Automorphism Group of an Inverse Fuzzy Automaton

Transcription

1 Annls f Pure nd Alied themtics Vl ISS: P nline Pulished n 8 Decemer 0 wwwreserchmthscirg Annls f Autmrhism Gru f n Inverse Fuzzy Autmtn Pmy Sestin nd P Jhnsn Dertment f themtics ry th Arts & Science llege nnthvdy Kerl E-mil: myse@gmilcm Alied Science nd Humnities Divisin Schl f Engineering chin University f Science & echnlgy chin- E-mil: tjhnsn@custcin Received 3 Decemer 0; cceted 7 Decemer 0 Astrct In this er we define n inverse fuzzy utmtn such tht the crresnding trnsitin mnid is n inverse mnid We ls study the structure f the utmrhism gru f n inverse fuzzy utmtn nd rve tht it is eul t sugru f the symmetric inverse mnid f ne ne rtil fuzzy trnsfrmtins n Keywrds: Inverse fuzzy utmt invertile fuzzy lnguges inverse mnid AS themtics Suject lssifictins 00: 8B0 Intrductin An utmtn is mdel f seuentil switching circuit with finite numer f stteswith stte chnging when it is sujected t n inut syml A seuentil mchine cnsits f tw min structures the trnsitin structure nd the utut structure Since ur interest is in the trnsitin structure f n utmtn nd its inut semigru ututs re nt cnsidered here Zdeh intrduced the cncet f fuzzy sets nd W G Wee intrduced fuzzy utmt he thery f inverse mnids were intrduced indeendently y Wgner nd Prestn vi the study f rtil ne-ne trnsfrmtins f set We define n inverse fuzzy utmtn such tht its trnsitin mnid is n inverse mnid nd study the structure f the utmrhism gru f n inverse fuzzy utmtn Preliminries his sectin resent sic definitins nd results t e used in the seuel A semigru S is sid t e n inverse semigru if fr every S there exist uniue S 67

2 Pmy Sestin nd P Jhnsn such tht nd We cll the inverse f nd dente y If S hs n identity then S is sid t e n inverse mnid Inverse mnids frm vriety defined y the identities Als nd Fr ny element f n inverse mnid is n idemtent nd idemtents in n inverse mnid cmmute An inverse mnid with single idemtent is gru As n nlgus t yleys therem fr grus Prestn nd Wgner rved tht n inverse mnid I is ismrhic t suinverse mnid f the mnid f ll ne-ne rtil trnsfrmtins n I A fuzzy utmtn n n lhet is 5-tule i τ where is finite set f sttes is finite set f inut symls nd is fuzzy suset f reresenting the trnsitin ming i is fuzzy suset f clled initil stte τ is fuzzy suset f clled finl stte is clled fuzzy finite stte mchine A fuzzy utmtn cn ls e reresented s five tule { u u } iτ where { u u } is the set f fuzzy trnsitin mtrices i [ i i i ] i [0] τ [ j j j ] j [0] fr k n n k n k cn e extended t the set y Λ 0 u { x x L i xx xk u} Let i τ e fuzzy utmtn We sy the trile is the fuzzy finite stte mchine sscited with Fr is clled n immedite successr f if there exist n such tht > 0 is clled successr f if there exist k x k x such tht x > 0 Let S e the set f ll successrs f Let he set f ll successrs f dented y S { S } where is fuzzy suset f is clled sumchine f if nd S is sid t e serted if S φ is sid t e cnnected if hs n serted rer sumchines Let nd e fuzzy finite stte mchines A ir f mings : nd : is clled hmmrhism written : if x x nd x is clled strng hmmrhism if 68

3 Autmrhism Gru f n Inverse Fuzzy Autmtn x { x t t t } nd x A hmmrhism is sid t e n ismrhism if nd re th ne ne nd nt If nd is the identity m then we write : is hmmrhism If is strng hmmrhism with ne-ne then x x nd x see [5] Let e fuzzy finite stte mchine nsider the set f ll strng hmmrhisms : dented y ED nd the set f ll strng ismrhisms frm y AU ED frm mnid under the ertin nd AU frm gru where the inverse f is msitin is sscitive nd identity is the ir f identity ms n nd If is the identity m n then we dente ED s End nd AU s Aut hen End is sumnid f ED nd Aut is sugru f AU Define cngruence s u v iff u v cngruence nd see [6] nsidering the cllectin f ll fuzzy finite stte utmt s ctegry F-AU with jects re fuzzy utmt ver finite set f sttes nd mrhisms s the utmt hmmrhisms etween them rresnds t every fuzzy utmt n hen is we get finite mnid nd every finite mnid is the trnsitin mnid f the miniml fuzzy utmtn recgnizing sme fuzzy lnguge which is clled the syntctic mnid f tht fuzzy lnguge see [6] 3 Inverse Fuzzy Autmt nd the Autmrhism Gru Definitin 3 A cnnected fuzzy utmtn is sid t e n inverse fuzzy utmtn if there exist uniue such tht nd herem 3 A fuzzy utmtn A i τ is inverse if nd nly if A is n inverse mnid Prf Suse A is n inverse fuzzy utmtn ie fr ech there exist uniue such tht nd A nd A [ ] [ ] nd [ ] [ ] hen [ ][ ][ ] [ ] nd [ ][ ][ ] [ ] A is n inverse mnid 69

4 Pmy Sestin nd P Jhnsn Lemm 3 If AU then fr ny u v u v u v Prf u v u v u v v since is ne-ne u v Let nd e tw fuzzy utmt nd let e mrhism etween them Let nd e the crresnding trnsitin mnids Let f : t defined y f [ u] [ u] u herem 3 Let nd e tw jects in the ctegry F AU nd let HO hen f f re semigru mrhisms nd f f f hus the ms nd f HO Prf f nd f re well defined y the ve lemm Let [ u][ v] where u v hen f [ u][ v] f [ uv] [ uv] [ u v] [ u][ v] f [ u] f [ v] S f is semigru mrhism f [ ] [ ] [ ] [ ] u u f u f f u We cn define cvrint functr F etween the ctegry f fuzzy utmt nd the ctegry f finite semigrus s F nd F f fr HO he set f ll inverse fuzzy utmt frm full suctegry f F AU nd F s defined ve is functr frm this suctegry t the ctegry f finite inverse mnids which is suctegry f finite mnids Definitin 3 A fuzzy utmtn is sid t e fithful if fr herem 33 Let e fithful fuzzy utmt Let e the trnsitin mnid nsider AU f ll utmrhisms n Let h : AU AU e m defined y h f hen h is gru hmmrhism nd Kerh Aut f 70

5 Autmrhism Gru f n Inverse Fuzzy Autmtn Prf Ker h { : h f } where f [ u] [ u] fr ll u u u is the identity m n hus Ker h Aut u [ u] [ u] u u u u u rllry By hmmrhism therem fr grus AU Aut is ismrhic t sugru f AU nsider the set f ll ne-ne rtil fuzzy trnsfrmtins n which is the symmetric inverse mnid f fuzzy trnsfrmtins dented s FI We cn cnsider FI s cllectin f fuzzy mtrices f crdinlity with tmst ne nnzer entry in ech rw nd clumn Fr ech FI there exist uniue inverse FI such tht Dm In mtrix frm it is the trnsse f the fuzzy mtrix crresnding t he trnsitin mnid f n inverse fuzzy utmtn is suinverse mnid f FI nsider { FI : } nd { FI : } where the cmsitin is the mx-min cmsitin f fuzzy mtrices Lemm 3 Let e fithful inverse fuzzy utmtn Let hen fr ny there exist uniue such tht Prf Since fr there exist such tht rve the uniueness suse there exist nther c such tht c hen since is ne-ne tmst ne nnzer entry will e there in ech rw nd clumn f the fuzzy mtrix crresnding t c c c c Let { Dm } nd { γ Dm γ } 7

6 Pmy Sestin nd P Jhnsn 7 herem 34 Let e fithful inverse fuzzy utmtn then Aut Prf Let Aut hen is the identity m n nd is ne-ne ming frm nt stisfying ie nsider since nversely let hen fr sme with nd nd clerly is ne-ne nd nt hus Aut herem 35 Let e fithful inverse fuzzy utmtn then AU Prf Let AU hen AU ie Euivlently Let Since is ne-ne nversely let hen y the lemm there exist such tht Define : s with hen is well defined ijectin frlet u t hen t c nd u d where c t nd d u hen d c nd s u t u t is nt fr ny nd y lemm there exist uniue such tht ie

7 Autmrhism Gru f n Inverse Fuzzy Autmtn 73 there exist n such tht w is hmmrhism fr let with ie with hus hus we hve rved tht AU herem 36 is nrml sugru f r euivlently Aut is nrml sugru f AU Prf Let Let Since there exist uniue such tht nd nsider hus is nrml sugru f rllry By the therem 5 nd the crllry is sugru f AU REFEREES A H liffrd nd G B Prestn he Algeric hery f Semigrus Amer th Sc 7 Prvidence RI 96 Gerge J Klir nd B Yun Fuzzy Sets nd Fuzzy Lgic hery nd Alictins Prentice-Hll Inc E Lee nd L A Zdeh te n Fuzzy Lnguges Infrmtin Sciences hing-hng Prk Sme remrks n the Autmt Hmmrhisms mm Kren th Sc Jhn dersn nd Devender S lik Fuzzy Autmt nd Lnguges hery nd Alictins hmn nd HllR 00 6 tjn Petkvic Vrieties f Fuzzy lnguges Prc in First Interntinl nference n Algeric Infrmtics Aristtle University f hesslniki L A Zdeh Fuzzy Sets Infrm nd ntrl