p Fu(])p Fq(;)17 b(x)p Fu([)p Fq(v)t Fu(]\))31 b(=)g Fq(x)p Fu([)p Fq(')3096 1322 y Fo(g)3136 1307 y Fu(\()p Fq(w)s Fu(\)])f(=)h Fq(x)p Fu([)p Fq(u)p Fu(].)49 b(Both)0 1423 y(graphs)28 b(ha)m(v)m(e)h(only)e(one)h (connected)i(comp)s(onen)m(t.)42 b(Our)27 b(de\014nitions)g(giv)m(e)h (ordered)g(sets)h Fp(j)p Fq(\013)q Fp(j)e Fu(=)h Fp(f)p Fu(1)p Fq(;)17 b Fu(2)p Fq(;)g Fu(3)p Fp(g)p Fu(,)0 1539 y Fp(j)p Fq(\014)6 b Fp(j)27 b Fu(=)g Fp(f)p Fu(1)p Fq(;)17 b Fu(2)p Fp(g)27 b Fu(=)g Fp(j)p Fq(\014)702 1555 y Fn(\(1\))796 1539 y Fp(j)p Fu(.)0 1698 y(F)-8 b(or)52 b Fl(e)-67 b Fq(')247 1714 y Fn(\()p Fo(l)q Fn(\))368 1698 y Fu(:)41 b Fp(j)p Fq(\014)519 1714 y Fn(\(1\))613 1698 y Fp(j)g(\000)-60 b(!)41 b(j)p Fq(\013)q Fp(j)e Fu(or)53 b Fl(e)-68 b Fq(')1189 1714 y Fn(\()p Fo(l)q Fn(\))1311 1698 y Fu(:)41 b Fp(f)p Fu(1)p Fq(;)17 b Fu(2)p Fp(g)40 b(\000)-60 b(!)41 b(f)p Fu(1)p Fq(;)17 b Fu(2)p Fq(;)g Fu(3)p Fp(g)38 b Fu(w)m(e)k(ha)m(v)m(e)f (the)g(c)m(hoice)g(among)e(three)i(\\order)0 1814 y(preserving)33 b(maps)g(of)f(ordered)h(m)m(ultisets")e(that)i(satisfy)f(the)h(\014rst) g(axiom)e(for)h(morphisms:)1075 1937 y Fq(w)1083 2132 y(u)p 1109 2042 4 59 v 1111 2042 a Fh(?)1270 1937 y Fq(w)1179 2042 y Fh(\011)1281 2132 y Fq(v)145 b(u)1723 1937 y(w)1732 2132 y(u)p 1758 2042 V 1759 2042 a Fh(?)1918 1937 y Fq(w)1929 2132 y(v)h(u)1998 2042 y Fh(R)2372 1937 y Fq(w)2380 2132 y(u)2544 2025 y Fh(H)2578 2042 y(H)-83 b(j)2567 1937 y Fq(w)2577 2132 y(v)146 b(u)2647 2042 y Fh(R)0 2279 y Fu(W)-8 b(e)33 b(w)m(an)m(t)g(to)g(establish)f(conditions)f(so)i (that)g(these)g(three)h(maps)e(de\014ne)i(morphisms)d(of)h(SDS.)0 2438 y(Case)i(1:)43 b(W)-8 b(e)33 b(ha)m(v)m(e)46 b Fl(e)-67 b Fq(')810 2454 y Fn(\(1\))904 2438 y Fu(\(1\))27 b(=)h(1)k(and)45 b Fl(e)-67 b Fq(')1495 2454 y Fn(\(1\))1589 2438 y Fu(\(2\))27 b(=)h(1.)43 b(So)33 b(the)g(follo)m(wing)c(diagrams)i(m)m(ust)i(comm)m (ute:)1026 2659 y Fq(k)1080 2622 y Fn(2)1611 2659 y Fq(k)1665 2622 y Fn(2)p 1148 2621 435 4 v 1499 2619 a Fh(-)1244 2577 y Fo(f)1278 2585 y Fj(\013)1318 2600 y Fi(1)1357 2577 y Fn(=)p Fo(f)1446 2585 y Fj(u)1046 3044 y Fq(k)533 b(k)p 1128 3011 474 4 v 1519 3009 a Fh(-)1189 3084 y Fo(g)1223 3096 y Fj(\014)1258 3111 y Fi(2)1296 3084 y Fo(g)1330 3096 y Fj(\014)1365 3111 y Fi(1)1403 3084 y Fn(=)p Fo(g)1494 3061 y Fi(2)1492 3101 y Fj(w)p 1071 2941 4 254 v 1073 2941 a Fh(?)908 2823 y Fe(pr)994 2845 y Fn(1)p 1656 2941 V 1658 2941 a Fh(?)1697 2823 y Fe(pr)1783 2845 y Fn(1)1611 2659 y Fq(k)1665 2622 y Fn(2)2196 2659 y Fq(k)2250 2622 y Fn(2)p 1733 2621 435 4 v 2084 2619 a Fh(-)1831 2577 y Fo(f)1865 2585 y Fj(\013)1905 2600 y Fi(2)1943 2577 y Fn(=)p Fo(f)2032 2585 y Fj(v)1630 3044 y Fq(k)h(k)p 1713 3011 474 4 v 2104 3009 a Fh(-)1921 3074 y Fn(id)p 2241 2941 4 254 v 2242 2941 a Fh(?)2281 2823 y Fe(pr)2368 2845 y Fn(1)2196 2659 y Fq(k)2250 2622 y Fn(2)2781 2659 y Fq(k)2835 2622 y Fn(2)p 2318 2621 435 4 v 2669 2619 a Fh(-)2414 2577 y Fo(f)2448 2585 y Fj(\013)2488 2600 y Fi(3)2526 2577 y Fn(=)p Fo(f)2615 2585 y Fj(u)2215 3044 y Fq(k)g(k)p 2298 3011 474 4 v 2689 3009 a Fh(-)2506 3074 y Fn(id)p 2826 2941 4 254 v 2827 2941 a Fh(?)2866 2823 y Fe(pr)2953 2845 y Fn(1)0 3241 y Fu(Conditions)41 b(for)h(the)g(lo)s(cal)e(maps)h(so)h(that)g (this)g(b)s(ecomes)g(a)g(morphism)e(of)i(SDS)f(are)h Fq(f)3411 3256 y Fo(u)3500 3241 y Fu(=)i(id)d(and)0 3357 y Fq(g)51 3321 y Fn(2)47 3382 y Fo(w)131 3357 y Fu(=)28 b(id)o(.)0 3516 y(Case)34 b(2:)43 b(W)-8 b(e)33 b(ha)m(v)m(e)46 b Fl(e)-67 b Fq(')810 3532 y Fn(\(1\))904 3516 y Fu(\(1\))27 b(=)h(1)k(and)45 b Fl(e)-67 b Fq(')1495 3532 y Fn(\(1\))1589 3516 y Fu(\(2\))27 b(=)h(3.)43 b(So)33 b(the)g(follo)m(wing)c(diagrams) i(m)m(ust)i(comm)m(ute:)1026 3736 y Fq(k)1080 3700 y Fn(2)1611 3736 y Fq(k)1665 3700 y Fn(2)p 1148 3699 435 4 v 1499 3697 a Fh(-)1244 3655 y Fo(f)1278 3663 y Fj(\013)1318 3678 y Fi(1)1357 3655 y Fn(=)p Fo(f)1446 3663 y Fj(u)1046 4121 y Fq(k)533 b(k)p 1128 4089 474 4 v 1519 4087 a Fh(-)1242 4135 y Fo(g)1276 4147 y Fj(\014)1311 4162 y Fi(1)1349 4135 y Fn(=)p Fo(g)1438 4143 y Fj(w)p 1071 4019 4 254 v 1073 4019 a Fh(?)908 3900 y Fe(pr)994 3922 y Fn(1)p 1656 4019 V 1658 4019 a Fh(?)1697 3900 y Fe(pr)1783 3922 y Fn(1)1611 3736 y Fq(k)1665 3700 y Fn(2)2196 3736 y Fq(k)2250 3700 y Fn(2)p 1733 3699 435 4 v 2084 3697 a Fh(-)1831 3655 y Fo(f)1865 3663 y Fj(\013)1905 3678 y Fi(2)1943 3655 y Fn(=)p Fo(f)2032 3663 y Fj(v)1630 4121 y Fq(k)h(k)p 1713 4089 474 4 v 2104 4087 a Fh(-)1921 4152 y Fn(id)p 2241 4019 4 254 v 2242 4019 a Fh(?)2281 3900 y Fe(pr)2368 3922 y Fn(1)2196 3736 y Fq(k)2250 3700 y Fn(2)2781 3736 y Fq(k)2835 3700 y Fn(2)p 2318 3699 435 4 v 2669 3697 a Fh(-)2414 3655 y Fo(f)2448 3663 y Fj(\013)2488 3678 y Fi(3)2526 3655 y Fn(=)p Fo(f)2615 3663 y Fj(u)2215 4121 y Fq(k)g(k)p 2298 4089 474 4 v 2689 4087 a Fh(-)2412 4135 y Fo(g)2446 4147 y Fj(\014)2481 4162 y Fi(1)2519 4135 y Fn(=)p Fo(g)2608 4143 y Fj(w)p 2826 4019 4 254 v 2827 4019 a Fh(?)2866 3900 y Fe(pr)2953 3922 y Fn(1)0 4297 y Fu(Conditions)33 b(for)h(the)h(lo)s(cal)d(maps)i (so)g(that)g(this)g(b)s(ecomes)h(a)f(morphism)e(of)i(SDS)g(are)g Fq(f)3303 4312 y Fo(u)3349 4297 y Fu(\()p Fq(x)p Fu([)p Fq(u)p Fu(])p Fq(;)17 b(x)p Fu([)p Fq(v)t Fu(]\))30 b(=)0 4413 y(\()p Fq(g)85 4428 y Fo(w)141 4413 y Fu(\()p Fq(x)p Fu([)p Fq(u)p Fu(]\))p Fq(;)17 b(x)p Fu([)p Fq(v)t Fu(]\).)0 4572 y(Case)34 b(3:)43 b(W)-8 b(e)33 b(ha)m(v)m(e)46 b Fl(e)-67 b Fq(')810 4587 y Fn(\(1\))904 4572 y Fu(\(1\))27 b(=)h(3)k(and)45 b Fl(e)-67 b Fq(')1495 4587 y Fn(\(1\))1589 4572 y Fu(\(2\))27 b(=)h(3.)43 b(So)33 b(the)g(follo)m(wing)c(diagrams) i(m)m(ust)i(comm)m(ute:)1026 4792 y Fq(k)1080 4756 y Fn(2)1611 4792 y Fq(k)1665 4756 y Fn(2)p 1148 4754 435 4 v 1499 4752 a Fh(-)1244 4711 y Fo(f)1278 4719 y Fj(\013)1318 4734 y Fi(1)1357 4711 y Fn(=)p Fo(f)1446 4719 y Fj(u)1046 5177 y Fq(k)533 b(k)p 1128 5145 474 4 v 1519 5143 a Fh(-)1336 5208 y Fn(id)p 1071 5075 4 254 v 1073 5075 a Fh(?)908 4956 y Fe(pr)994 4978 y Fn(1)p 1656 5075 V 1658 5075 a Fh(?)1697 4956 y Fe(pr)1783 4978 y Fn(1)1611 4792 y Fq(k)1665 4756 y Fn(2)2196 4792 y Fq(k)2250 4756 y Fn(2)p 1733 4754 435 4 v 2084 4752 a Fh(-)1831 4711 y Fo(f)1865 4719 y Fj(\013)1905 4734 y Fi(2)1943 4711 y Fn(=)p Fo(f)2032 4719 y Fj(v)1630 5177 y Fq(k)h(k)p 1713 5145 474 4 v 2104 5143 a Fh(-)1921 5208 y Fn(id)p 2241 5075 4 254 v 2242 5075 a Fh(?)2281 4956 y Fe(pr)2368 4978 y Fn(1)2196 4792 y Fq(k)2250 4756 y Fn(2)2781 4792 y Fq(k)2835 4756 y Fn(2)p 2318 4754 435 4 v 2669 4752 a Fh(-)2414 4711 y Fo(f)2448 4719 y Fj(\013)2488 4734 y Fi(3)2526 4711 y Fn(=)p Fo(f)2615 4719 y Fj(u)2215 5177 y Fq(k)g(k)p 2298 5145 474 4 v 2689 5143 a Fh(-)2359 5218 y Fo(g)2393 5230 y Fj(\014)2428 5245 y Fi(2)2466 5218 y Fo(g)2500 5230 y Fj(\014)2535 5245 y Fi(1)2573 5218 y Fn(=)p Fo(g)2664 5194 y Fi(2)2662 5234 y Fj(w)p 2826 5075 4 254 v 2827 5075 a Fh(?)2866 4956 y Fe(pr)2953 4978 y Fn(1)0 5374 y Fu(Conditions)41 b(for)h(the)g(lo)s(cal)e(maps)h(so)h(that)g(this)g (b)s(ecomes)g(a)g(morphism)e(of)i(SDS)f(are)h Fq(f)3411 5389 y Fo(u)3500 5374 y Fu(=)i(id)d(and)0 5491 y Fq(g)51 5454 y Fn(2)47 5515 y Fo(w)131 5491 y Fu(=)28 b(id)o(.)0 5650 y(\(5\))38 b(If)g Fq(k)s Fu([)p Fq(a)p Fu(])f(=)g([0)p Fq(;)17 b Fu(1])36 b Fp(\032)i Fm(R)49 b Fu(is)37 b(the)i(closed)f (unit)f(in)m(terv)-5 b(al)37 b(and)h Fq(k)s Fu([)p Fq(b)p Fu(])g(=)f Fp(f)p Fu(0)p Fq(;)2818 5610 y Fn(1)p 2818 5627 36 4 v 2818 5684 a(2)2863 5650 y Fq(;)17 b Fu(1)p Fp(g)p Fu(,)39 b(then)g(an)m(y)f(SDS)g(with)0 5767 y(state)30 b(spaces)h Fq(k)s Fu([)p Fq(a)p Fu(])f(can)g(b)s(e)g(considered)g(as)g (a)f(probabilistic)e(or)i(fuzzy)i(dynamical)d(system,)j(where)g(eac)m (h)p eop %%Page: 11 11 11 10 bop 1235 251 a Fr(SEQUENTIAL)32 b(D)n(YNAMICAL)g(SYSTEMS)1158 b(11)0 450 y Fu(v)m(ertex)31 b(has)f(states)g(b)s(et)m(w)m(een)h(0)e (and)g(1.)42 b(Some)29 b(in)m(teresting)g(lo)s(cal)e(up)s(date)i (functions)g Fq(f)3242 465 y Fo(a)3312 450 y Fu(:)e Fq(k)3420 414 y Fo(n)3495 450 y Fp(\000)-60 b(!)28 b Fq(k)3694 414 y Fo(n)3770 450 y Fu(are)0 566 y(the)j(ones)g(where)g(the)g Fq(a)p Fu(-th)f(comp)s(onen)m(t)g(is)g(the)g(pro)s(duct)h(of)e(all)g (states)i(in)e(a)h(1-neigh)m(b)s(orho)s(o)s(d)e(of)i Fq(x)3715 581 y Fo(a)3757 566 y Fu(.)43 b(A)0 683 y(discretization)29 b(of)h(suc)m(h)i(a)f(system)g(is)f(obtained)h(b)m(y)g(taking)f(the)h (iden)m(tit)m(y)f(maps)h(for)f Fq(')3245 698 y Fo(g)3315 683 y Fu(and)43 b Fl(e)-67 b Fq(')p Fu(,)31 b(and)g(as)0 801 y(maps)h(b)s(et)m(w)m(een)j(the)e(states)g(the)g(maps)g(that)f (send)i(0)e(to)g(0,)h(1)f(to)g(1,)h(and)f(all)f(other)h(v)-5 b(alues)33 b(to)3552 762 y Fn(1)p 3552 778 36 4 v 3552 835 a(2)3597 801 y Fu(.)0 965 y Fy(Theorem)40 b(2.6.)j Ft(The)36 b(se)-5 b(quential)37 b(dynamic)-5 b(al)36 b(systems)h Fu(\()p Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(a)p Fu(]\))p Fq(;)g Fu(\()p Fq(f)2684 980 y Fo(a)2725 965 y Fu(\))p Fq(;)g(\013)q Fu(\))36 b Ft(to)-5 b(gether)37 b(with)g(the)h(mor-)0 1081 y(phisms)c(of)g(SDS)g(form)h(a)f(c)-5 b(ate)g(gory)35 b Fy(SDS)p Ft(.)0 1355 y(Pr)-5 b(o)g(of.)41 b Fu(The)34 b(asso)s(ciativit)m(y)d(and)i(unit)f(la)m(ws)g(are)h (easily)f(c)m(hec)m(k)m(ed.)46 b(So)33 b(the)g(pro)s(of)e(follo)m(ws)h (from)250 b Fd(\003)0 1586 y Fy(Prop)s(osition)36 b(and)i(De\014nition) e(2.7.)42 b Ft(L)-5 b(et)513 1797 y Fq(')28 b Fu(:)g Fp(F)37 b Fu(=)873 1716 y Fl(\000)918 1797 y Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(a)p Fu(]\))p Fq(;)g Fu(\()p Fq(f)1393 1812 y Fn(1)1432 1797 y Fq(;)g(:)g(:)g(:)f(;)h(f)1699 1812 y Fo(n)1746 1797 y Fu(\))p Fq(;)g(\013)1891 1716 y Fl(\001)1963 1797 y Fp(\000)-57 b(!)28 b(G)34 b Fu(=)2307 1716 y Fl(\000)2353 1797 y Fq(G;)17 b Fu(\()p Fq(k)s Fu([)p Fq(b)p Fu(]\))p Fq(;)g Fu(\()p Fq(g)2828 1812 y Fn(1)2867 1797 y Fq(;)g(:)g(:)g(:)f(;)h(g)3133 1812 y Fo(m)3199 1797 y Fu(\))p Fq(;)g(\014)3342 1716 y Fl(\001)0 2007 y Ft(and)506 2176 y Fq( )32 b Fu(:)27 b Fp(G)34 b Fu(=)852 2095 y Fl(\000)897 2176 y Fq(G;)17 b Fu(\()p Fq(k)s Fu([)p Fq(b)p Fu(]\))p Fq(;)g Fu(\()p Fq(g)1372 2191 y Fn(1)1411 2176 y Fq(;)g(:)g(:)g(:)f(;)h(g)1677 2191 y Fo(m)1743 2176 y Fu(\))p Fq(;)g(\014)1886 2095 y Fl(\001)1959 2176 y Fp(\000)-57 b(!)27 b(H)i Fu(=)2322 2095 y Fl(\000)2368 2176 y Fq(H)r(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(c)p Fu(]\))p Fq(;)g Fu(\()p Fq(h)2859 2191 y Fn(1)2898 2176 y Fq(;)g(:)g(:)g(:)f(;)h(h)3173 2191 y Fo(r)3211 2176 y Fu(\))p Fq(;)g(\015)3349 2095 y Fl(\001)0 2366 y Ft(b)-5 b(e)35 b(two)f(morphisms)g(of)g(SDS.)g(Then)g(the)h(c)-5 b(omp)g(osition)1626 2576 y Fq( )26 b Fp(\016)c Fq(')27 b Fu(:)h Fp(F)37 b(\000)-57 b(!)27 b(H)0 2786 y Ft(is)35 b(a)f(morphism)g(of)g(SDS,)g(wher)-5 b(e)35 b(the)f(c)-5 b(omp)g(osition)34 b Fq( )26 b Fp(\016)c Fq(')35 b Ft(c)-5 b(onsists)34 b(of)241 2992 y Fu(-)41 b Ft(the)34 b(c)-5 b(omp)g(osite)34 b(of)h(the)g(asso)-5 b(ciate)g(d)34 b(gr)-5 b(aph)34 b(morphisms)f Fq(')2478 3007 y Fo(g)2518 2992 y Fq( )2581 3007 y Fo(g)2649 2992 y Fu(:)28 b Fq(H)35 b Fp(\000)-57 b(!)28 b Fq(F)14 b Ft(,)241 3108 y Fu(-)41 b Ft(the)34 b(family)h(of)f(maps)0 3318 y Fu(\(2\))941 b(\()p Fq( )26 b Fp(\016)c Fq(')p Fu(\))1367 3333 y Fo(s)1403 3318 y Fu([)p Fq(c)p Fu(])28 b(:=)g Fq( )1721 3333 y Fo(s)1758 3318 y Fu([)p Fq(c)p Fu(])p Fq(')1918 3333 y Fo(s)1955 3318 y Fu([)p Fq( )2045 3333 y Fo(g)2085 3318 y Fu(\()p Fq(c)p Fu(\)])35 b Ft(for)g(al)5 b(l)34 b Fq(c)28 b Fp(2)g Fq(H)r(;)241 3547 y Fu(-)41 b Ft(and)24 b(the)h(c)-5 b(omp)g(osite)24 b(of)g(the)h(c)-5 b(orr)g(esp)g(onding)23 b(families)h(of)h(or)-5 b(der)24 b(pr)-5 b(eserving)24 b(maps)3365 3520 y Fl(e)3348 3547 y Fq( )3411 3562 y Fn(\()p Fo(l)q Fn(\))3520 3547 y Fu(:)k Fp(j)p Fq(\015)3654 3562 y Fn(\()p Fo(l)q Fn(\))3734 3547 y Fp(j\000)-57 b(!)o(j)p Fq(\014)6 b Fp(j)315 3663 y Ft(and)46 b Fl(e)-67 b Fq(')568 3678 y Fn(\()p Fo(l)617 3659 y Fg(0)639 3678 y Fn(\))699 3663 y Fu(:)27 b Fp(j)p Fq(\014)836 3678 y Fn(\()p Fo(l)885 3659 y Fg(0)908 3678 y Fn(\))939 3663 y Fp(j\000)-57 b(!j)p Fq(\013)q Fp(j)p Ft(.)0 3937 y(Pr)-5 b(o)g(of.)41 b Fu(The)34 b(comp)s(osite)d(of)h(the)h(graph)g(morphisms) e(is)h(ob)m(viously)g(again)g(a)g(graph)g(morphism.)0 4096 y(Since)h(connected)i(comp)s(onen)m(ts)e(of)g(the)g(graph)g Fq(H)41 b Fu(are)33 b(mapp)s(ed)f(in)m(to)h(connected)h(comp)s(onen)m (ts)g(of)f(the)0 4212 y(graph)i Fq(G)p Fu(,)i(w)m(e)f(\014nd)g(a)f (connected)j(comp)s(onen)m(t)d Fq(G)1885 4228 y Fn(\()p Fo(l)1934 4209 y Fg(0)1957 4228 y Fn(\))2024 4212 y Fu(in)m(to)g(whic)m (h)h(the)g(connected)h(comp)s(onen)m(t)e Fq(H)3718 4228 y Fn(\()p Fo(l)q Fn(\))3834 4212 y Fu(is)0 4351 y(mapp)s(ed)30 b(hence)i(w)m(e)f(get)956 4325 y Fl(e)939 4351 y Fq( )1002 4366 y Fn(\()p Fo(l)q Fn(\))1111 4351 y Fu(:)d Fp(j)p Fq(\015)1245 4366 y Fn(\()p Fo(l)q Fn(\))1325 4351 y Fp(j\000)-60 b(!j)p Fq(\014)1553 4366 y Fn(\()p Fo(l)1602 4347 y Fg(0)1624 4366 y Fn(\))1656 4351 y Fp(j)30 b Fu(so)g(that)g(the) h(order)f(preserving)h(maps)f(can)g(b)s(e)h(comp)s(osed)0 4467 y(to)1317 4650 y Fm(])1307 4679 y Fu(\()p Fq(' )t Fu(\))1514 4708 y Fn(\()p Fo(l)q Fn(\))1623 4679 y Fu(:)d Fp(j)p Fq(\015)1757 4694 y Fn(\()p Fo(l)q Fn(\))1837 4679 y Fp(j)1905 4586 y Ff(e)1892 4604 y Fo( )1938 4618 y Fi(\()p Fj(l)p Fi(\))1893 4679 y Fp(\000)-60 b(!)28 b(j)p Fq(\014)2121 4694 y Fn(\()p Fo(l)2170 4675 y Fg(0)2192 4694 y Fn(\))2224 4679 y Fp(j)2289 4599 y Ff(e)-49 b Fo(')2325 4618 y Fi(\()p Fj(l)2369 4604 y Fg(0)2392 4618 y Fi(\))2291 4679 y Fp(\000)-60 b(!)39 b(j)p Fq(\013)q Fp(j)p Fq(:)0 4924 y Fu(The)h(\014rst)f(axiom)e(is)h(easily)g(v)m (eri\014ed:)57 b(let)38 b Fq(H)1709 4939 y Fn(\()p Fo(l)q Fn(\))1829 4924 y Fu(b)s(e)h(a)f(connected)j(comp)s(onen)m(t)d(of)h Fq(H)8 b Fu(.)61 b(Let)39 b Fq(j)44 b Fp(2)39 b(j)p Fq(\015)3765 4939 y Fn(\()p Fo(l)q Fn(\))3845 4924 y Fp(j)p Fu(.)0 5040 y(Then)34 b Fq(')319 5055 y Fo(g)359 5040 y Fq( )422 5055 y Fo(g)462 5040 y Fu(\()p Fq(\015)551 5055 y Fo(j)587 5040 y Fu(\))28 b(=)f Fq(')820 5055 y Fo(g)860 5040 y Fu(\()p Fq(\014)965 5054 y Ff(e)953 5071 y Fo( )999 5085 y Fi(\()p Fj(l)p Fi(\))1072 5071 y Fn(\()p Fo(j)t Fn(\))1163 5040 y Fu(\))h(=)f Fq(\013)1403 5071 y Ff(e)-48 b Fo(')1440 5090 y Fi(\()p Fj(l)1484 5076 y Fg(0)1507 5090 y Fi(\))1547 5054 y Ff(e)1535 5071 y Fo( )1581 5085 y Fi(\()p Fj(l)p Fi(\))1653 5071 y Fn(\()p Fo(j)t Fn(\))1772 5040 y Fu(=)28 b Fq(\013)1946 5061 y Fc(])1938 5080 y Fn(\()p Fo(' )r Fn(\))2087 5100 y Fi(\()p Fj(l)p Fi(\))2163 5040 y Fu(\()p Fq(j)6 b Fu(\).)0 5233 y(No)m(w)33 b(observ)m(e)i(that)964 5417 y Fp(h)p Fu(\()p Fq( )26 b Fp(\016)c Fq(')p Fu(\))1304 5381 y Fk(\003)1343 5417 y Fu(\()p Fq(x)p Fu(\))p Fq(;)17 b(c)p Fp(i)83 b Fu(=)27 b(\()p Fq( )f Fp(\016)c Fq(')p Fu(\))2086 5432 y Fo(s)2123 5417 y Fu([)p Fq(c)p Fu(]\()p Fp(h)p Fq(x;)17 b Fu(\()p Fq( )26 b Fp(\016)c Fq(')p Fu(\))2696 5432 y Fo(g)2736 5417 y Fu(\()p Fq(c)p Fu(\))p Fp(i)p Fu(\))1682 5533 y(=)27 b Fq( )1848 5548 y Fo(s)1886 5533 y Fu([)p Fq(c)p Fu(])p Fq(')2046 5548 y Fo(s)2082 5533 y Fu([)p Fq( )2172 5548 y Fo(g)2213 5533 y Fu(\()p Fq(c)p Fu(\)]\()p Fp(h)p Fq(x;)17 b(')2598 5548 y Fo(g)2638 5533 y Fq( )2701 5548 y Fo(g)2741 5533 y Fu(\()p Fq(c)p Fu(\))p Fp(i)p Fu(\))1682 5649 y(=)27 b Fq( )1848 5664 y Fo(s)1886 5649 y Fu([)p Fq(c)p Fu(]\()p Fp(h)p Fq(')2123 5613 y Fk(\003)2162 5649 y Fu(\()p Fq(x)p Fu(\))p Fq(;)17 b( )2400 5664 y Fo(g)2440 5649 y Fu(\()p Fq(c)p Fu(\))p Fp(i)p Fu(\))1682 5765 y(=)27 b Fp(h)p Fq( )1891 5729 y Fk(\003)1931 5765 y Fq(')1995 5729 y Fk(\003)2034 5765 y Fu(\()p Fq(x)p Fu(\))p Fq(;)17 b(c)p Fp(i)p eop %%Page: 12 12 12 11 bop 0 251 a Fr(12)894 b(REINHARD)23 b(LA)n(UBENBA)n(CHER)h(AND)h (BODO)g(P)-6 b(AREIGIS)0 450 y Fu(whic)m(h)33 b(implies)d(\()p Fq( )c Fp(\016)c Fq(')p Fu(\))911 414 y Fk(\003)978 450 y Fu(=)28 b Fq( )1149 414 y Fk(\003)1188 450 y Fq(')1252 414 y Fk(\003)1292 450 y Fu(.)43 b(So)32 b(w)m(e)i(ha)m(v)m(e)g(to)e (sho)m(w)i(that)e(the)h(diagram)1607 660 y Fq(k)1661 624 y Fo(n)2192 660 y Fq(k)2246 624 y Fo(n)p 1736 627 428 4 v 2080 625 a Fh(-)1898 582 y Fo(f)1932 590 y Fj(\013)1972 606 y(i)1597 1050 y Fq(k)1651 1014 y Fo(m)2182 1050 y Fq(k)2236 1014 y Fo(m)p 1746 1017 408 4 v 2071 1015 a Fh(-)1858 910 y Ff(Q)1937 960 y Fo(g)1971 972 y Fj(\014)2006 987 y(j)1612 1440 y Fq(k)1666 1404 y Fo(r)2196 1440 y Fq(k)2250 1404 y Fo(r)p 1733 1408 435 4 v 2084 1406 a Fh(-)1852 1319 y Ff(Q)1930 1369 y Fo(h)1971 1377 y Fj(\015)2003 1385 y(u)p 1656 947 4 254 v 1658 947 a Fh(?)1537 838 y Fo(')1583 814 y Fg(\003)p 2241 947 V 2242 947 a Fh(?)2281 838 y Fo(')2327 814 y Fg(\003)p 1656 1337 V 1658 1337 a Fh(?)1534 1227 y Fo( )1582 1204 y Fg(\003)p 2241 1337 V 2242 1337 a Fh(?)2281 1227 y Fo( )2329 1204 y Fg(\003)0 1575 y Fu(comm)m(utes.)67 b(The)42 b(middle)c(arro)m(w)j(can)g(b)s(e)g (decomp)s(osed)g(in)m(to)2424 1501 y Fl(Q)2519 1604 y Fo(l)2561 1501 y Fl(Q)p 2655 1566 46 3 v 103 x Fo(')2701 1622 y Fi(\()p Fj(l)p Fi(\))2773 1604 y Fn(\()p Fo(i)p Fn(\))2873 1575 y Fq(g)2920 1590 y Fo(\014)2960 1600 y Fj(j)3037 1575 y Fu(=)g Fq(g)3201 1590 y Fo(\014)3241 1598 y Fj(t)3289 1575 y Fq(:)17 b(:)g(:)f(g)3467 1590 y Fo(\014)3507 1599 y Fi(1)3586 1575 y Fu(and)40 b(for)0 1712 y(eac)m(h)33 b Fq(g)266 1727 y Fo(\014)306 1737 y Fj(j)375 1712 y Fu(w)m(e)h(get)e(asso)s(ciated)h(diagrams)d(from)i Fq( )k Fu(so)d(that)f(the)h(total)e(diagram)681 1932 y Fq(k)735 1895 y Fo(n)3118 1932 y Fq(k)3172 1895 y Fo(n)p 810 1899 2280 4 v 3007 1897 a Fh(-)1898 1853 y Fo(f)1932 1861 y Fj(\013)1972 1877 y(i)671 2321 y Fq(k)725 2285 y Fo(m)1646 2321 y Fq(k)1700 2285 y Fo(m)p 820 2289 798 4 v 1535 2287 a Fh(-)1165 2241 y Fo(g)1199 2253 y Fj(\014)1234 2268 y Fi(1)2133 2321 y Fq(k)2187 2285 y Fo(m)3108 2321 y Fq(k)3162 2285 y Fo(m)p 2282 2289 V 2997 2287 a Fh(-)2631 2243 y Fo(g)2665 2255 y Fj(\014)2700 2268 y(t)685 2711 y Fq(k)739 2675 y Fo(r)1660 2711 y Fq(k)1714 2675 y Fo(r)p 806 2679 825 4 v 1548 2677 a Fh(-)975 2552 y Ff(Q)1041 2622 y Fj(l)1077 2552 y Ff(Q)p 1144 2583 43 3 v 73 x Fj( )1187 2645 y Fi(\()p Fj(l)p Fi(\))1259 2625 y(\(1\))1353 2601 y Fo(h)1394 2609 y Fj(\015)1426 2626 y(j)1882 2287 y Fu(.)16 b(.)g(.)2148 2711 y Fq(k)2202 2675 y Fo(r)3123 2711 y Fq(k)3177 2675 y Fo(r)p 2269 2679 825 4 v 3011 2677 a Fh(-)2441 2552 y Ff(Q)2507 2622 y Fj(l)2543 2552 y Ff(Q)p 2610 2583 43 3 v 73 x Fj( )2652 2645 y Fi(\()p Fj(l)p Fi(\))2724 2625 y(\()p Fj(t)p Fi(\))2812 2601 y Fo(h)2853 2609 y Fj(\015)2885 2626 y(j)1882 2677 y Fu(.)g(.)g(.)p 730 2219 4 254 v 731 2219 a Fh(?)610 2109 y Fo(')656 2085 y Fg(\003)p 3167 2219 V 3169 2219 a Fh(?)3208 2109 y Fo(')3254 2085 y Fg(\003)p 730 2609 V 731 2609 a Fh(?)608 2499 y Fo( )656 2475 y Fg(\003)p 1705 2609 V 1706 2609 a Fh(?)1583 2499 y Fo( )1631 2475 y Fg(\003)p 2192 2609 V 2194 2609 a Fh(?)2070 2499 y Fo( )2118 2475 y Fg(\003)p 3167 2609 V 3169 2609 a Fh(?)3208 2499 y Fo( )3256 2475 y Fg(\003)0 2870 y Fu(comm)m(utes.)41 b(F)-8 b(rom)22 b(the)i(considerations)g(w)m(e)h(made)f(ab)s(out)f(the) i(comp)s(osition)c(of)2999 2844 y Fl(e)2983 2870 y Fq( )3046 2886 y Fn(\()p Fo(l)q Fn(\))3151 2870 y Fu(and)37 b Fl(e)-68 b Fq(')3396 2886 y Fn(\()p Fo(l)3445 2867 y Fg(0)3468 2886 y Fn(\))3523 2870 y Fu(it)23 b(follo)m(ws)0 2986 y(that)31 b(the)g(pro)s(duct)g(on)f(the)i(lo)m(w)m(er)e(line)g(is)1571 2912 y Fl(Q)1665 3016 y Fo(l)1708 2912 y Fl(Q)p 1802 2960 95 3 v 104 x Fo(' )1896 3033 y Fi(\()p Fj(l)p Fi(\))1969 3016 y Fn(\()p Fo(i)p Fn(\))2068 2986 y Fq(h)2124 3001 y Fo(\015)2160 3011 y Fj(j)2228 2986 y Fu(b)m(y)i(using)e(the)h(fact)g (from)e(Remark)i(2.2)f(that)0 3123 y(lo)s(cal)g(functions)j(on)f (di\013eren)m(t)h(connected)h(comp)s(onen)m(ts)f(comm)m(ute:)405 3312 y Fl(Y)458 3525 y Fo(l)586 3312 y Fl(Y)p 549 3484 49 3 v 549 3540 a Fo( )598 3557 y Fi(\()p Fj(l)p Fi(\))670 3540 y Fn(\()p Fo(t)p Fn(\))767 3407 y Fq(h)823 3422 y Fo(\015)859 3432 y Fj(j)913 3407 y Fq(:)17 b(:)g(:)1044 3312 y Fl(Y)1097 3525 y Fo(l)1229 3312 y Fl(Y)p 1188 3484 V 1188 3540 a Fo( )1236 3557 y Fi(\()p Fj(l)p Fi(\))1308 3540 y Fn(\(1\))1415 3407 y Fq(h)1471 3422 y Fo(\015)1507 3432 y Fj(j)1572 3407 y Fu(=)1675 3312 y Fl(Y)1728 3525 y Fo(l)1819 3207 y Fl(0)1819 3386 y(@)2078 3312 y(Y)p 1907 3484 V 1907 3540 a Fo( )1955 3557 y Fi(\()p Fj(l)p Fi(\))2027 3540 y Fn(\()p Fo(t)p Fn(\))p Fo(:::)p 2166 3484 V( )2215 3557 y Fi(\()p Fj(l)p Fi(\))2287 3540 y Fn(\(1\))2394 3407 y Fq(h)2450 3422 y Fo(\015)2486 3432 y Fj(j)2523 3207 y Fl(1)2523 3386 y(A)2638 3407 y Fu(=)2741 3312 y Fl(Y)2794 3525 y Fo(l)2885 3207 y Fl(0)2885 3386 y(@)3032 3312 y(Y)p 2972 3484 95 3 v 2972 3540 a Fo(' )3067 3557 y Fi(\()p Fj(l)p Fi(\))3139 3540 y Fn(\()p Fo(i)p Fn(\))3235 3407 y Fq(h)3291 3422 y Fo(\015)3327 3432 y Fj(j)3364 3207 y Fl(1)3364 3386 y(A)3468 3407 y Fq(;)0 3729 y Fu(where)p 273 3648 67 4 v 25 w Fq( )340 3753 y Fn(\()p Fo(l)q Fn(\))420 3729 y Fu(\()p Fq(t)p Fu(\))g Fq(:)g(:)g(:)p 679 3648 V 16 w( )746 3753 y Fn(\()p Fo(l)q Fn(\))827 3729 y Fu(\(1\))23 b(is)g(the)h(concatenation)e(of)h(the)h (corresp)s(onding)f(sub)m(w)m(ords.)43 b(This)23 b(is)g(a)g(sub)m(w)m (ord)0 3876 y(of)32 b Fq(\015)38 b Fu(since)455 3849 y Fl(e)439 3876 y Fq( )502 3891 y Fn(\()p Fo(l)q Fn(\))615 3876 y Fu(is)33 b(order)f(preserving)i(and)e(\()p Fq(\014)1720 3891 y Fo(t)1750 3876 y Fq(;)17 b(:)g(:)g(:)f(;)h(\014)2024 3891 y Fn(1)2063 3876 y Fu(\))32 b(is)g(a)h(sub)m(w)m(ord)h(of)e Fq(\014)6 b Fu(.)0 4035 y(Hence)34 b(the)f(diagrams)1509 4199 y Fq(k)1563 4163 y Fo(n)2289 4199 y Fq(k)2343 4163 y Fo(n)p 1639 4166 623 4 v 2178 4164 a Fh(-)1898 4121 y Fo(f)1932 4129 y Fj(\013)1972 4145 y(i)1514 4589 y Fq(k)1568 4553 y Fo(r)2294 4589 y Fq(k)2348 4553 y Fo(r)p 1635 4556 630 4 v 2182 4554 a Fh(-)1690 4429 y Ff(Q)1757 4500 y Fj(l)1792 4429 y Ff(Q)p 1859 4461 84 3 v 74 x Fj(' )1942 4523 y Fi(\()p Fj(l)p Fi(\))2014 4503 y(\()p Fj(i)p Fi(\))2100 4479 y Fo(h)2141 4487 y Fj(\015)2173 4503 y(j)p 1558 4486 4 254 v 1560 4486 a Fh(?)1300 4376 y Fn(\()p Fo( )r Fk(\016)p Fo(')p Fn(\))1483 4352 y Fg(\003)p 2338 4486 V 2340 4486 a Fh(?)2379 4376 y Fn(\()p Fo( )r Fk(\016)p Fo(')p Fn(\))2562 4352 y Fg(\003)0 4703 y Fu(comm)m(ute.)3408 b Fd(\003)0 4926 y Fu(Since)36 b(the)g(comp)s(osition)e(of)h(morphisms) g(of)h(SDS)f(is)h(de\014ned)h(b)m(y)g(comp)s(osition)d(of)h(certain)h (set)g(maps,)0 5043 y(isomorphisms)h(of)h(SDS)h(consists)h(of)e (certain)h(bijectiv)m(e)f(set)i(maps.)62 b(This)39 b(is)g(useful)g(to)f (construct)i(or)0 5159 y(iden)m(tify)32 b(isomorphisms.)42 b(In)32 b(fact)h(Example)f(2.5)g(\(2\))g(w)m(as)i(constructed)g(in)e (suc)m(h)i(a)e(w)m(a)m(y)-8 b(.)1563 5434 y(3.)49 b Fv(St)-7 b(a)g(te)38 b(Sp)-7 b(a)n(ces)0 5651 y Fu(An)m(y)40 b(function)e Fq(f)50 b Fu(:)38 b Fq(k)820 5615 y Fo(n)906 5651 y Fp(\000)-60 b(!)38 b Fq(k)1115 5615 y Fo(n)1201 5651 y Fu(de\014nes)j(a)d(\014nite) h(directed)g(graph)g(with)f(v)m(ertex)j(set)f Fq(k)3277 5615 y Fo(n)3363 5651 y Fu(and)f(directed)0 5767 y(edges)30 b(\()p Fq(x;)17 b(f)11 b Fu(\()p Fq(x)p Fu(\)\))30 b(for)f(all)e Fq(x)h Fp(2)g Fq(k)1161 5731 y Fo(n)1208 5767 y Fu(,)i(called)e(the)i Ft(state)i(sp)-5 b(ac)g(e)29 b Fu(of)g Fq(f)38 b Fu(:)28 b Fq(k)2486 5731 y Fo(n)2561 5767 y Fp(\000)-60 b(!)27 b Fq(k)2759 5731 y Fo(n)2806 5767 y Fu(.)43 b(A)29 b(morphism)f(from)g Fq(f)38 b Fu(:)28 b Fq(k)3853 5731 y Fo(n)p eop %%Page: 13 13 13 12 bop 1235 251 a Fr(SEQUENTIAL)32 b(D)n(YNAMICAL)g(SYSTEMS)1158 b(13)0 450 y Fp(\000)-60 b(!)28 b Fq(k)199 414 y Fo(n)278 450 y Fu(to)33 b Fq(g)d Fu(:)e Fq(k)584 414 y Fo(m)679 450 y Fp(\000)-60 b(!)27 b Fq(k)877 414 y Fo(m)976 450 y Fu(is)33 b(a)f(comm)m(utativ)m(e)f(diagram)1691 669 y Fq(k)1745 633 y Fo(n)2081 669 y Fq(k)2135 633 y Fo(n)p 1820 636 233 4 v 1969 634 a Fh(-)1907 596 y Fq(f)1681 1059 y(k)1735 1023 y Fo(m)2071 1059 y Fq(k)2125 1023 y Fo(m)p 1830 1026 213 4 v 1960 1024 a Fh(-)1911 986 y Fq(g)p 1740 956 4 254 v 1741 956 a Fh(?)1646 864 y Fq(h)p 2130 956 V 2131 956 a Fh(?)2170 864 y Fq(h)2226 1024 y(:)0 1190 y Fu(In)f(this)f(w)m(a)m(y)i(ev)m(ery)g(morphism)d(of)h ('functions')h(induces)g(a)f(morphism)f(of)h(the)h(asso)s(ciated)f (state)h(spaces)0 1306 y(in)37 b(the)h(category)f(of)g(directed)h (graphs.)59 b(So)37 b(w)m(e)i(ha)m(v)m(e)g(a)e(co)m(v)-5 b(arian)m(t)37 b(functor)g(from)g('functions')g(to)h(the)0 1422 y(full)31 b(sub)s(category)i Fy(S)g Fu(of)f(state)h(spaces)h(in)e (the)h(category)g(of)f(directed)h(graphs.)0 1581 y(Let)41 b Fp(F)51 b Fu(=)42 b(\()p Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(a)p Fu(]\))p Fq(;)g Fu(\()p Fq(f)937 1596 y Fn(1)976 1581 y Fq(;)g(:)g(:)g(:)f(;)h(f)1243 1596 y Fo(n)1289 1581 y Fu(\))p Fq(;)g(\013)q Fu(\))40 b(b)s(e)h(an)g(SDS)g (with)f(up)s(date)h(function)f Fq(f)3025 1596 y Fo(\013)3117 1581 y Fu(:)h Fq(k)3239 1545 y Fo(n)3328 1581 y Fp(\000)-59 b(!)41 b Fq(k)3541 1545 y Fo(n)3588 1581 y Fu(,)i Fq(f)3706 1596 y Fo(\013)3797 1581 y Fu(:=)0 1698 y Fq(f)48 1713 y Fo(\013)93 1722 y Fi(1)149 1698 y Fq(:)17 b(:)g(:)f(f)328 1713 y Fo(\013)373 1721 y Fj(n)420 1698 y Fu(.)43 b(In)33 b(this)f(section)h(w)m(e)h(sho)m(w)g(that)e(there)h(is)f(a)h(co)m(v)-5 b(arian)m(t)32 b(functor)1639 1863 y Fp(S)j Fu(:)28 b Fy(SDS)g Fp(\000)-59 b(!)27 b Fy(S)p Fq(;)0 2029 y Fu(giv)m(en)33 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y(\(1\))1257 3700 y Fo(g)1291 3712 y Fj(\014)1326 3727 y(j)1901 3806 y Fq(:)g(:)g(:)p 1541 3774 323 4 v 1781 3772 a Fh(-)p 2036 3774 V 411 w(-)2377 3806 y Fq(k)2431 3770 y Fo(m)3060 3806 y Fq(k)3114 3770 y Fo(m)p 2526 3774 506 4 v 2948 3772 a Fh(-)2538 3650 y Ff(Q)2605 3720 y Fj(l)2641 3650 y Ff(Q)p 2707 3692 41 3 v 2707 3720 a Fj(')2748 3741 y Fi(\()p Fj(l)p Fi(\))2820 3720 y(\()p Fj(r)r Fi(\))2914 3700 y Fo(g)2948 3712 y Fj(\014)2983 3727 y(j)p 778 3703 4 254 v 780 3703 a Fh(?)659 3593 y Fo(')705 3570 y Fg(\003)p 1461 3703 V 1463 3703 a Fh(?)1502 3593 y Fo(')1548 3570 y Fg(\003)p 2436 3703 V 2437 3703 a Fh(?)2476 3593 y Fo(')2522 3570 y Fg(\003)p 3118 3703 V 3120 3703 a Fh(?)3159 3593 y Fo(')3205 3570 y Fg(\003)0 3937 y Fu(comm)m(utes.)43 b(It)33 b(remains)f(to)g(sho)m(w) i(that)995 4103 y Fq(g)1042 4118 y Fo(\014)1117 4103 y Fu(=)27 b Fq(g)1267 4118 y Fo(\014)1307 4126 y Fj(s)1361 4103 y Fq(:)17 b(:)g(:)f(g)1539 4118 y Fo(\014)1579 4127 y Fi(1)1644 4103 y Fu(=)1748 4008 y Fl(Y)1801 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Fk(2)p Fo(\014)1305 4828 y Fi(\()p Fj(l)p Fi(\))1393 4696 y Fq(g)1440 4711 y Fo(\014)1480 4721 y Fj(j)1544 4696 y Fu(=)1746 4602 y Fl(Y)1647 4818 y Fo(\014)1687 4828 y Fj(j)1719 4818 y Fk(2)p 1766 4780 V Fo(')1812 4835 y Fi(\()p Fj(l)p Fi(\))1884 4818 y Fn(\()p Fo(r)r Fn(\))1990 4696 y Fq(g)2037 4711 y Fo(\014)2077 4721 y Fj(j)2129 4696 y Fq(:)17 b(:)g(:)2360 4602 y Fl(Y)2260 4818 y Fo(\014)2300 4828 y Fj(j)2333 4818 y Fk(2)p 2380 4780 V Fo(')2426 4835 y Fi(\()p Fj(l)p Fi(\))2498 4818 y Fn(\(1\))2604 4696 y Fq(g)2651 4711 y Fo(\014)2691 4721 y Fj(j)2727 4696 y Fq(:)0 4984 y Fu(In)23 b(fact)g(the)g Fq(g)t Fu('s)g(ma)m(y)f(b)s(e)h(group)s(ed)g (together)g(according)g(to)f(connected)j(comp)s(onen)m(ts)e(without)f (c)m(hanging)0 5100 y(their)34 b(pro)s(duct.)50 b(But)35 b(this)f(equation)h(holds)f(since)h(the)g(map)50 b(~)-65 b Fq(')2385 5116 y Fn(\()p Fo(l)q Fn(\))2497 5100 y Fu(:)32 b Fp(j)p Fq(\014)2639 5116 y Fn(\()p Fo(l)q Fn(\))2719 5100 y Fp(j)f(\000)-59 b(!)30 b(j)p Fq(\013)q Fp(j)k Fu(is)g(order)h(preserving.)0 5216 y(The)f(rest)f(of)f(the)h(statemen)m (t)g(follo)m(ws)e(trivially)-8 b(.)2022 b Fd(\003)0 5436 y Fu(Th)m(us)34 b Fq(')f Fu(induces)g(a)f(morphism)f(of)h(state)h (spaces)1488 5602 y Fp(S)7 b Fu(\()p Fq(')p Fu(\))28 b(:)g Fp(S)7 b Fu(\()p Fp(F)j Fu(\))28 b Fp(\000)-59 b(!)27 b(S)7 b Fu(\()p Fp(G)f Fu(\))p Fq(:)0 5767 y Fu(This)33 b(pro)m(v)m(es)h(the)f(follo)m(wing)d(theorem.)p eop %%Page: 14 14 14 13 bop 0 251 a Fr(14)894 b(REINHARD)23 b(LA)n(UBENBA)n(CHER)h(AND)h (BODO)g(P)-6 b(AREIGIS)0 450 y Fy(Theorem)37 b(3.2.)42 b Ft(Passage)34 b(to)h(state)g(sp)-5 b(ac)g(es)34 b(induc)-5 b(es)34 b(a)h(c)-5 b(ovariant)34 b(functor)1638 612 y Fp(S)h Fu(:)28 b Fy(SDS)g Fp(\000)-57 b(!)27 b Fy(S)p Fq(:)0 817 y Fu(Observ)m(e)34 b(that)f(this)f(functor)h(extracts)h(and) e(clari\014es)g(the)h(dynamic)f(b)s(eha)m(vior)g(of)g(an)h(SDS.)763 1071 y(4.)49 b Fv(Pr)n(oducts)37 b(of)i(Sequential)e(D)m(ynamical)h (Systems)0 1288 y Fy(Theorem)f(4.1.)42 b Ft(The)34 b(c)-5 b(ate)g(gory)35 b Fy(SDS)g Ft(has)g(\014nite)f(pr)-5 b(o)g(ducts.)0 1503 y(Pr)g(o)g(of.)41 b Fu(Let)33 b Fp(F)38 b Fu(=)29 b(\()p Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(a)p Fu(]\))p Fq(;)g Fu(\()p Fq(f)1201 1518 y Fn(1)1240 1503 y Fq(;)g(:)g(:)g(:)f(;)h(f)1507 1518 y Fo(n)1553 1503 y Fu(\))p Fq(;)g(\013)q Fu(\))33 b(and)g Fp(G)i Fu(=)28 b(\()p Fq(G;)17 b Fu(\()p Fq(k)s Fu([)p Fq(b)p Fu(]\))p Fq(;)g Fu(\()p Fq(g)2670 1518 y Fn(1)2709 1503 y Fq(;)g(:)g(:)g(:)f(;)h(g)2975 1518 y Fo(m)3041 1503 y Fu(\))p Fq(;)g(\014)6 b Fu(\))32 b(b)s(e)h(SDS.)g(De\014ne)0 1619 y(the)g Ft(pr)-5 b(o)g(duct)1051 1740 y Fp(H)28 b Fu(=)g(\()p Fq(H)r(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(c)p Fu(]\))p Fq(;)g Fu(\()p Fq(h)1796 1755 y Fn(1)1835 1740 y Fq(;)g(:)g(:)g(:)f(;)h(h)2110 1755 y Fo(n)p Fn(+)p Fo(m)2274 1740 y Fu(\))p Fq(;)g(\015)5 b Fu(\))27 b(=)h Fp(F)k(\002)22 b(G)0 1882 y Fu(as)41 b(follo)m(ws.)69 b(Let)41 b Fq(H)50 b Fu(=)43 b Fq(F)1042 1870 y Fu(_)1023 1882 y Fp([)p Fq(G)e Fu(b)s(e)h(the)f(disjoin)m(t)g(union)f(of)h(the)h (graphs)f Fq(F)55 b Fu(and)42 b Fq(G)p Fu(,)h(with)e(v)m(ertex)i(set)0 1998 y Fq(V)57 2013 y Fo(F)135 1986 y Fu(_)115 1998 y Fp([)q Fq(V)239 2013 y Fo(G)298 1998 y Fu(.)i(Observ)m(e)35 b(that)e(the)g(graph)g(comp)s(onen)m(ts)h Fq(F)47 b Fu(and)33 b Fq(G)g Fu(are)g(disconnected.)46 b(De\014ne)34 b Fq(k)s Fu([)p Fq(c)p Fu(])29 b(:=)g Fq(k)s Fu([)p Fq(a)p Fu(])0 2114 y(for)j(an)m(y)h Fq(c)28 b Fu(=)f Fq(a)h Fp(2)g Fq(V)736 2129 y Fo(F)823 2114 y Fp(\022)g Fq(V)985 2129 y Fo(H)1084 2114 y Fu(and)33 b Fq(k)s Fu([)p Fq(c)p Fu(])28 b(:=)g Fq(k)s Fu([)p Fq(b)p Fu(])33 b(for)f(an)m(y)h Fq(c)28 b Fu(=)f Fq(b)i Fp(2)f Fq(V)2492 2129 y Fo(G)2578 2114 y Fp(\022)h Fq(V)2741 2129 y Fo(H)2808 2114 y Fu(.)43 b(No)m(w)33 b(de\014ne)1321 2276 y Fq(h)1377 2291 y Fo(c)1440 2276 y Fu(:)28 b Fq(k)1549 2235 y Fo(n)p Fn(+)p Fo(m)1741 2276 y Fu(=)f Fq(k)1898 2235 y Fo(n)1968 2276 y Fp(\002)22 b Fq(k)2121 2235 y Fo(m)2215 2276 y Fp(\000)-59 b(!)27 b Fq(k)2414 2235 y Fo(n)p Fn(+)p Fo(m)0 2438 y Fu(as)38 b(follo)m(ws.)58 b(F)-8 b(or)37 b Fq(c)f Fu(=)h Fq(a)g Fp(2)f Fq(V)1117 2453 y Fo(F)1214 2438 y Fu(de\014ne)j Fq(h)1557 2453 y Fo(c)1628 2438 y Fu(:=)e Fq(f)1816 2453 y Fo(a)1883 2438 y Fp(\002)26 b Fu(id,)39 b(and)f(for)f Fq(c)f Fu(=)h Fq(b)g Fp(2)g Fq(V)2912 2453 y Fo(G)3008 2438 y Fu(w)m(e)i(set)g Fq(h)3371 2453 y Fo(c)3442 2438 y Fu(:=)e(id)25 b Fp(\002)h Fq(g)3839 2453 y Fo(b)3873 2438 y Fu(.)0 2554 y(Finally)-8 b(,)30 b(w)m(e)k(de\014ne)f(the)g(up)s (date)g(sc)m(hedule)h(as)f(the)g(concatenation)f(of)g(w)m(ords)1781 2716 y Fq(\015)h Fu(=)27 b Fq(\013)q(\014)6 b(:)0 2921 y Fu(Next)33 b(w)m(e)h(de\014ne)g(a)e(pro)5 b(jection)32 b(morphism)1571 3082 y(pr)1663 3106 y Fk(G)1741 3082 y Fu(:)c Fp(F)j(\002)23 b(G)34 b(\000)-60 b(!)28 b(G)6 b Fq(;)0 3244 y Fu(and)30 b(similarly)d(a)i(pro)5 b(jection)30 b(in)m(to)g Fp(F)10 b Fu(.)42 b(Let)30 b(pr)1734 3268 y Fk(G)t Fo(;g)1867 3244 y Fu(:)e Fq(G)g Fp(\000)-60 b(!)27 b Fq(F)k Fp([)17 b Fq(G)30 b 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b(sho)m(ws)i(that)e(the)h(second)g(condition)e(of)h(a)g (morphism)e(is)i(satis\014ed)h(and)f(that)g(pr)3339 4326 y Fk(G)3429 4302 y Fu(is)g(indeed)g(a)0 4418 y(morphism)31 b(of)h(SDS.)0 4577 y(It)40 b(remains)e(to)h(v)m(erify)h(the)g(univ)m (ersal)f(prop)s(ert)m(y)h(of)f(the)h(pro)s(duct.)65 b(Supp)s(ose)40 b(w)m(e)h(are)e(giv)m(en)h(an)f(SDS)0 4694 y Fp(K)33 b Fu(=)e(\()p Fq(K)r(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(d)p Fu(]\))p Fq(;)g Fu(\()p Fq(k)751 4709 y Fo(d)790 4694 y Fu(\))p Fq(;)g(\016)t Fu(\))34 b(and)h(morphisms)f Fq(')d Fu(:)h Fp(K)h(\000)-60 b(!)31 b(F)44 b Fu(and)35 b Fq( )h Fu(:)31 b Fp(K)i(\000)-60 b(!)31 b(G)6 b Fu(.)51 b(W)-8 b(e)35 b(need)h(to)e(sho)m(w)i(that)0 4810 y(there)d(is)f(a)h (unique)g(morphism)1604 4931 y Fq(!)e Fu(:)d Fp(K)h(\000)-60 b(!)27 b(F)32 b(\002)23 b(G)6 b Fq(;)0 5072 y Fu(suc)m(h)34 b(that)e(pr)524 5096 y Fk(F)607 5072 y Fp(\016)22 b Fq(!)31 b Fu(=)c Fq(')33 b Fu(and)f(pr)1253 5096 y Fk(G)1325 5072 y Fp(\016)22 b Fq(!)31 b Fu(=)c Fq( )t Fu(.)44 b(De\014ne)1595 5234 y Fq(!)1656 5249 y Fo(g)1723 5234 y Fu(:)28 b Fq(F)36 b Fp([)23 b Fq(G)k Fp(\000)-59 b(!)27 b Fq(K)0 5396 y Fu(to)47 b(b)s(e)g(equal)f(to)h Fq(')750 5411 y Fo(g)790 5396 y Fu(,)k(resp.)87 b Fq( )1218 5411 y Fo(g)1259 5396 y Fu(,)50 b(on)d(the)g(comp)s(onen)m(t)g Fq(F)14 b Fu(,)50 b(resp.)88 b Fq(G)p Fu(.)e(De\014ne)48 b Fq(!)3191 5411 y Fo(s)3227 5396 y Fu([)p Fq(c)p Fu(])k(:=)h Fq(')3595 5411 y Fo(s)3631 5396 y Fu([)p Fq(a)p Fu(])47 b(for)0 5512 y Fq(c)34 b Fu(=)f Fq(a)h Fp(2)g Fq(V)427 5527 y Fo(F)522 5512 y Fu(and)i Fq(!)776 5527 y Fo(s)813 5512 y Fu([)p Fq(c)p Fu(])e(:=)f Fq( )1142 5527 y Fo(s)1179 5512 y Fu([)p Fq(b)p Fu(])k(for)f Fq(c)d Fu(=)h Fq(b)g Fp(2)g Fq(V)1881 5527 y Fo(G)1940 5512 y Fu(.)55 b(Since)36 b(a)g(connected)h(comp)s(onen)m(t)g(of)e Fp(F)f(\002)25 b(G)43 b Fu(is)35 b(a)0 5628 y(connected)c(comp)s(onen)m(t)f(of)g (either)f Fp(F)39 b Fu(or)30 b Fp(G)36 b Fu(the)30 b(order)g (preserving)g(map)k Fl(e)-60 b Fq(!)2811 5644 y Fn(\()p Fo(l)q Fn(\))2922 5628 y Fu(is)29 b(determined)g(b)m(y)i(either)12 5767 y Fl(e)-67 b Fq(')64 5783 y Fn(\()p Fo(l)q Fn(\))177 5767 y Fu(or)313 5741 y Fl(e)297 5767 y Fq( )360 5783 y Fn(\()p Fo(l)q Fn(\))441 5767 y Fu(.)p eop %%Page: 15 15 15 14 bop 1235 251 a Fr(SEQUENTIAL)32 b(D)n(YNAMICAL)g(SYSTEMS)1158 b(15)0 450 y Fu(Then)23 b(clearly)e(\(pr)675 474 y Fk(F)736 450 y Fp(\016)p Fq(!)t Fu(\))889 465 y Fo(g)956 450 y Fu(=)27 b Fq(')1123 465 y Fo(g)1185 450 y Fu(and)22 b(\(pr)1494 474 y Fk(G)1544 450 y Fp(\016)p Fq(!)t Fu(\))1697 465 y Fo(g)1764 450 y Fu(=)27 b Fq( )1930 465 y Fo(g)1971 450 y Fu(.)40 b(F)-8 b(urthermore)21 b(\(pr)2722 474 y Fk(F)2783 450 y Fp(\016)p Fq(!)t Fu(\))2936 465 y Fo(s)2972 450 y Fu([)p Fq(a)p Fu(])28 b(=)f(pr)3301 474 y Fk(F)7 b Fo(;s)3414 450 y Fu([)p Fq(a)p Fu(])p Fq(!)3580 465 y Fo(s)3617 450 y Fu([pr)3736 474 y Fk(F)g Fo(;g)3853 450 y Fu(\()p Fq(a)p Fu(\)])28 b(=)0 570 y Fq(!)61 585 y Fo(s)98 570 y Fu([)p Fq(a)p Fu(])g(=)f Fq(')398 585 y Fo(s)435 570 y Fu([)p Fq(a)p Fu(])33 b(for)f(all)e Fq(a)e Fp(2)g Fq(V)1087 585 y Fo(F)1178 570 y Fu(and)33 b(similarly)c(\(pr)1897 593 y Fk(G)1969 570 y Fp(\016)22 b Fq(!)t Fu(\))2144 585 y Fo(s)2180 570 y Fu([)p Fq(b)p Fu(])28 b(=)g Fq( )2470 585 y Fo(s)2507 570 y Fu([)p Fq(b)p Fu(].)0 729 y(It)33 b(is)f(clear)g(no)m(w)h(that)f Fq(!)k Fu(is)c(a)h(morphism)d(uniquely)j(determined)f(b)m(y)i Fq(')e Fu(and)h Fq( )t Fu(.)819 b Fd(\003)0 900 y Fy(Lemma)30 b(4.2.)37 b Ft(L)-5 b(et)29 b Fp(F)37 b Fu(=)28 b(\()p Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(a)p Fu(]\))p Fq(;)g Fu(\()p Fq(f)1489 915 y Fo(a)1530 900 y Fu(\))p Fq(;)g(\013)q Fu(\))28 b Ft(b)-5 b(e)29 b(an)g(SDS.)f(If)g Fq(\013)2389 915 y Fo(i)2446 900 y Ft(and)h Fq(\013)2692 915 y Fo(i)p Fn(+1)2839 900 y Ft(ar)-5 b(e)29 b(in)g(di\013er)-5 b(ent)28 b(c)-5 b(onne)g(cte)g(d)0 1016 y(c)g(omp)g(onents)34 b(of)g Fq(F)14 b Ft(,)35 b(then)1099 1177 y Fu(\()p Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(a)p Fu(]\))p Fq(;)g Fu(\()p Fq(f)1612 1192 y Fo(a)1653 1177 y Fu(\))p Fq(;)g Fu(\()p Fq(\013)1835 1192 y Fn(1)1874 1177 y Fq(;)g(:)g(:)g(:)f(;)h(\013)2155 1192 y Fo(i)p Fn(+1)2273 1177 y Fq(;)g(\013)2379 1192 y Fo(i)2407 1177 y Fq(;)g(:)g(:)g(:)f(;)h(\013)2688 1192 y Fo(r)2725 1177 y Fu(\)\))0 1337 y Ft(is)35 b(isomorphic)e(to)i Fp(F)10 b Ft(.)0 1551 y(Pr)-5 b(o)g(of.)41 b Fu(De\014ne)d Fq(')e Fu(:)h Fp(F)46 b(\000)-60 b(!)36 b Fu(\()p Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(a)p Fu(]\))p Fq(;)g Fu(\()p Fq(f)1553 1566 y Fo(a)1594 1551 y Fu(\))p Fq(;)g Fu(\()p Fq(\013)1776 1566 y Fn(1)1815 1551 y Fq(;)g(:)g(:)g(:)f(;)h(\013)2096 1566 y Fo(i)p Fn(+1)2214 1551 y Fq(;)g(\013)2320 1566 y Fo(i)2348 1551 y Fq(;)g(:)g(:)g(:)f(;)h(\013)2629 1566 y Fo(r)2667 1551 y Fu(\)\).)58 b(Use)39 b Fq(')3085 1566 y Fo(g)3161 1551 y Fu(:=)d(\(id)g(:)g Fq(F)50 b Fp(\000)-60 b(!)36 b Fq(F)14 b Fu(\))0 1667 y(and)34 b Fq(')255 1682 y Fo(s)292 1667 y Fu([)p Fq(a)p Fu(])d(=)f(id)o(.)48 b(Let)34 b Fq(\013)928 1682 y Fo(i)987 1667 y Fp(2)c Fq(F)1146 1683 y Fn(\(1\))1275 1667 y Fu(and)k Fq(\013)1528 1682 y Fo(i)p Fn(+1)1677 1667 y Fp(2)c Fq(F)1836 1683 y Fn(\(2\))1931 1667 y Fu(.)48 b(Then)35 b(let)46 b Fl(e)-67 b Fq(')2469 1683 y Fn(\(2\))2563 1667 y Fu(\()p Fq(i)p Fu(\))30 b(:=)g Fq(i)24 b Fu(+)f(1)34 b(and)46 b Fl(e)-67 b Fq(')3329 1683 y Fn(\(1\))3423 1667 y Fu(\()p Fq(i)23 b Fu(+)h(1\))29 b(:=)i Fq(i)0 1784 y Fu(and)45 b Fl(e)-67 b Fq(')254 1799 y Fn(\()p Fo(l)q Fn(\))362 1784 y Fu(:=)28 b(id)k(otherwise.)44 b(This)32 b(is)g(ob)m(viously)h(a)f(canonical)f (isomorphism.)916 b Fd(\003)0 1997 y Fu(Th)m(us)39 b(the)e(up)s(date)h (sc)m(hedule)g Fq(\013)g Fu(of)f(an)m(y)g(SDS)g Fp(F)47 b Fu(ma)m(y)37 b(b)s(e)g(rearranged)g(according)g(to)g(the)g(connected) 0 2114 y(comp)s(onen)m(ts)c(of)f Fq(F)46 b Fu(and)33 b(this)f(giv)m(es)h(a)g(canonically)d(isomorphic)h Fq(S)6 b(D)s(S)g Fu(.)0 2252 y Fy(Theorem)37 b(4.3.)190 b Fu(\(1\))41 b Ft(A)n(n)32 b(SDS)f(is)g(inde)-5 b(c)g(omp)g(osable)30 b(\(w.r.t.)43 b(pr)-5 b(o)g(ducts\))32 b(if)f(and)g(only)h(if)g(the)f 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Fu(where)0 3377 y Fp(G)k Fu(is)32 b(indecomp)s(osable)f(or)h(connected.)46 b(This)32 b(holds)h(since)1018 3554 y(Mor\()p Fp(F)10 b Fq(;)17 b Fp(G)1417 3569 y Fn(\(1\))1533 3554 y Fp(\002)22 b Fq(:)17 b(:)g(:)22 b Fp(\002)h(G)1928 3569 y Fn(\()p Fo(r)r Fn(\))2021 3554 y Fu(\))2086 3526 y Fp(\030)2087 3558 y Fu(=)2192 3459 y Fl(Y)2336 3554 y Fu(Mor\()p Fp(F)10 b Fq(;)17 b Fp(G)2735 3569 y Fn(\()p Fo(i)p Fn(\))2817 3554 y Fu(\))p Fq(:)1153 3821 y Fu(5.)49 b Fv(Decomposition)37 b(of)i(Morphisms)0 4038 y Fy(Theorem)47 b(5.1.)g Ft(L)-5 b(et)43 b Fp(F)53 b Fu(=)43 b Fp(F)1198 4053 y Fn(1)1265 4038 y Fp(\002)29 b Fq(:)17 b(:)g(:)28 b Fp(\002)h(F)1692 4053 y Fo(n)1781 4038 y Ft(with)43 b(inde)-5 b(c)g(omp)g(osable)41 b(c)-5 b(omp)g(onents)42 b Fp(F)3293 4053 y Fo(i)3364 4038 y Ft(and)h(let)g Fp(G)49 b Ft(b)-5 b(e)0 4154 y(inde)g(c)g(omp)g (osable.)79 b(L)-5 b(et)47 b Fq(')k Fu(:)f Fp(F)60 b(\000)-57 b(!)50 b(G)j Ft(b)-5 b(e)47 b(a)g(morphism.)80 b(Then)46 b(ther)-5 b(e)47 b(is)g(a)g(uniquely)g(determine)-5 b(d)0 4270 y(c)g(omp)g(onent)34 b Fp(F)558 4285 y Fo(i)620 4270 y Ft(and)h(a)f(uniquely)h(determine)-5 b(d)34 b(morphism)g Fq(')2301 4285 y Fo(i)2357 4270 y Fu(:)27 b Fp(F)2483 4285 y Fo(i)2539 4270 y Fp(\000)-57 b(!)27 b(G)41 b Ft(such)35 b(that)1115 4431 y Fu(\()p Fq(')28 b Fu(:)g Fp(F)37 b(\000)-57 b(!)27 b(G)6 b Fu(\))28 b(=)f(\()p Fq(')1892 4446 y Fo(i)1920 4431 y Fu(pr)2013 4455 y Fo(i)2068 4431 y Fu(:)h Fp(F)37 b(\000)-57 b(!)28 b(F)2452 4446 y Fo(i)2507 4431 y Fp(\000)-57 b(!)27 b(G)6 b Fu(\))p Fq(:)0 4592 y Ft(Ther)-5 b(efor)g(e)1100 4787 y Fu(Mor\()p Fp(F)1386 4802 y Fn(1)1447 4787 y Fp(\002)23 b Fq(:)17 b(:)g(:)22 b Fp(\002)g(F)1855 4802 y Fo(n)1902 4787 y Fq(;)17 b Fp(G)6 b Fu(\))2076 4759 y Fp(\030)2077 4791 y Fu(=)2217 4662 y Fo(n)2183 4692 y Fl([)2181 4902 y Fo(i)p Fn(=1)2312 4787 y Fu(Mor\()p Fp(F)2598 4802 y Fo(i)2626 4787 y Fq(;)17 b Fp(G)6 b Fu(\))p Fq(:)0 5087 y Ft(Pr)-5 b(o)g(of.)41 b Fu(These)h(assertions)e(follo)m(w)e (from)g(the)i(construction)g(of)f(the)i(pro)s(duct)f(and)f(the)h(fact)g (that)g(the)0 5203 y(image)31 b(of)h(a)g(connected)i(graph)f(is)f (connected.)2071 b Fd(\003)0 5417 y Fu(An)30 b(indecomp)s(osable)e(SDS) h(m)m(ust)h(b)s(e)g(considered)g(in)f(some)g(sense)i(as)f(an)f (autonomous)g(system.)44 b(So)29 b(an)0 5533 y(SDS)38 b(ma)m(y)g(b)s(e)h(considered)g(as)f(a)g(parallel)e(system)j(of)f(sev)m (eral)h(connected)h(comp)s(onen)m(ts.)61 b(The)39 b(ab)s(o)m(v)m(e)0 5649 y(theorem)f(also)f(implies)f(that)i(ev)m(ery)i(morphism)c Fq(')h Fu(:)g Fp(F)47 b(\000)-59 b(!)36 b(G)6 b Fu(,)40 b(where)g Fp(F)47 b Fu(and)38 b Fp(G)45 b Fu(are)38 b(arbitary)f(SDS,)0 5766 y(can)47 b(b)s(e)g(describ)s(ed)h(b)m(y)f(a)g(family)d(of)j (morphisms)e Fq(')2041 5781 y Fo(j)2129 5766 y Fu(:)52 b Fp(F)2280 5781 y Fo(i)2304 5791 y Fj(j)2393 5766 y Fp(\000)-60 b(!)52 b(G)2621 5781 y Fo(j)2657 5766 y Fu(,)f(where)d(the) f Fp(F)3285 5781 y Fo(i)3309 5791 y Fj(j)3392 5766 y Fu(are)g(suitable)p eop %%Page: 16 16 16 15 bop 0 251 a Fr(16)894 b(REINHARD)23 b(LA)n(UBENBA)n(CHER)h(AND)h (BODO)g(P)-6 b(AREIGIS)0 450 y Fu(indecomp)s(osable)27 b(comp)s(onen)m(ts)j(of)e Fp(F)10 b Fu(,)29 b(and)f(the)h Fp(G)1885 465 y Fo(j)1951 450 y Fu(run)g(through)f(all)f Fq(r)k Fu(indecomp)s(osable)c(comp)s(onen)m(ts)0 566 y(of)32 b Fp(G)6 b Fu(.)44 b(So)32 b(w)m(e)i(could)e(write)1589 697 y Fq(')c Fu(=)f(\()p Fq(')1886 712 y Fn(1)1925 697 y Fq(;)17 b(:)g(:)g(:)f(;)h(')2208 712 y Fo(r)2246 697 y Fu(\))p Fq(:)0 848 y Fu(Therefore)34 b(it)d(is)h(su\016cien)m(t)i (only)e(to)g(study)i(morphisms)d(b)s(et)m(w)m(een)k(indecomp)s(osable)c (SDS.)1612 1130 y(6.)48 b Fv(Equalizers)0 1347 y Fy(Example)38 b(6.1.)43 b Fu(W)-8 b(e)34 b(w)m(an)m(t)h(to)e(giv)m(e)h(an)g(example)f (of)h(a)f(non)m(trivial)f(equalizer)i(in)f(the)h(category)g Fy(SDS)p Fu(.)0 1464 y(This)28 b(will)e(also)h(giv)m(e)h(us)g(some)g (examples)g(and)g(sho)m(w)i(the)e(great)g(v)-5 b(ariet)m(y)28 b(of)f(morphisms)g(b)s(et)m(w)m(een)j(fairly)0 1580 y(small)g(SDS.)0 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2976 y Fu([)p Fq(b)p Fu(])28 b(=)g Fq(')1542 2940 y Fk(0)1565 3000 y Fo(s)1602 2976 y Fu([)p Fq(b)p Fu(])g(:=)g(id)f(:)g Fq(k)k Fp(\000)-59 b(!)27 b Fq(k)s Fu(.)0 3135 y(In)40 b(order)h(to)e(c)m(hec)m(k)k(that)d (these)h(are)f(morphisms)f(w)m(e)i(only)e(ha)m(v)m(e)j(to)e(sho)m(w)h (the)f(second)i(prop)s(ert)m(y)e(of)0 3251 y(morphisms,)31 b(namely)h(that)g(the)h(t)m(w)m(o)h(diagrams)1201 3447 y Fq(k)339 b(k)p 1284 3414 279 4 v 1480 3412 a Fh(-)1334 3380 y Fo(g)1368 3392 y Fj(b)1399 3380 y Fn(=id)1201 3837 y Fq(k)g(k)p 1284 3804 V 1480 3802 a Fh(-)1329 3760 y Fo(h)1370 3736 y Fi(2)1370 3783 y Fj(b)1404 3760 y Fn(=id)p 1227 3734 4 254 v 1228 3734 a Fh(?)994 3624 y Fo(')1040 3601 y Fg(\003)1076 3624 y Fn(=id)p 1617 3734 V 1618 3734 a Fh(?)1657 3624 y Fo(')1703 3601 y Fg(\003)1739 3624 y Fn(=id)2254 3447 y Fq(k)g(k)p 2337 3414 279 4 v 2533 3412 a Fh(-)2387 3380 y Fo(g)2421 3392 y Fj(b)2452 3380 y Fn(=id)2254 3837 y Fq(k)g(k)p 2337 3804 V 2533 3802 a Fh(-)2447 3783 y Fn(id)p 2280 3734 4 254 v 2282 3734 a Fh(?)2047 3624 y Fo(')2093 3601 y Fg(\003)2129 3624 y Fn(=id)p 2670 3734 V 2672 3734 a Fh(?)2711 3624 y Fo(')2757 3601 y Fg(\003)2792 3624 y Fn(=id)0 3975 y Fu(comm)m(ute.)41 b(The)29 b(\014rst)g(diagram)d (arises)i(from)f(the)h(coun)m(terimage)g(of)f(the)i(\014rst)g(letter)e (of)h Fq(\014)33 b Fu(=)28 b(\()p Fq(b;)17 b(b)p Fu(\))28 b(con-)0 4091 y(sisting)h(of)g(t)m(w)m(o)h(instances)h(of)e(the)h (letter)g Fq(b)p Fu(,)h(and)e(the)i(second)g(diagram)c(arises)j(from)e (the)j(coun)m(terimage)0 4207 y(of)h(the)h(second)h(letter)e(of)g Fq(\014)h Fu(=)28 b(\()p Fq(b;)17 b(b)p Fu(\))33 b(that)f(is)h(empt)m (y)-8 b(.)43 b(Similarly)29 b(w)m(e)34 b(sho)m(w)f(for)f Fq(')3042 4171 y Fk(0)3098 4207 y Fu(that)g(the)h(diagrams)1179 4408 y Fq(k)339 b(k)p 1262 4375 279 4 v 1458 4373 a Fh(-)1312 4341 y Fo(g)1346 4353 y Fj(b)1377 4341 y Fn(=id)1179 4798 y Fq(k)g(k)p 1262 4765 V 1458 4763 a Fh(-)1372 4744 y Fn(id)p 1204 4695 4 254 v 1206 4695 a Fh(?)949 4589 y Fo(')995 4565 y Fg(0)1017 4558 y(\003)1053 4589 y Fn(=id)p 1594 4695 V 1596 4695 a Fh(?)1635 4589 y Fo(')1681 4565 y Fg(0)1703 4558 y(\003)1739 4589 y Fn(=id)2277 4408 y Fq(k)g(k)p 2360 4375 279 4 v 2556 4373 a Fh(-)2410 4341 y Fo(g)2444 4353 y Fj(b)2475 4341 y Fn(=id)2277 4798 y Fq(k)g(k)p 2360 4765 V 2556 4763 a Fh(-)2404 4721 y Fo(h)2445 4698 y Fi(2)2445 4744 y Fj(b)2480 4721 y Fn(=id)p 2302 4695 4 254 v 2304 4695 a Fh(?)2047 4589 y Fo(')2093 4565 y Fg(0)2115 4558 y(\003)2151 4589 y Fn(=id)p 2692 4695 V 2694 4695 a Fh(?)2733 4589 y Fo(')2779 4565 y Fg(0)2801 4558 y(\003)2837 4589 y Fn(=id)0 4935 y Fu(comm)m(ute.)0 5094 y(Let)37 b Fp(K)g Fu(=)404 5013 y Fl(\000)449 5094 y Fu(\()p Fq(b)p Fu(\))p Fq(;)17 b Fu(\()p Fq(k)s Fu(\))p Fq(;)g Fu(\()p Fq(l)851 5109 y Fo(b)921 5094 y Fu(=)35 b(id)o(\))p Fq(;)17 b Fu(\()p Fq(b)p Fu(\))1312 5013 y Fl(\001)1395 5094 y Fu(b)s(e)38 b(an)f(SDS)g(and)g(de\014ne)h(a)f(morphism)f Fq(\023)g Fu(:)f Fp(K)i(\000)-60 b(!)35 b(G)44 b Fu(b)m(y)38 b Fq(\023)p Fu(\()p Fq(b)p Fu(\))e(=)g Fq(b)0 5212 y Fu(\(rev)m(erse)f (direction!\))42 b(and)28 b Fl(e)-50 b Fq(\023)p Fu(\(1\))27 b(=)h(1)f(=)c Fl(e)-50 b Fq(\023)p Fu(\(2\).)44 b(Since)32 b(the)h(diagram)1728 5411 y Fq(k)339 b(k)p 1811 5378 279 4 v 2007 5376 a Fh(-)1867 5346 y Fo(l)1888 5358 y Fj(b)1919 5346 y Fn(=id)1728 5801 y Fq(k)g(k)p 1811 5768 V 2007 5766 a Fh(-)1858 5724 y Fo(g)1894 5700 y Fi(2)1892 5747 y Fj(b)1928 5724 y Fn(=id)p 1753 5698 4 254 v 1755 5698 a Fh(?)1541 5594 y Fo(\023)1566 5571 y Fg(\003)1602 5594 y Fn(=id)p 2143 5698 V 2145 5698 a Fh(?)2184 5594 y Fo(\023)2209 5571 y Fg(\003)2245 5594 y Fn(=id)p eop %%Page: 17 17 17 16 bop 1235 251 a Fr(SEQUENTIAL)32 b(D)n(YNAMICAL)g(SYSTEMS)1158 b(17)0 450 y Fu(comm)m(utes,)33 b Fq(\023)g Fu(is)f(a)g(morphism)f(and) h(w)m(e)i(ha)m(v)m(e)g(morphisms)1490 646 y Fp(K)320 b(G)p 1596 618 262 4 v 1775 616 a Fh(-)1714 598 y Fo(\023)2266 646 y Fp(H)p 1980 599 258 4 v 2155 597 a Fh(-)2085 565 y Fo(')p 1980 638 V 2154 636 a Fh(-)2074 706 y Fo(')2120 682 y Fg(0)2383 616 y Fq(:)0 851 y Fu(W)-8 b(e)41 b(ha)m(v)m(e)h Fq('\023)g Fu(=)g Fq(')731 815 y Fk(0)754 851 y Fq(\023)f Fu(since)h Fp(K)g Fu(has)f(only)f(one)h(elemen)m(t)g(comp)s(onen)m(ts.) 68 b(W)-8 b(e)42 b(claim)c(that)j(\()p Fp(K)q Fq(;)17 b(\023)p Fu(\))41 b(is)f(an)0 967 y(equalizer)32 b(of)g(the)h(pair)f (\()p Fq(';)17 b(')1099 931 y Fk(0)1122 967 y Fu(\).)0 1128 y(Let)34 b Fp(F)k Fu(=)391 1047 y Fl(\000)437 1128 y Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(a)p Fu(]\))p Fq(;)g Fu(\()p Fq(f)912 1143 y Fo(i)940 1128 y Fu(\))p Fq(;)g(\013)1085 1047 y Fl(\001)1163 1128 y Fu(b)s(e)34 b(an)f(SDS)h(and)f Fq( )g Fu(:)d Fp(F)38 b(\000)-59 b(!)28 b(G)40 b Fu(b)s(e)34 b(a)f(morphism)e(suc)m(h)k(that)f Fq(' )f Fu(=)c Fq(')3783 1092 y Fk(0)3806 1128 y Fq( )t Fu(.)0 1244 y(W)-8 b(e)44 b(ha)m(v)m(e)i(to)e(sho)m(w)h(that)f(there)g (is)g(a)g(unique)g(morphism)f Fq(\027)53 b Fu(:)48 b Fp(F)56 b(\000)-59 b(!)47 b(K)e Fu(suc)m(h)h(that)d Fq(\023\027)55 b Fu(=)47 b Fq( )t Fu(.)78 b(Let)0 1370 y Fq( )63 1385 y Fo(g)104 1370 y Fu(\()p Fq(b)p Fu(\))40 b(:=)g Fq(a)g Fp(2)h Fq(V)659 1385 y Fo(F)717 1370 y Fu(,)h(and)e(let)1148 1344 y Fl(e)1131 1370 y Fq( )t Fu(\(1\))g(=)g Fq(i)g Fu(and)1765 1344 y Fl(e)1749 1370 y Fq( )t Fu(\(2\))f(=)h Fq(j)6 b Fu(,)42 b(i.e.)65 b Fq(\013)2463 1385 y Fo(i)2531 1370 y Fu(=)40 b Fq(a)g Fu(=)g Fq(\013)2916 1385 y Fo(j)2953 1370 y Fu(.)65 b(F)-8 b(rom)38 b Fq(' )44 b Fu(=)c Fq(')3659 1334 y Fk(0)3682 1370 y Fq( )k Fu(w)m(e)0 1504 y(get)181 1478 y Fl(e)165 1504 y Fq( )16 b Fl(e)-67 b Fq(')31 b Fu(=)451 1478 y Fl(e)434 1504 y Fq( )517 1482 y Fl(e)501 1504 y Fq(')565 1476 y Fk(0)623 1504 y Fu(and)k(hence)h Fq(i)c Fu(=)1277 1478 y Fl(e)1260 1504 y Fq( )16 b Fl(e)-67 b Fq(')p Fu(\(1\))31 b(=)1671 1478 y Fl(e)1654 1504 y Fq( )1737 1482 y Fl(e)1721 1504 y Fq(')1785 1476 y Fk(0)1808 1504 y Fu(\(1\))g(=)h Fq(j)6 b Fu(.)49 b(F)-8 b(urthermore)35 b(w)m(e)g(ha)m(v)m(e)h Fq( )3201 1468 y Fk(\003)3272 1504 y Fu(=)31 b Fq( )3442 1519 y Fo(s)3480 1504 y Fu([)p Fq(b)p Fu(]pr)3667 1528 y Fo(a)3740 1504 y Fu(:)h Fq(k)3853 1468 y Fo(n)0 1621 y Fp(\000)-60 b(!)28 b Fq(k)s Fu([)p Fq(a)p Fu(])g Fp(\000)-60 b(!)28 b Fq(k)s Fu([)p Fq(b)p Fu(],)33 b(and)g(w)m(e)g(ha)m(v)m(e)h(comm)m(utativ)m(e)e(diagrams)1113 1822 y Fq(k)1167 1786 y Fo(n)1503 1822 y Fq(k)1557 1786 y Fo(n)p 1242 1789 233 4 v 1391 1787 a Fh(-)1323 1756 y Fo(f)1357 1764 y Fj(a)1136 2212 y Fq(k)339 b(k)p 1219 2180 279 4 v 1415 2178 a Fh(-)1269 2146 y Fo(g)1303 2158 y Fj(b)1334 2146 y Fn(=id)p 1162 2109 4 254 v 1163 2109 a Fh(?)1040 2000 y Fo( )1088 1976 y Fg(\003)p 1552 2109 V 1553 2109 a Fh(?)1592 2000 y Fo( )1640 1976 y Fg(\003)1871 2178 y Fu(and)2296 1822 y Fq(k)2350 1786 y Fo(n)2686 1822 y Fq(k)2740 1786 y Fo(n)p 2425 1789 233 4 v 2575 1787 a Fh(-)2494 1750 y Fo(f)2528 1769 y Fj(a)2562 1755 y Fg(0)2319 2212 y Fq(k)g(k)p 2402 2180 279 4 v 2598 2178 a Fh(-)2512 2159 y Fn(id)p 2345 2109 4 254 v 2347 2109 a Fh(?)2223 2000 y Fo( )2271 1976 y Fg(\003)p 2735 2109 V 2737 2109 a Fh(?)2775 2000 y Fo( )2823 1976 y Fg(\003)0 2344 y Fu(for)32 b(all)f Fq(a)336 2308 y Fk(0)387 2344 y Fp(6)p Fu(=)c Fq(a)33 b Fu(in)f Fq(F)14 b Fu(.)0 2503 y(No)m(w)34 b(de\014ne)g Fq(\027)i Fu(:)28 b Fp(F)39 b(\000)-60 b(!)29 b(K)34 b Fu(b)m(y)g Fq(\027)1195 2518 y Fo(g)1235 2503 y Fu(\()p Fq(b)p Fu(\))c(:=)e Fq(a)p Fu(,)34 b Fq(\027)1673 2518 y Fo(s)1710 2503 y Fu([)p Fq(b)p Fu(])29 b(=)g Fq( )2002 2518 y Fo(s)2039 2503 y Fu([)p Fq(b)p Fu(],)34 b(and)i Fl(e)-57 b Fq(\027)6 b Fu(\(1\))29 b(:=)f Fq(i)h Fu(=)g Fq(j)6 b Fu(.)45 b(This)34 b(is)e(ob)m(viously)i(the)0 2619 y(only)27 b(c)m(hoice)i(if)d(w)m(e)j (w)m(an)m(t)g(to)e(get)h Fq(\023\027)35 b Fu(=)27 b Fq( )t Fu(.)42 b(Then)29 b(w)m(e)g(ha)m(v)m(e)g Fq(\027)2243 2583 y Fk(\003)2311 2619 y Fu(=)e Fq(\027)2462 2634 y Fo(s)2499 2619 y Fu([)p Fq(b)p Fu(]pr)2687 2643 y Fo(a)2756 2619 y Fu(:)h Fq(k)2865 2583 y Fo(n)2940 2619 y Fp(\000)-60 b(!)28 b Fq(k)i Fu(and)e(the)h(diagrams)1299 2821 y Fq(k)1353 2785 y Fo(n)1689 2821 y Fq(k)1743 2785 y Fo(n)p 1428 2788 233 4 v 1577 2786 a Fh(-)1509 2754 y Fo(f)1543 2762 y Fj(a)1322 3211 y Fq(k)339 b(k)p 1405 3178 279 4 v 1601 3176 a Fh(-)1458 3149 y Fo(l)1479 3157 y Fj(a)1517 3149 y Fn(=id)p 1348 3108 4 254 v 1349 3108 a Fh(?)1235 3005 y Fo(\027)1274 2981 y Fg(\003)p 1738 3108 V 1739 3108 a Fh(?)1778 3005 y Fo(\027)1817 2981 y Fg(\003)2110 2821 y Fq(k)2164 2785 y Fo(n)2500 2821 y Fq(k)2554 2785 y Fo(n)p 2240 2788 233 4 v 2389 2786 a Fh(-)2309 2748 y Fo(f)2343 2767 y Fj(a)2377 2753 y Fg(0)2134 3211 y Fq(k)g(k)p 2216 3178 279 4 v 2412 3176 a Fh(-)2326 3157 y Fn(id)p 2159 3108 4 254 v 2161 3108 a Fh(?)2047 3005 y Fo(\027)2086 2981 y Fg(\003)p 2549 3108 V 2551 3108 a Fh(?)2590 3005 y Fo(\027)2629 2981 y Fg(\003)0 3343 y Fu(comm)m(ute,)32 b(hence)i Fq(\023)f Fu(is)f(the)h(unique)g(morphism)e(suc)m(h)j(that)e Fq(\023\027)j Fu(=)28 b Fq( )t Fu(.)43 b(So)32 b(\()p Fp(K)q Fq(;)17 b(\023)p Fu(\))33 b(is)f(an)h(equalizer.)0 3528 y(F)-8 b(or)32 b(the)h(next)g(t)m(w)m(o)h(remarks)e(w)m(e)i(will)c (assume)j Fq(Z)i Fu(=)27 b Fp(f)p Fq(k)s(;)17 b Fu(id)2217 3543 y Fo(k)2260 3528 y Fp(g)p Fu(.)0 3669 y Fy(Remark)39 b(6.2.)k Fu(W)-8 b(e)34 b(w)m(an)m(t)i(to)e(sho)m(w)h(that,)g(in)e (general,)h(there)i(are)e(no)g(equalizers)g(in)g(the)h(category)f(of)0 3785 y(SDS.)f(Let)f Fp(G)i Fu(=)28 b(\()p Fq(G;)17 b Fu(\()p Fq(k)s Fu([)p Fq(b)p Fu(]\))p Fq(;)g Fu(\()p Fq(g)1127 3800 y Fo(b)1161 3785 y Fu(\))p Fq(;)g(\014)6 b Fu(\))31 b(and)i Fp(H)c Fu(=)e(\()p Fq(H)r(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(c)p Fu(]\))p Fq(;)g Fu(\()p Fq(h)2308 3800 y Fo(c)2343 3785 y Fu(\))p Fq(;)g(\015)5 b Fu(\))32 b(b)s(e)g(the)h(follo)m(wing)d(SDS)241 3969 y(-)41 b Fq(V)372 3984 y Fo(G)458 3969 y Fu(:=)28 b Fp(f)p Fq(a;)17 b(b;)g(c;)g(d)p Fp(g)p Fu(,)32 b Fq(E)1137 3984 y Fo(G)1224 3969 y Fu(:=)27 b Fp(ff)p Fq(a;)17 b(b)p Fp(g)p Fq(;)g Fp(f)p Fq(c;)g(d)p Fp(gg)p Fu(,)241 4085 y(-)41 b Fq(k)s Fu([)p Fq(b)p Fu(])28 b(:=)g Fq(k)i Fu(=)e Fp(f)p Fu(0)p Fq(;)17 b Fu(1)p Fp(g)31 b Fu(for)h(all)f Fq(b)d Fp(2)g Fq(V)1586 4100 y Fo(G)1645 4085 y Fu(,)241 4202 y(-)41 b Fq(g)362 4217 y Fo(b)423 4202 y Fu(:=)28 b(id)k(for)g(all)e Fq(b)e Fp(2)g Fq(V)1172 4217 y Fo(G)1232 4202 y Fu(,)241 4318 y(-)41 b Fq(\014)33 b Fu(:=)27 b(\()p Fq(a;)17 b(b;)g(c;)g(d)p Fu(\))27 b(=)h(\()p Fq(b)1136 4333 y Fn(1)1176 4318 y Fq(;)17 b(b)1261 4333 y Fn(2)1300 4318 y Fq(;)g(b)1385 4333 y Fn(3)1425 4318 y Fq(;)g(b)1510 4333 y Fn(4)1550 4318 y Fu(\).)241 4502 y(-)41 b Fq(V)372 4517 y Fo(H)466 4502 y Fu(:=)28 b Fp(f)p Fq(a)p Fp(g)p Fu(,)k Fq(E)879 4517 y Fo(F)966 4502 y Fu(:=)c Fp(;)p Fu(,)241 4618 y(-)41 b Fq(k)s Fu([)p Fq(c)p Fu(])27 b(:=)h Fq(k)j Fu(=)c Fp(f)p Fu(0)p Fq(;)17 b Fu(1)p Fp(g)32 b Fu(for)g(all)e Fq(c)e Fp(2)g Fq(V)1587 4633 y Fo(H)1654 4618 y Fu(,)241 4735 y(-)41 b Fq(h)371 4750 y Fo(c)433 4735 y Fu(:=)28 b(id)k(for)g(all)e Fq(c)e Fp(2)g Fq(V)1183 4750 y Fo(H)1250 4735 y Fu(,)241 4851 y(-)41 b Fq(\015)32 b Fu(:=)c(\()p Fq(a)p Fu(\))g(=)f(\()p Fq(c)867 4866 y Fn(1)907 4851 y Fu(\).)0 5035 y(Let)33 b Fq(';)17 b( )31 b Fu(:)d Fp(G)34 b(\000)-60 b(!)27 b(H)34 b Fu(b)s(e)f(t)m(w)m(o)g(morphisms)e(giv)m(en)i(b)m(y)241 5219 y(-)41 b Fq(')379 5234 y Fo(g)419 5219 y Fu(\()p Fq(a)p Fu(\))27 b(:=)h Fq(a)p Fu(,)33 b Fq( )878 5234 y Fo(g)918 5219 y Fu(\()p Fq(a)p Fu(\))28 b(:=)g Fq(c)p Fu(,)241 5335 y(-)41 b Fq(')379 5350 y Fo(s)415 5335 y Fu([)p Fq(a)p Fu(])28 b(:=)g(id)f(=:)h Fq( )982 5350 y Fo(s)1019 5335 y Fu([)p Fq(a)p Fu(],)241 5467 y(-)53 b Fl(e)-67 b Fq(')o Fu(\(1\))28 b(:=)f(1,)786 5440 y Fl(e)770 5467 y Fq( )t Fu(\(1\))g(:=)h(3.)0 5651 y(W)-8 b(e)37 b(w)m(an)m(t)h(to)e(\014nd)h(an)g(equalizer)f Fq(\032)f Fu(:)g Fp(E)43 b(\000)-59 b(!)34 b(G)43 b Fu(of)36 b(these)i(t)m(w)m(o)g(morphisms)d(\(i.e.)55 b Fq('\032)35 b Fu(=)g Fq( )t(\032)i Fu(and)g(\()p Fp(E)9 b Fq(;)17 b(\032)p Fu(\))0 5767 y(univ)m(ersal)32 b(w.r.t.)44 b(this)33 b(prop)s(ert)m(y\).)p eop %%Page: 18 18 18 17 bop 0 251 a Fr(18)894 b(REINHARD)23 b(LA)n(UBENBA)n(CHER)h(AND)h (BODO)g(P)-6 b(AREIGIS)0 450 y Fu(As)33 b(a)f(test)i(ob)5 b(ject)33 b(w)m(e)h(use)f(the)g(SDS)g Fp(F)k Fu(=)27 b(\()p Fq(F)s(;)17 b Fu(\()p Fq(k)s Fu([)p Fq(a)p Fu(]\))p Fq(;)g Fu(\()p Fq(f)2132 465 y Fo(a)2173 450 y Fu(\))p Fq(;)g(\013)q Fu(\))32 b(de\014ned)i(as)f(follo)m(ws)241 631 y(-)41 b Fq(V)372 646 y Fo(F)458 631 y Fu(:=)27 b Fp(f)p Fq(a;)17 b(b;)g(d)p Fp(g)p Fu(,)32 b Fq(E)1050 646 y Fo(F)1137 631 y Fu(:=)c Fp(ff)p Fq(a;)17 b(b)p Fp(g)p Fq(;)g Fp(f)p Fq(a;)g(d)p Fp(gg)p Fu(,)241 748 y(-)41 b Fq(k)s Fu([)p Fq(a)p Fu(])28 b(:=)f Fq(k)k Fu(=)d Fp(f)p Fu(0)p Fq(;)17 b Fu(1)p Fp(g)31 b Fu(for)h(all)f Fq(a)c Fp(2)i Fq(V)1606 763 y Fo(F)1664 748 y Fu(,)241 864 y(-)41 b Fq(f)363 879 y Fo(a)432 864 y Fu(:=)28 b(id)j(for)h(all)f Fq(a)d Fp(2)g Fq(V)1191 879 y Fo(F)1249 864 y Fu(,)241 980 y(-)41 b Fq(\013)28 b Fu(:=)g(\()p Fq(a;)17 b(b;)g(d)p Fu(\))27 b(=)g(\()p Fq(a)1062 995 y Fn(1)1102 980 y Fq(;)17 b(a)1197 995 y Fn(2)1236 980 y Fq(;)g(a)1331 995 y Fn(3)1371 980 y Fu(\).)0 1161 y(The)33 b(follo)m(wing)c(t)m(w)m(o)j(morphisms)f Fq(\033)n(;)17 b(\034)39 b Fu(:)28 b Fp(F)37 b(\000)-59 b(!)27 b(G)38 b Fu(should)32 b(serv)m(e)h(as)f(test)h(morphisms.)42 b(They)33 b(are)f(giv)m(en)0 1278 y(b)m(y)241 1459 y(-)41 b Fq(\033)370 1474 y Fo(g)410 1459 y Fu(\()p Fq(a)p Fu(\))28 b(=)f Fq(\033)723 1474 y Fo(g)764 1459 y Fu(\()p Fq(c)p Fu(\))h(:=)f Fq(a)p Fu(,)33 b Fq(\033)1206 1474 y Fo(g)1247 1459 y Fu(\()p Fq(b)p Fu(\))28 b(:=)f Fq(b)p Fu(,)33 b Fq(\033)1678 1474 y Fo(g)1719 1459 y Fu(\()p Fq(d)p Fu(\))27 b(=)h Fq(d)p Fu(,)315 1575 y Fq(\034)357 1590 y Fo(g)397 1575 y Fu(\()p Fq(a)p Fu(\))g(=)f Fq(\034)697 1590 y Fo(g)738 1575 y Fu(\()p Fq(c)p Fu(\))g(:=)h Fq(a)p Fu(,)33 b Fq(\034)1167 1590 y Fo(g)1207 1575 y Fu(\()p Fq(b)p Fu(\))28 b(:=)g Fq(d)p Fu(,)k Fq(\034)1635 1590 y Fo(g)1676 1575 y Fu(\()p Fq(d)p Fu(\))27 b(=)h Fq(b)p Fu(,)241 1691 y(-)41 b Fq(\033)370 1706 y Fo(s)407 1691 y Fu([)p Fq(a)p Fu(])28 b(:=)g(id)e(=:)i Fq(\034)952 1706 y Fo(s)989 1691 y Fu([)p Fq(a)p Fu(])33 b(for)f(all)f Fq(a)d Fp(2)g Fq(V)1642 1706 y Fo(G)1701 1691 y Fu(,)241 1808 y(-)42 b Fl(e)-56 b Fq(\033)t Fu(\(1\))27 b(:=)h(1,)34 b Fl(e)-57 b Fq(\033)t Fu(\(2\))27 b(:=)h(2,)34 b Fl(e)-57 b Fq(\033)t Fu(\(3\))28 b(:=)f(1,)35 b Fl(e)-57 b Fq(\033)t Fu(\(4\))27 b(:=)h(3,)316 1924 y Fl(e)-56 b Fq(\034)11 b Fu(\(1\))27 b(:=)h(1,)34 b Fl(e)-57 b Fq(\034)12 b Fu(\(2\))27 b(:=)h(3,)34 b Fl(e)-57 b Fq(\034)12 b Fu(\(3\))27 b(:=)h(1,)34 b Fl(e)-57 b Fq(\034)12 b Fu(\(4\))27 b(:=)h(2.)0 2105 y(Then)34 b(it)d(is)h(easy)i(to)e(see)i(that)e Fq(\033)t(')c Fu(=)f Fq(\033)t( )37 b Fu(and)c Fq(\034)11 b(')28 b Fu(=)f Fq(\034)11 b( )t Fu(.)0 2264 y(Assume)34 b(that)g(w)m(e)g(ha)m (v)m(e)h(an)f(equalizer)f Fq(\032)c Fu(:)h Fp(E)37 b(\000)-59 b(!)29 b(G)39 b Fu(of)33 b Fq(')h Fu(and)f Fq( )t Fu(.)47 b(Then)34 b Fq(\032)2809 2279 y Fo(g)2883 2264 y Fu(m)m(ust)g(ha)m(v)m (e)h(the)f(follo)m(wing)0 2383 y(images)23 b Fq(\032)362 2398 y Fo(g)403 2383 y Fu(\()p Fq(a)p Fu(\))28 b(=)f Fq(\032)711 2398 y Fo(g)752 2383 y Fu(\()p Fq(c)p Fu(\))g(=:)i(\026)-50 b Fq(a)25 b Fu(\(since)g Fq(\032)1423 2398 y Fo(g)1463 2383 y Fq(')1527 2398 y Fo(g)1595 2383 y Fu(=)i Fq(\032)1748 2398 y Fo(g)1789 2383 y Fq( )1852 2398 y Fo(g)1892 2383 y Fu(\),)g Fq(\032)2034 2398 y Fo(g)2074 2383 y Fu(\()p Fq(b)p Fu(\))h(=:)2346 2357 y(\026)2350 2383 y Fq(b)p Fu(,)f(and)e Fq(\032)2677 2398 y Fo(g)2717 2383 y Fu(\()p Fq(d)p Fu(\))i(=:)3019 2357 y(\026)3002 2383 y Fq(d)d Fu(in)g Fq(V)3240 2398 y Fo(E)3300 2383 y Fu(.)41 b(F)-8 b(urthermore)0 2499 y(let)44 b Fl(e)-61 b Fq(\032)q Fu(\(1\))38 b(=:)h Fq(i)536 2514 y Fn(1)576 2499 y Fu(,)46 b Fl(e)-61 b Fq(\032)q Fu(\(2\))38 b(=:)h Fq(i)1032 2514 y Fn(2)1071 2499 y Fu(,)47 b Fl(e)-61 b Fq(\032)p Fu(\(3\))39 b(=:)g Fq(i)1528 2514 y Fn(3)1567 2499 y Fu(,)i(and)k Fl(e)-61 b Fq(\032)q Fu(\(4\))38 b(=:)h Fq(i)2220 2514 y Fn(4)2259 2499 y Fu(.)63 b(Then)41 b(w)m(e)f(ha)m(v)m(e)g Fq(i)3025 2514 y Fn(1)3104 2499 y Fu(=)k Fl(e)-61 b Fq(\032)q Fu(\(1\))38 b(=)44 b Fl(e)-60 b Fq(\032)12 b Fl(e)-67 b Fq(')p Fu(\(1\))38 b(=)6 2631 y Fl(e)-61 b Fq(\032)67 2604 y Fl(e)50 2631 y Fq( )t Fu(\(1\))28 b(=)34 b Fl(e)-60 b Fq(\032)p Fu(\(3\))28 b(=)g Fq(i)715 2646 y Fn(3)755 2631 y Fu(.)45 b(Since)38 b Fl(e)-60 b Fq(\032)33 b Fu(is)g(order)g(preserving)h(on)e(the)i (connected)h(comp)s(onen)m(ts,)e(w)m(e)h(get)f Fq(i)3653 2646 y Fn(1)3721 2631 y Fp(\024)c Fq(i)3860 2646 y Fn(2)0 2747 y Fu(and)h Fq(i)220 2762 y Fn(1)288 2747 y Fp(\024)e Fq(i)426 2762 y Fn(4)466 2747 y Fu(.)42 b(Since)31 b(all)d(of)i Fq(i)1063 2762 y Fn(1)1102 2747 y Fq(;)17 b(i)1179 2762 y Fn(2)1219 2747 y Fq(;)g(i)1296 2762 y Fn(4)1366 2747 y Fu(map)29 b(in)m(to)h(a)g(connected)h(comp)s(onen)m(t)g(of)f(the)g (graph)g Fq(E)37 b Fu(\(they)31 b(map)0 2863 y(in)m(to)g(\026)-50 b Fq(a;)287 2837 y Fu(\026)291 2863 y Fq(b;)393 2837 y Fu(\026)376 2863 y Fq(d)30 b Fu(resp.\))43 b(w)m(e)32 b(m)m(ust)e(ha)m(v)m(e)i Fq(i)1375 2878 y Fn(2)1445 2863 y Fu(and)f Fq(i)1666 2878 y Fn(4)1736 2863 y Fu(comparable)e(in)g(the)i (order)g(of)f Fp(j)p Fq(\016)t Fp(j)p Fu(,)g(the)h(up)s(date)f(sc)m (hedule)0 2979 y(of)i Fp(E)9 b Fu(.)0 3138 y(The)48 b(morphisms)f Fq(\033)k Fu(and)d Fq(\034)59 b Fu(ha)m(v)m(e)48 b(unique)g (factorizations)e Fq(\033)57 b Fu(=)c Fq(\032\033)2674 3153 y Fn(0)2762 3138 y Fu(and)47 b Fq(\034)65 b Fu(=)53 b Fq(\032\034)3294 3153 y Fn(0)3381 3138 y Fu(through)48 b(the)0 3255 y(equalizer.)43 b(No)m(w)33 b Fl(e)-57 b Fq(\033)722 3270 y Fn(0)762 3255 y Fu(\()p Fq(i)833 3270 y Fn(2)873 3255 y Fu(\))27 b(=)j Fl(e)-57 b Fq(\033)t Fu(\(2\))27 b(=)h(2)i(and)j Fl(e)-57 b Fq(\033)1679 3270 y Fn(0)1719 3255 y Fu(\()p Fq(i)1790 3270 y Fn(4)1830 3255 y Fu(\))27 b(=)j Fl(e)-57 b Fq(\033)t Fu(\(4\))27 b(=)h(3.)42 b(Since)33 b Fl(e)-57 b Fq(\033)2740 3270 y Fn(0)2811 3255 y Fu(is)30 b(order)h(preserving)h(w)m(e)g(get)0 3371 y Fq(i)33 3386 y Fn(2)100 3371 y Fq(<)c(i)237 3386 y Fn(4)277 3371 y Fu(.)0 3530 y(With)34 b(the)h(same)g(argumen)m(t)f (for)g Fq(\034)46 b Fu(w)m(e)36 b(get)f Fq(i)1692 3545 y Fn(2)1763 3530 y Fq(>)c(i)1903 3545 y Fn(4)1978 3530 y Fu(a)j(con)m(tradiction.)49 b(Hence)36 b(there)f(cannot)g(exist)g(an) 0 3646 y(equalizer)d(for)g Fq(')p Fu(,)h Fq( )t Fu(.)0 3785 y Fy(Remark)24 b(6.3.)34 b Fu(In)22 b(this)f(con)m(text)i(and)f (with)g Fq(Z)34 b Fu(=)28 b Fp(f)p Fq(k)s(;)17 b Fu(id)2071 3800 y Fo(k)2113 3785 y Fp(g)22 b Fu(it)f(migh)m(t)f(app)s(ear)h(as)h (if)f Fp(O)31 b Fu(=)3252 3704 y Fl(\000)3298 3785 y Fu(\()p Fq(a)p Fu(\))p Fq(;)17 b Fu(\()p Fq(k)s Fu(\))p Fq(;)g Fu(\(id)o(\))p Fq(;)g Fu(\()p Fq(a)p Fu(\))3971 3704 y Fl(\001)0 3901 y Fu(is)33 b(an)g(initial)c(ob)5 b(ject)34 b(in)f Fy(SDS)p Fu(.)h(This)f(is)g(not)g(the)h(case)g(since)f (this)g(SDS)g(do)s(es)h(not)f(admit)f(a)h(morphism)0 4017 y(in)m(to)f(an)m(y)h(SDS)f(with)h(up)s(date)g(functions)f(not)h (the)g(iden)m(tit)m(y)f(on)g(the)h(diagonal)e(in)g Fq(k)3131 3981 y Fo(n)3178 4017 y Fu(,)i(the)g(diagram)1728 4188 y Fq(k)339 b(k)p 1811 4155 279 4 v 2007 4153 a Fh(-)1921 4135 y Fn(id)1704 4578 y Fq(k)1758 4542 y Fo(n)2094 4578 y Fq(k)2148 4542 y Fo(n)p 1834 4545 233 4 v 1983 4543 a Fh(-)1911 4511 y Fo(f)1945 4519 y Fj(\013)p 1753 4475 4 254 v 1755 4475 a Fh(?)1635 4383 y Fu(\001)p 2143 4475 V 2145 4475 a Fh(?)2184 4383 y Fu(\001)0 4704 y(do)s(es)26 b(not)g(comm)m(ute.)41 b(T)-8 b(o)25 b(complete)h(our)f(study)i(of)f (pro)s(ducts)g(w)m(e)h(ha)m(v)m(e,)h(ho)m(w)m(ev)m(er,)i(the)c(empt)m (y)h(pro)s(duct.)0 4842 y Fy(Lemma)37 b(6.4.)42 b Ft(The)34 b(SDS)1030 4762 y Fl(\000)1076 4842 y Fp(;)p Fq(;)17 b Fp(;)27 b Fu(=)g(\(\))p Fq(;)17 b Fp(;)28 b Fu(=)f(\(\))p Fq(;)17 b Fp(;)27 b Fu(=)h(\(\))2028 4762 y Fl(\001)2108 4842 y Ft(is)35 b(a)f(\014nal)h(obje)-5 b(ct)34 b(in)h Fy(SDS)p Ft(.)0 5057 y(Pr)-5 b(o)g(of.)41 b Fu(There)34 b(is)e(a)g(unique)h(morphism)e(of)h(graphs)h Fp(;)28 b(\000)-60 b(!)27 b Fq(F)47 b Fu(and)32 b(the)h(diagram)1704 5242 y Fq(k)1758 5206 y Fo(n)2094 5242 y Fq(k)2148 5206 y Fo(n)p 1834 5209 233 4 v 1983 5207 a Fh(-)1920 5175 y Fo(f)1954 5185 y Fj(i)1680 5622 y Fp(f\003g)240 b(f\003g)p 1858 5599 184 4 v 1959 5597 a Fh(-)1921 5579 y Fn(id)p 1753 5529 4 254 v 1755 5529 a Fh(?)p 2143 5529 V 307 w(?)0 5767 y Fu(comm)m(utes.)3370 b Fd(\003)p eop %%Page: 19 19 19 18 bop 1235 251 a Fr(SEQUENTIAL)32 b(D)n(YNAMICAL)g(SYSTEMS)1158 b(19)687 450 y Fu(7.)48 b Fv(Simula)-7 b(tions)39 b(and)f(their)g (Effects)f(on)h(St)-7 b(a)g(te)37 b(Sp)-7 b(a)n(ces)0 667 y Fu(As)37 b(w)m(e)g(men)m(tioned)e(in)h(the)g(in)m(tro)s(duction,) g(w)m(e)h(consider)f(a)g(morphism)e Fq(')f Fu(:)h Fp(F)43 b(\000)-59 b(!)33 b(G)42 b Fu(as)36 b(a)g(sim)m(ulation)0 784 y(if)d Fq(')i Fu(is)f(a)g(monomorphism)e({)i(then)i Fp(G)k Fu(is)34 b(sim)m(ulated)g(b)m(y)h Fp(F)44 b Fu({)34 b(or,)h(if)f Fq(')g Fu(is)g(an)h(epimorphism)d({)i(then)i Fp(F)0 900 y Fu(is)d(sim)m(ulated)e(b)m(y)j Fp(G)6 b Fu(.)45 b(W)-8 b(e)33 b(will)e(only)i(consider)g(those)h(monomorphisms) c Fq(')j Fu(where)h Fq(')3188 915 y Fo(g)3261 900 y Fu(is)f(surjectiv)m (e)h(on)0 1016 y(the)f(set)g(of)f(v)m(ertices,)i(and)f(the)g Fq(')1233 1031 y Fo(s)1270 1016 y Fu([)p Fq(b)p Fu(])g(are)f(injectiv)m (e.)44 b(W)-8 b(e)33 b(will)d(call)h(them)h Ft(inje)-5 b(ctive)34 b(monomorphisms)p Fu(.)0 1163 y Fy(Lemma)51 b(7.1.)d Ft(If)e Fq(')i Fu(:)h Fp(F)58 b(\000)-58 b(!)49 b(G)j Ft(is)46 b(an)f(inje)-5 b(ctive)46 b(monomorphism,)g(then)g Fq(')3032 1126 y Fk(\003)3120 1163 y Fu(:)i Fq(k)3249 1126 y Fo(n)3345 1163 y Fp(\000)-57 b(!)48 b Fq(k)3567 1126 y Fo(m)3680 1163 y Ft(is)d(an)0 1279 y(inje)-5 b(ctive)34 b(map.)0 1512 y(Pr)-5 b(o)g(of.)41 b Fu(Let)33 b Fq(')537 1476 y Fk(\003)576 1512 y Fu(\()p Fq(x)669 1527 y Fn(1)709 1512 y Fq(;)17 b(:)g(:)g(:)f(;)h(x)983 1527 y Fo(n)1030 1512 y Fu(\))28 b(=)f Fq(')1263 1476 y Fk(\003)1302 1512 y Fu(\()p Fq(y)1388 1527 y Fn(1)1427 1512 y Fq(;)17 b(:)g(:)g(:)f(;)h (y)1694 1527 y Fo(n)1740 1512 y Fu(\).)44 b(Then)1025 1610 y Fl(\000)1070 1691 y Fq(')1134 1706 y Fo(s)1171 1691 y Fu([)p Fq(b)1239 1706 y Fn(1)1279 1691 y Fu(]\()p Fq(x)p Fu([)p Fq(')1490 1706 y Fo(g)1531 1691 y Fu(\()p Fq(b)1610 1706 y Fn(1)1650 1691 y Fu(\)]\))p Fq(;)17 b(:)g(:)g(:)e(;)i(')2035 1706 y Fo(s)2072 1691 y Fu([)p Fq(b)2140 1706 y Fo(m)2207 1691 y Fu(]\()p Fq(x)p Fu([)p Fq(')2418 1706 y Fo(g)2459 1691 y Fu(\()p Fq(b)2538 1706 y Fo(m)2604 1691 y Fu(\)]\))2707 1610 y Fl(\001)1025 1810 y Fu(=)1128 1729 y Fl(\000)1174 1810 y Fq(')1238 1825 y Fo(s)1275 1810 y Fu([)p Fq(b)1343 1825 y Fn(1)1383 1810 y Fu(]\()p Fq(y)t Fu([)p Fq(')1591 1825 y Fo(g)1630 1810 y Fu(\()p Fq(b)1709 1825 y Fn(1)1749 1810 y Fu(\)]\))p Fq(;)g(:)g(:)g(:)f(;)h(')2135 1825 y Fo(s)2171 1810 y Fu([)p Fq(b)2239 1825 y Fo(m)2306 1810 y Fu(]\()p Fq(y)t Fu([)p Fq(')2514 1825 y Fo(g)2553 1810 y Fu(\()p Fq(b)2632 1825 y Fo(m)2699 1810 y Fu(\)]\))2802 1729 y Fl(\001)2848 1810 y Fq(:)0 1989 y Fu(Hence,)40 b(for)c(all)g Fq(i)f Fu(=)h(1)p Fq(;)17 b(:)g(:)g(:)e(;)i(m)p Fu(,)39 b(w)m(e)f(get)f Fq(')1594 2004 y Fo(s)1631 1989 y Fu([)p Fq(b)1699 2004 y Fo(i)1728 1989 y Fu(]\()p Fq(x)p Fu([)p Fq(')1939 2004 y Fo(g)1979 1989 y Fu(\()p Fq(b)2058 2004 y Fo(i)2087 1989 y Fu(\)]\))e(=)h Fq(')2401 2004 y Fo(s)2437 1989 y Fu([)p Fq(b)2505 2004 y Fo(i)2534 1989 y Fu(]\()p Fq(y)t Fu([)p Fq(')2742 2004 y Fo(g)2781 1989 y Fu(\()p Fq(b)2860 2004 y Fo(i)2889 1989 y Fu(\)]\).)57 b(Since)38 b(the)f Fq(')3572 2004 y Fo(s)3609 1989 y Fu([)p Fq(b)3677 2004 y Fo(i)3706 1989 y Fu(])g(are)0 2105 y(injectiv)m(e)h(w)m(e)h(get)g (that)f Fq(x)p Fu([)p Fq(')1074 2120 y Fo(g)1114 2105 y Fu(\()p Fq(b)1193 2120 y Fo(i)1222 2105 y Fu(\)])f(=)g Fq(y)t Fu([)p Fq(')1580 2120 y Fo(g)1619 2105 y Fu(\()p Fq(b)1698 2120 y Fo(i)1727 2105 y Fu(\)].)60 b(No)m(w,)41 b Fq(')2200 2120 y Fo(g)2278 2105 y Fu(is)c(surjectiv)m(e,)k(so)e(w)m (e)g(get)f(\()p Fq(x)3388 2120 y Fn(1)3428 2105 y Fq(;)17 b(:)g(:)g(:)f(;)h(x)3702 2120 y Fo(n)3749 2105 y Fu(\))37 b(=)0 2221 y(\()p Fq(y)86 2236 y Fn(1)125 2221 y Fq(;)17 b(:)g(:)g(:)f(;)h(y)392 2236 y Fo(n)438 2221 y Fu(\).)3320 b Fd(\003)0 2454 y Fu(Considering)32 b(epimorphisms,)g(w)m(e)i(will)d (only)h(consider)h(those)h Fq(')28 b Fu(:)h Fp(F)38 b(\000)-60 b(!)28 b(G)6 b Fu(,)33 b(where)i Fq(')3238 2469 y Fo(g)3311 2454 y Fu(is)d(injectiv)m(e)h(on)0 2571 y(the)g(set)g(of)f(v)m (ertices,)i(and)f(the)g Fq(')1233 2586 y Fo(s)1270 2571 y Fu([)p Fq(b)p Fu(])g(are)f(surjectiv)m(e.)45 b(W)-8 b(e)33 b(will)d(call)h(them)i Ft(surje)-5 b(ctive)34 b(epimorphisms)p Fu(.)0 2717 y Fy(Lemma)k(7.2.)43 b Ft(If)35 b Fq(')29 b Fu(:)h Fp(F)39 b(\000)-57 b(!)29 b(G)42 b Ft(is)36 b(a)f(surje)-5 b(ctive)36 b(epimorphism,)e(then)i Fq(')2748 2681 y Fk(\003)2816 2717 y Fu(:)30 b Fq(k)2927 2681 y Fo(n)3004 2717 y Fp(\000)-58 b(!)30 b Fq(k)3207 2681 y Fo(m)3309 2717 y Ft(is)36 b(a)f(surje)-5 b(ctive)0 2833 y(map.)0 3067 y(Pr)g(o)g(of.)41 b Fu(Let)33 b(\()p Fq(y)t Fu([)p Fq(b)631 3082 y Fn(1)670 3067 y Fu(])p Fq(;)17 b(:)g(:)g(:)f(;)h(y)t Fu([)p Fq(b)1036 3082 y Fo(m)1102 3067 y Fu(]\))28 b Fp(2)g Fq(k)1343 3031 y Fo(m)1409 3067 y Fu(.)44 b(Then)33 b(there)h(is)e(an)g Fq(m)p Fu(-tuple)781 3165 y Fl(\000)826 3245 y Fq(z)t Fu([)p Fq(')966 3260 y Fo(g)1007 3245 y Fu(\()p Fq(b)1086 3260 y Fn(1)1126 3245 y Fu(\)])p Fq(;)17 b(:)g(:)g(:)f(;)h(z)t Fu([)p Fq(')1550 3260 y Fo(g)1590 3245 y Fu(\()p Fq(b)1669 3260 y Fo(m)1736 3245 y Fu(\)])1801 3165 y Fl(\001)1875 3245 y Fp(2)28 b Fq(k)s Fu([)p Fq(')2114 3260 y Fo(g)2154 3245 y Fu(\()p Fq(b)2233 3260 y Fn(1)2273 3245 y Fu(\)])22 b Fp(\002)h Fq(:)17 b(:)g(:)k Fp(\002)i Fq(k)s Fu([)p Fq(')2841 3260 y Fo(g)2881 3245 y Fu(\()p Fq(b)2960 3260 y Fo(m)3027 3245 y Fu(\)])p Fq(;)0 3434 y Fu(suc)m(h)42 b(that)446 3353 y Fl(\000)492 3434 y Fq(')556 3449 y Fo(s)593 3434 y Fu([)p Fq(b)661 3449 y Fn(1)701 3434 y Fu(]\()p Fq(z)t Fu([)p Fq(')906 3449 y Fo(g)947 3434 y Fu(\()p Fq(b)1026 3449 y Fn(1)1066 3434 y Fu(\)]\))p Fq(;)17 b(:)g(:)g(:)e(;)i(')1451 3449 y Fo(s)1488 3434 y Fu([)p Fq(b)1556 3449 y Fo(m)1623 3434 y Fu(]\()p Fq(z)t Fu([)p Fq(')1828 3449 y Fo(g)1869 3434 y Fu(\()p Fq(b)1948 3449 y Fo(m)2015 3434 y Fu(\)]\))2118 3353 y Fl(\001)2204 3434 y Fu(=)40 b(\()p Fq(y)t Fu([)p Fq(b)2478 3449 y Fn(1)2517 3434 y Fu(])p Fq(;)17 b(:)g(:)g(:)f(;)h(y)t Fu([)p Fq(b)2883 3449 y Fo(m)2949 3434 y Fu(]\),)42 b(since)f(the)f Fq(')3569 3449 y Fo(s)3606 3434 y Fu([)p Fq(b)3674 3449 y Fo(i)3703 3434 y Fu(])g(are)0 3552 y(surjectiv)m(e.)k(De\014ne)32 b Fq(x)p Fu([)p Fq(a)908 3567 y Fo(i)937 3552 y Fu(])27 b(:=)h Fq(z)t Fu([)p Fq(')1262 3567 y Fo(g)1303 3552 y Fu(\()p Fq(b)1382 3567 y Fo(j)1419 3552 y Fu(\)])g Fp(2)g Fq(k)s Fu([)p Fq(a)1738 3567 y Fo(i)1766 3552 y Fu(],)k(if)e Fq(')2004 3567 y Fo(g)2044 3552 y Fu(\()p Fq(b)2123 3567 y Fo(j)2160 3552 y Fu(\))d(=)h Fq(a)2380 3567 y Fo(i)2408 3552 y Fu(,)j(and)h Fq(x)p Fu([)p Fq(a)2788 3567 y Fo(i)2816 3552 y Fu(])g(arbitrary)e(if)f Fq(a)3426 3567 y Fo(i)3494 3552 y Fq(=)-61 b Fp(2)29 b Fu(Im)o(\()p Fq(')3795 3567 y Fo(g)3835 3552 y Fu(\).)0 3668 y(The)e Fq(x)p Fu([)p Fq(a)327 3683 y Fo(i)356 3668 y Fu(])h(=)f Fq(z)t Fu([)p Fq(')654 3683 y Fo(g)695 3668 y Fu(\()p Fq(b)774 3683 y Fo(j)811 3668 y Fu(\)])f(are)g(w)m(ell)g(de\014ned)h (since)g Fq(')1876 3683 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b(\000)-57 b(!)45 b(G)51 b Ft(b)-5 b(e)44 b(an)g(inje)-5 b(ctive)43 b(monomorphism,)i(and)f(let)g Fp(S)7 b Fu(\()p Fq(')p Fu(\))46 b(:)g Fp(S)7 b Fu(\()p Fp(F)j Fu(\))0 4383 y Fp(\000)-57 b(!)27 b(S)7 b Fu(\()p Fp(G)f Fu(\))36 b Ft(b)-5 b(e)34 b(the)h(gr)-5 b(aph)35 b(morphism)e(of)i(the)g(asso)-5 b(ciate)g(d)33 b(state)j(sp)-5 b(ac)g(es.)43 b(Then)148 4573 y Fu(\(1\))f Fp(S)7 b Fu(\()p Fq(')p Fu(\))35 b Ft(maps)f(e)-5 b(ach)34 b(limit)h(cycle)f(of)h Fp(S)7 b Fu(\()p Fp(F)j Fu(\))35 b Ft(bije)-5 b(ctively)34 b(onto)h(a)g(limit)f(cycle)h(of)g Fp(S)7 b Fu(\()p Fp(G)f Fu(\))p Ft(;)148 4689 y Fu(\(2\))42 b Fp(S)7 b Fu(\()p Fq(')p Fu(\))35 b Ft(is)g(inje)-5 b(ctive)34 b(on)g(the)h(set)g(of)g (limit)f(cycles;)148 4805 y Fu(\(3\))42 b Fp(S)7 b Fu(\()p Fq(')p Fu(\))35 b Ft(maps)f(tr)-5 b(ansients)35 b(inje)-5 b(ctively)34 b(into)h(tr)-5 b(ansients,)34 b(pr)-5 b(eserving)34 b(endp)-5 b(oints.)0 5039 y(Pr)g(o)g(of.)41 b Fu(The)34 b(three)f(statemen)m(ts)g(are)g(clear,)f(since)h Fp(S)7 b Fu(\()p 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Ft(onto)g(the)g(set)g(of)f (limit)h(cycles)f(of)h Fp(S)7 b Fu(\()p Fp(G)f Fu(\))p Ft(;)p eop %%Page: 20 20 20 19 bop 0 251 a Fr(20)894 b(REINHARD)23 b(LA)n(UBENBA)n(CHER)h(AND)h (BODO)g(P)-6 b(AREIGIS)148 450 y Fu(\(3\))42 b Ft(if)33 b(two)h(no)-5 b(des)33 b(of)g(a)h(tr)-5 b(ansient)33 b(in)h Fp(S)7 b Fu(\()p Fp(F)j Fu(\))34 b Ft(of)f(distanc)-5 b(e)33 b Fq(n)20 b Fu(+)g(1)33 b Ft(ar)-5 b(e)34 b(mapp)-5 b(e)g(d)32 b(into)i(the)g(same)f(no)-5 b(de)315 566 y(of)38 b Fp(S)7 b Fu(\()p Fp(G)f Fu(\))p Ft(,)40 b(then)f(the)f(p)-5 b(ath)39 b(b)-5 b(etwe)g(en)37 b(the)i(two)g(no)-5 b(des)37 b(in)i Fp(S)7 b Fu(\()p Fp(F)j Fu(\))38 b Ft(is)h(mapp)-5 b(e)g(d)37 b(onto)i(a)f(limit)g(cycle)315 683 y(of)c(length)h Fq(t)g Ft(in)f Fp(S)7 b Fu(\()p Fp(G)f Fu(\))p Ft(,)36 b(wher)-5 b(e)34 b Fq(t)h Ft(divides)f Fq(n)p Ft(.)0 892 y(Pr)-5 b(o)g(of.)41 b Fu(Again,)31 b(the)h(three)h(statemen)m(ts)f (follo)m(w)e(from)h(the)h(fact)f(that)h Fp(S)7 b Fu(\()p Fq(')p Fu(\))28 b(:)g Fp(S)7 b Fu(\()p Fp(F)j Fu(\))28 b Fp(\000)-60 b(!)28 b(S)7 b Fu(\()p Fp(G)f Fu(\))32 b(is)f(surjec-)0 1008 y(tiv)m(e)i(on)f(the)h(set)g(of)g(v)m(ertices.) 2720 b Fd(\003)0 1217 y Fu(The)32 b(t)m(w)m(o)g(prop)s(ositions)e(sho)m (w)i(what)g(kind)f(of)g(information)d(ab)s(out)i(the)i(dynamical)d(b)s (eha)m(vior)i(of)g(some)0 1333 y(SDS)h(is)g(preserv)m(ed)k(if)31 b(it)h(is)g(sim)m(ulated)f(b)m(y)i(another)g(SDS.)1660 1573 y Fv(References)214 1731 y Fx([1])42 b(C.)25 b(L.)g(Barrett,)g(R.) h(Thord,)f(and)g(C.)h(M.)f(Reidys,)h(Kno)n(wledge)e(and)h(Net)n(w)n (orks)f(in)i(a)f(Dynamical)g(Econom)n(y)-7 b(,)24 b(in)344 1831 y Fs(Simulations)33 b(in)g(De)l(cision)h(Making)h(for)f(So)l (cio-te)l(chnic)l(al)h(Systems)p Fx(,)c(M.)h(Bec)n(kman)e(and)i(B.)f (Johansson)e(and)344 1930 y(F.)f(Snic)n(k)-5 b(ars)26 b(and)i(R.)f(Thord)g(\(eds.\),)h(Springer)f(V)-7 b(erlag,)27 b(New)g(Y)-7 b(ork,)27 b(1998.)214 2030 y([2])42 b(C.)c(L.)g(Barrett,)i (and)e(C.)h(M.)f(Reidys,)j(Elemen)n(ts)d(of)g(a)g(Theory)f(of)h(Sim)n (ulation,)j(I:)d(Sequen)n(tial)g(CA)h(Ov)n(er)344 2130 y(Random)27 b(Graphs,)g Fs(Appl.)k(Math.)g(and)f(Comput.)e Fb(98)p Fx(,)g(pp.)g(241{259,)c(1999.)214 2229 y([3])42 b(C.)24 b(L.)g(Barrett,)g(H.)h(S.)f(Mortv)n(eit,)h(and)f(C.)g(M.)h (Reidys,)f(Elemen)n(ts)g(of)g(a)g(Theory)f(of)i(Sim)n(ulation)f(I)r(I:) g(Sequen)n(tial)344 2329 y(Dynamical)j(Systems,)g Fs(Appl.)k(Math.)g (and)g(Comp.)e Fb(107)e Fx(\(2{3\),)g(pp.)h(121{136,)c(1999.)214 2429 y([4])42 b(C.)19 b(L.)f(Barrett,)i(H.)f(S.)g(Mortv)n(eit,)h(and)f (C.)g(M.)g(Reidys,)i(Elemen)n(ts)d(of)h(a)f(Theory)g(of)h(Sim)n (ulation)f(I)r(I)r(I:)i(Equiv)-5 b(alence)344 2528 y(of)27 b(SDS,)i Fs(Appl.)i(Math.)g(and)f(Comp.)f Fb(122)p Fx(,)e(pp.)h (325{340,)c(2001.)214 2628 y([5])42 b(L.)27 b(Garcia,)g(A.)h(Jarrah,)d (R.)j(Laub)r(en)n(bac)n(her,)e(A)i(Classi\014cation)e(of)i(Finite)g (Dynamical)g(Systems,)f(preprin)n(t.)214 2727 y([6])42 b(R.)35 b(Laub)r(en)n(bac)n(her)f(and)h(B.)g(P)n(areigis,)g(Equiv)-5 b(alence)34 b(Relations)h(on)g(Finite)h(Dynamical)f(Systems,)i Fs(A)l(dv.)g(in)344 2827 y(Appl.)31 b(Math.)e Fb(26)e Fx(\(2001\))f(237{251.)0 3057 y Fw(Vir)n(ginia)31 b(Bioinf)n(orma)-6 b(tics)33 b(Institute,)e(Bla)n(cksbur)n(g,)g(V)-10 b(A,)30 b(24061)0 3199 y Fs(E-mail)h(addr)l(ess)7 b Fx(:)38 b Fa(reinhard@vbi.vt)o(.ed)o(u)0 3411 y Fw(Ma)-6 b(thema)g(tisches)37 b(Institut,)e(Universit)1527 3404 y(\177)1524 3411 y(at)g(M)1744 3404 y(\177)1741 3411 y(unchen,)h(Theresienstr.)54 b(39,)35 b(D-80333)g(M)3343 3404 y(\177)3340 3411 y(unchen,)h(Ger-)0 3511 y(many)0 3654 y Fs(E-mail)31 b(addr)l(ess)7 b Fx(:)38 b Fa(pareigis@rz.mat)o(hem)o(at)o(ik.)o(un)o(i-)o(mue)o(nc)o(hen)o(.d)o (e)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF