An Application of Linear Automata to Near Rings

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1 ppled Mhemcs,, 3, hp://dxdoorg/436/m33 Publshed Ole November (hp://wwwscrporg/ourl/m) pplco o Ler uom o Ner Rgs Sog You, Yu Feg, Mg Co, Ypg We School o Mhemcs d Compuer Scece, Hube Uversy, Wuh, Ch Eml: yousog@63com Receved ugus 3, ; revsed Sepember 3, ; cceped Ocober 7, BSTRCT I hs pper, we hve esblshed me coeco bewee er-gs d ler uom, d ob he ollowg resuls: ) For er-rg N here exss ler GS S wh N N S () N, s bel, (b) N hs dey, (c) There s some d N d such h N s geered by {,d}; ) Le h : S S be GS- N h q h q h or ll epmorphsm The here exss er-rg epmorphsm h rom N S o S wh qq d N S ; 3) Le QBFG,,,, be G The () : Q N : Q,, B, F Q, G Q s ccessble, (b) Q = N(), (c) / : Q /,, B, F, Q wh F q, : Fq, d G, : Gq, q s reduced, (d) / s mml Keywords: Ler uom; ccessble; GS-Homomorphsm; Ner-Rg Iroduco uom coss o pus, ses, d oupus, ogeher wh mps whch descrbe how ew pus ec he se d he oupu sem-uomo s rple S QF,,, where Q d re ses, clled he se se d pu se, d F s uco rom Q Q, clled he se-rso uco I Q s group, we cll S group-semuomo d bbreve hs by GS uom coss o pus, ses, d oupus, ogeher wh mps whch descrbe how ew pus ec he se d he oupu semuomo s rple S QF,,, where Q d re ses, clled he se se d he pu se, d F s uco rom Q Q, clled he se-rso uco I Q s group (we lwys wre ddvely), we cll S group-semuomo d bbreve hs by GS For q Q d we erpree F q, s he ew se obed rom he old se q by me o he pu [] I S QF,, s semuomo, we ge colleco o mppgs rom Q o Q, oe or ech, whch re gve by q : Fq, Hece descrbes he eec o he pu o he se se Q o S I he pu s ollowed by he pu, he semuomo moves rom he se qq rs o q d he o q We exed (s usul) o he ree mood over cossg o ll e sequeces o elemes o, cludg he empy sequece, d ge, e he mp s moomorphsm rom o he rsormo mood over Q wh dq I he cse o GS s, we re lso ble o sudy he superposo (deed powsely) o wo smuleous pus, Hece s url o cosder d ll o s sums d producs (composo o mps) The obvous rmewor or h s, o course, he srucure o er rg Le S QF,, be GS, The suber-rg N S o M Q geered by dq d ll s s clled he sycc er-rg o S Thus N S s lwys er-rg wh dey I Q s e, he N S s e, oo [] Dscusso ) The homomorphsm cse Le Q d be ddve groups wh zero d F homomorphsm rom he drec produc Q We he cll QF,, homomorphc Becuse o GS q F q, F q,, Fq, F, q o, we ge, where s homomorphsm (e dsrbuve eleme N(Q)), whle s he mp wh cos vlue I o pu c chge he zero se, e or ll, he N S obvously s dsrbuvely geered er-rg, cossg o -sums o powers o whch re edomorphsms, we lso ge dsrbuvely geered er-rg F s ddve he rs compoe For homomorphc GSs oe sees by Copyrgh ScRes

2 S F YOU ET L 65 duco h, where he mp brces s cos Ech power s homomorphsm [3] ) The ler cse s specl cse o he homomorphsm cse whch Q d re bel groups (or more geerlly, R-modules or some rg R) d where F s ler Le Q d be ree R-modules wh e bse X, Y respecvely Le X, Y m The he co o F c be descrbed by mm -mrx Z z over R we replce ech eleme o Q d o by s decomposo duces decomposo o Z such h F q, Z q, z z m z m z m q z z z z m mm m, m m, m : BqC We he ge q I, B q B C B C C prculr, C, we ge q d B q N S s rg, geered by B d he u mrx I [4] o he oher hd, B, he q We C ge C * yhow, ech (d hece ech or ) s e mp rom Q o Q I Q s ree o X wh X he we c exed he de o mrx represeos rom ler mps o e mps Le be e mp The decomposes s c where s homomorphsm d c s cos Le F be he mrx or wh respec o X Ive symbol e wh eeeee d er re e or ll r R The F c e Esblshes somorphsm bewee M (Q) (ll e o Q) d suber-rg o ll mrces over R e [3] 3 M Resuls Theorem Le S QF,, be homomorphc GS, The NS : N Proo N NS s cler Coversely suces o show h N s er-rg, sce obvously N cos ll d d Q I c, we show h N s suber-rg o M(Q) Te N, g N I s cler h gn So cosder g : g Hece we oly loo he ls expresso (), le * The We rs ocus our eo o or mome d pu N Thereore we ge wh By duco, hs s N Le S QF,, be homomorphc The zero-sym- : N N S cosss merc pr N S S, d o ll e sums o elemes o he orm c c 3,,,, d c d wh I c, ll elemes c c re N S Cog N S The versely, e g By sdrd group heory, we c rrge g o sums d dereces o elemes o he orm c c, where c s he sum o some s [5] I S be ler The (wh : d ) N S z z z z Z ( s o e- gve eger ), Hece N M Q geered by d, Sce rg, S s he suber-rg o M Q s N S s rg, oo [6] We c d group Q such h N s somorphc o M Q Le be dex se or N Le F q, : q The suber-rg N o N, e N N N S wh S QF,, Sce every errg c be embedded er-rg wh dey, we ge every er-rg c be embedded he er-rg o some GS [7] Theorem For er-rg N here exss ler S () N, s bel, (b) N hs dey, (c) There s some d N d such h N s geered by, d Proo Le N be er-rg wh ()-(c), we ow h N s somorphc o suber-rg N o M N, [] GS S wh N N Copyrgh ScRes

3 66 S F YOU ET L Le d d be he mges o d d N Sce d s dsrbuve, d s edomorphsm o N, d d N s geered by d d d, whece N N zd zd zd z Z ( s o egve eger) Now le, : Q, : N, d F q, : qd The QF,, s ler GS, Sce N, s bel Sce d we ge N NS Furhermore, e N c S We ge wh d N N c Ths shows N c S N c (wh cos vlue c) s c Hece NS N N Coversely, every c Nc N S sce c c I s cusomry lgebrc uom heory o cosder he semgroup-epmorphsm NS gve by The de o smuleous pus ebles us o rser hs epmorphsm rom semgroups o errgs We c, or sce, erpre s beg he complex pu pu sequece ogeher wh he smuleous pu ( double sregh) We exed o he ree er-rg over I w,, s word we dee : w,, F q, : q Thus, d w we ge exeded smuleous sequel GS S : Q,, F Le I be { s he zero mp} The I s er-rg del d we ge by he homomorphsm heorem: I NS NS I we hd used rgh er-rgs, we would hve N S -somorphc o I Hece N S c be vewed s homomorphc mge o I s, however, mpossble o gve ce cocl orm or ll elemes o possble rele comes rom he observo h oe mgh replce by v, he ree lgebr vrey v o er-rgs cog N S (or sce, oe mgh e v s he vrey geered by N S ) eo! I lredy bers some ddve srucure, hs ew ddo c (d mos cses wll) be dere rom he gve ddo! I prculr, our ew ddo s oe d o I he ler cse we sw h N S s e errg Sce he clss o ll e er-rgs s ow o orm vrey, mes sese o loo ree e errgs, he more so sce we ow how hs mosers loo le Le be se, * he ree mood over d he ree e er-rg over The every eleme o s e sum o elemes wh I c Sce xy z xy xz, x y z xz xz yz yz z d x y xy yx y re lws he vrey o e er-rgs, we c brg ll expressos o -sums o elemes whch re producs o elemes (observe h we use le er-rgs!) Le S QF,, be GS d he ree errg o q Q s ccessble rom q Q here s some wh q q S s ccessble ech se q s ccessble rom ech oher se N S s o oly er-rg, bu lso operes o Q obvously Q s N S group v q he usul meg q s ccessble rom q q q NS lervely, Q c be vewed s -group v q : q We hve S s ccessble Q s N : NS -group wh N Q I c, S s ccessble he obvously N Q Coversely, suppose h Q N N C I q Q he qn qn qnc qn NC qn Q Q, d S s show o be ccessble I mgh be mos useul o exme he reloshp bewee geerors, prmvy d ccessbly more closely Now we loo cosrucos o semuom d her correspodg sycc er-rgs Le S QF,, d S Q,, F be GS wh decl pu ses group homomorphsm h: Q Q s clled GS-homomorphsm hq hq holds or ll qq d (wh q : F, q o course) Theorem 3 Le h : S S be GS-epmorphsm The here exss er-rg epmorphsm h rom N S o N S wh hq hqh or ll q Q d N S Proo I N S, s word w w,, w,,,, The hqw hq w by duco o he legh o w Dee h w : w h s well-deed sce w w, mples hq w hqw hqw hq w, or ll q Q Sce h s surecve, w w ollows Obvously, h s er-rg epmorphsm d hq hqw hq w hqh s lso rue or ll q Q d N S uomo s quuple QBFG,,,,, where QF,, s semuomo, B se (he oupu se) d G: Q B uco (clled he oupu uco o ) I Q s group, s clled groupuomo (bbreved by G) We cll homomorphc G Q,, B re groups d F, G re homomorphsms s clled ler G or ler uomo or ler sequel mche Q,, B re R-modules or some rg R d F, G re R-ler mps [] I my cses, however, oupus do ply essel role For sce, oe ws o coec wo (or more) uom seres For dog h, cosder QBFG,,,, d Q, B, C, F, G The Copyrgh ScRes

4 S F YOU ET L 67 Q B Q C Seres coeco s oupus o shll be he pus o : QQ,, C, F, G F qg,, : F q,, F q, Gqg, d More ormlly, s G q, q, : G Gq,, q I d re ler G he N s er-rg N sn prs o he orm, he cos-mp-prs, G, p ( s o egve eger), wh p : Q M Q, wh s he ddvely geered by ll ( s o egve eger),, d ll C q Gq, Le * d B * deoe he ree moods over d B, respecvely For q Q le : B be deed by :, : Gq,,, : Gq, GFq,, sfq, d proceed ducvely wh GFq, q,, q, : B s clled he sequel (pu-oupu-) uco o q I s G, s : s s clled he sequel uco o Furhermore, cll qq, Q equvle ses ( q q ) (e q d q duce he sme pu-oupu-behvour) I mgh me sese o exed s q rom o B, where d B re he ree er-rgs [] vrey whch cos he oe geered by N we dee : Gq, G q, I QBFG,,,, s homomorphc we ge or qq,, q Q: I q q he s q s q Le qq The q q; s G qq, G q, G q, G, qq qq, ),,, qq qq,, Fq,, F,, qq,, Fq,, F,, G q G q G G qq s s s G F q s G F q s qq d so o, hece q q, whece qq qq Smlrly, q q he s q d N,,,,,,,, q G q G F q G F q G q s d duco shows q q We here ore ge Theorem 4 Le be homomorphc G The s cogruece relo he N -group Q d () Q : q Q q s del o N Q ; (b) Gq, or ll q Q We mgh s wh q q mes del Theorem 5 Le be homomorphc d g : Q B, q qg Gq, The q q For y o egve eger, q g q g Proo Le q q We use duco o d sr wh I he S G q, G q, G, qg G, q Sce S S qg qg q q we ge Now suppose heorem 5 holds or ll words o legh : The or ll, Sq Sq, hece Gq, Gq,, we hve, Gq, Gq, Smlrly,,,, G q G G,,, G q G q G G,, hece Gq, Gq, d we ge q G q g The coverse s show smlrly G QBFG,,,, s reduced s he equly I s ccessble (e (Q,, F)s ccessble) d reduced he s clled mml [] Obvously, homomorphc G s reduced G, we hve Q,, BFG,, be G The Corollry 6 Le () : :,,,, s ccessble; (b) Q N ; (c) / : Q/,, B, F, Q wh F q, : Fq, d G q, : G q, Q N Q B F Q G Q s reduced; (d) / s mml The proos re srghorwrd I loog or crer o decde gve G s mml or o, we obvously hve o vew Q o oly s N -group bu lso hve o cre bou B Copyrgh ScRes

5 68 S F YOU ET L Corollry 7 Le be homomorphc G The s [] G Plz, Ner Rgs, Norh-Holld, mserdm, 977 reduced N Q hs o o-zero dels p wh [3] S F You, M Co d Y J Feg, Semuom d pg Ner Rgs, Quve Logc d So Compug, Vol Proo I Q 5,, pp N hs o such dels he Q d s reduced So suppose h coversely s [4] S F You, H Y Zho, Y J Feg d M Co, preduced d h PN Q hs GP, pg plco o Euler Grph o PI o M(C), ppled or ll p P I p P, we see by smlr rgumes Mhemcs, Vol 3, No 7,, pp 89-8 h p, hece p, whece P [5] S F You, pplco o Euler Grph o Poly- From corollry 7 we ge oml Idey, IEEE Proceedgs o he Ierol Coerece o Compuol Iellgece d Corollry 8 Le be homomorphc G The Sowre Egeerg (CSE ), Wuh, 9- Des mml N Q s geered by d does o cember co o-zero dels whch re hled by g [6] S F You, e l, Euler Grph d Polyoml Idees o Mrx Rgs, dvces Mhemcs, Vol 3, REFERENCES No 4, 3, pp [7] S F You, The Prmvy o Exeded Cerod Exeso o Prme GPI-Rgs, dvces Mhemcs, Vol [] S Eleberg, uom, Lguge, d Mches, cdemc Press, New Yor, 974 9, No 4,, pp Copyrgh ScRes