Deduction of Fuzzy Autocatalytic Set to Omega Algebra and Transformation Semigroup

Transcription

1 World Academy of Scece Egeerg Techology Iteratoal Joural of Mathematcal Computatoal Physcal Electrcal Computer Egeerg Vol:4 No: 00 Deducto of Fuzzy Autocatalytc Set to Omega Algebra Trasformato Semgroup Lew Saw Yee Tahr Ahmad Iteratoal Scece Ide Mathematcal Computatoal Sceces Vol:4 No: 00 wasetorg/publcato/4570 Abstract I ths paper the Fuzzy Autocatalytc Set () s composed to Omega Algebra by embeddg the membershp alue of fuzzy edge coectty usg the property of traste affty The the Omega Algebra of s a trasformato semgroup whch s a specal class of semgroup s show Keywords Fuzzy autocatalytc set omega algebra semgroup trasformato semgroup I INTRODUCTION HE cocept of autocatalytc set (ACS) was frst T troduced the cotet of catalytcally teractg molecules [ ] Howeer Ja Krsha hae formalzed the autocatalytc set terms of graph [] Tahr et al defed Fuzzy Autocatalytc Set () [4] as subgraph whch each of those odes has at least oe comg lk wth membershp alue μ( e ) (0] e E The ertces of the graph correspod to the arables a drected lk from erte to erte dcates that arable catalyzes the producto of arable Fuzzy Graph Type G F as a fuzzy graph where both the erte edge sets are crsp but the edges hae fuzzy heads tals Sabarah show that the Fuzzy Graph Type- s a [5] wth the followg deftos: Defto [4]: Let e E The fuzzy head of e deotes as he ( ) the fuzzy tal te ( ) are fuctos of e such that h: E [0] t : E [0] for e E Fuzzy edge coectty s a tuple (( te) he ( )) the set of all fuzzy edge coectty s deoted as C {( t( e) h( e)) : e E} The membershp alue of fuzzy edge coectty s deoted as μ ( e ) m{ t( e ) h( e )} Authors are wth Theoretcal Computatoal Modellg for Comple Systems Research Group (TCM) Departmet of Mathematcs & Ibu Sa Isttute for Fudametal Scece Studes Faculty of Scece Uest Tekolog Malaysa 80 UTM Skuda Johor Malaysa (e-mal: aoraee@yahoocom tahr@busautmmy) Defto [5]: Let deoted the fuzzy edge coectty betwee ode CF ode the: 0 f e E CF μ( e ) f II OMEGA ALGEBRA AND GRAPH A operato s a acto or procedure whch produces a ew alue from oe or more put alue N-ary algebrac operato o the set M s a fucto of the form : M M K M M I others word the -ary or arty of a fucto or operato s the umber of argumet or opers of the fucto I the fucto : M M for some set M s a operato s ts arty Therefore omega algebra ( -algebra) s a set wth the certa system of operatos e -algebra { k k {0 K }} whch s defed o oe basc set s called oe-sorted algebrac system [6] For eample : M M M : M M M M M : M M L M M Whe a system s defed as a complete drected graph t meas for ay two elemets the system there est two coectos betwee them e coects to coects to M M M The term coecto here depeds o the releacy of system oe gog to defe Thus f the -algebras appled the completed drected graph t s terpreted as the mappg of the Cartesa product of ay set of ertces to oe of ts erte - algebra was physcally terpreted as for ay elemet a arbtrary system the coecto betwee these elemets wll produce oe of these elemets as a fal product Fgure ges the llustrato eplag the omega algebra of fe elemets For eample M { 4 5} for ay M the - algebra { k k { 4 5}} for Iteratoal Scholarly Scetfc Research & Ioato 4() 00 scholarwasetorg/9997/4570

2 World Academy of Scece Egeerg Techology Iteratoal Joural of Mathematcal Computatoal Physcal Electrcal Computer Egeerg Vol:4 No: 00 Iteratoal Scece Ide Mathematcal Computatoal Sceces Vol:4 No: 00 wasetorg/publcato/4570 : k : m k : 4 m k There s o repetto of 5 : m k erte s assumed ths system Fg -algebrafor M III OMEGA ALGEBRA OF The herarchcal relatoshp betwee cycles rreducble subgraph ACSs s show by Ja Krsha [ 7] as below: Proposto [ 7]: PI: All cycles are rreducble subgraphs all rreducble subgraphs are ACSs PII: Not all ACSs are rreducble subgraphs ot all rreducble subgraphs are cycles s defed by troducg fuzzy cocept to the autocatalytc set hece Proposto I II also hold for : Proposto : PFI: All cycles are rreducble subgraphs all rreducble subgraphs are s PFII: Not all s are rreducble subgraphs ot all rreducble subgraphs are cycles As PFII that ot all are rreducble subgraph Further we let our to be a rreducble () Ths leads to the followg terms () V { } s a set of ertces of whch eery erte hae at least oe comg edges wth membershp alue μ( e) (0] e E from oe erte belogg to the same () k( ) represets 4 5 catalyzed the producto of hae k-ary relato whch through k-ary Cartesa product of * operato or there est some such that operato hae k-ary catalytc relato - Theorem : The set of operatos of form the -algebra of e { k k K } s a by PI PFI As the smplest s -cycle [] whch s a loop ths -cycle s also a Suppose that V s the set of M as Secto II thus operatos that est the set of V are: uary operato: : V V such that V V bary operato: : V V V such that for V V : V V V V : V V terary operato: or such that for k V ( ) k V through -ary operato: : V V V V such that p Ths set of p p k operatos form the omega algebra of e { k k K } I ths - algebra represetato of { k k K } s a set of - algebra operatos whch s a set of path from a erte to tself the legth of k or to aother erte through the path wth legth of k s a set of omega operatos that physcally meas catalytc relato amog the elemet of Net the ( Θ ) s a semgroup s show wth the followg defto of the bary operato: Defto : A operato Θ s defed for whch whe V y the for { } p q Iteratoal Scholarly Scetfc Research & Ioato 4() 00 scholarwasetorg/9997/4570

3 World Academy of Scece Egeerg Techology Iteratoal Joural of Mathematcal Computatoal Physcal Electrcal Computer Egeerg Vol:4 No: 00 Iteratoal Scece Ide Mathematcal Computatoal Sceces Vol:4 No: 00 wasetorg/publcato/4570 Θ y Θy( p q) Θq for some s { } s ( q) s where s q q Theorem : ( Θ ) s a semgroup Let { k k K } Cosder y( ) h z( ) q Θ y Θ y ( g h) Θ h g h r for some r h p q Net ( Θ ) Θ ( Θy ) Θ z y z g h p q ( ) Θ h z( p q) s ( h ) z( p q ) Θ for some s Θ h q t for some t h q q Θ( yθ z) Θ( y Θ z ) g h p q Θ( Θ ) h q r h q ( ) Θ ( ) for some r Θ q w for some w q q Hece ( Θ ) Θ Θ( Θ ) y z y z The closure of Θ meas a catalytc reacto wll produce a elemet whch s oe of the chemcal elemets or arables the clcal waste cerato process [4] Furthermore the assocatty mples that t s free regards to the order of the catalytc reactos Lemma : ( Θ ) s reflee Θ Θ Θ for some s s ( ) Lemma : ( Θ ) s traste Let y z therefore Θ Θ y y( p q) Θ for some m q m ( q) q Θ Θ y z y( p q) z( g h) Thus Θ for some q h ( q h) h Θ z Θ z ( g h) Θ h r ( h) for some r h Ufortuately ( Θ ) does ot obsere the symmetry property Let the Θ Θ y y y( p q) Θ q s ( q) q for some s Θ Θ y y( p q) Θ q t ( q ) for some t Hece Θ s ot a symmetry sce Θy yθ Cosequetly ( Θ ) s ot a equalece relato Selecto of desrable membershp alue of fuzzy coectty of operatos s subect to ts applcato If the membershp alue of fuzzy edge k Iteratoal Scholarly Scetfc Research & Ioato 4() 00 scholarwasetorg/9997/4570

4 World Academy of Scece Egeerg Techology Iteratoal Joural of Mathematcal Computatoal Physcal Electrcal Computer Egeerg Vol:4 No: 00 Iteratoal Scece Ide Mathematcal Computatoal Sceces Vol:4 No: 00 wasetorg/publcato/4570 coectty for k( ) s deoted as ( k( )) μ ( ) (0] k μ such that Suppose catalysts the producto of catalysts the producto of elemet t s lkely that elemet also catalysts the producto of elemet e there s a ozero traste affty betwee [8] Sce the FASC s defed as a rreducble preously thus eery erte s beg coected to aother erte the same The assocatty of s cosstet wth traste affty I other words fuzzy edge coectty s the weakest lk for a partcular path I geeral for ay dfferet paths that coectg ay two ertces fuzzy edge coectty of a path whch s mamal amog those paths s chose to be the membershp alue for fuzzy edge coectty of that par of ertces Defto 4: e path of Let ( ) K k k The membershp alue for fuzzy edge coectty of s defed as μ( ) m( ( )) ϖ where ϖ { μ ( ) μ ( ) K ( ) ( k ) ( ) k k μ( ) μ( )} ( k ) k k Defto 5: The mamal membershp alue of fuzzy edge coectty betwee s defed as μ ( ) ma ( ) where s ay possble path betwee Therefore for a ge the membershp alue of fuzzy edge coectty betwee ay par of ertces s uquely defed Hece for k K wth order of wth ther uque membershp alue of fuzzy edge coectty obtaed the followg table: TABLE I MEMBERSHIP VALUE OF FUZZY EDGE CONNECTIVITY BETWEEN ANY PAIR OF VERTICES L ( μ ( )) k ( μ ( )) k ( μ ( )) L ( k( ) μ ( )) k ( μ ( )) L M k M M M O M ( μ ( )) k( ) IV TRANSFORMATION SEMIGROUP OF It s obous that sce all rreducble subgraphs are s Wth the proe defto of semgroup of { k k K } the trasformato semgroup of wll be eamed ( μ ( )) L ( k( ) μ ( )) k( ) Defto 6 [9]: A trasformato semgroup X ( Q S) whch cosst of a fte set Q a subsemgroup S of PF(Q) The elemets of Q are called states Q tself s called the uderlyg set of X The elemets of S are called trasformatos of X whle S tself s called the acto semgroup of X (see Fg ) Notce that PF(Q) s a partal fucto oer Q e Q' Qwhere Q' Q S Θ s a semgroup ( ) Iteratoal Scholarly Scetfc Research & Ioato 4() 00 4 scholarwasetorg/9997/4570

5 World Academy of Scece Egeerg Techology Iteratoal Joural of Mathematcal Computatoal Physcal Electrcal Computer Egeerg Vol:4 No: 00 Iteratoal Scece Ide Mathematcal Computatoal Sceces Vol:4 No: 00 wasetorg/publcato/4570 Theorem : Fg Trasformato semgroup respecte sets A trasformato semgroup of ( S ) where set of ts s a tuple s called state or the uderlyg S s called trasformato of tself s called the acto semgroup of whle S Recall let Θ defed o such that for ay elemet y the Θy The ( S ) s show to be a semgroup Net Q s a set of path of all s reealed Θ s a fucto Θ: ' But ' s ackowledged sce ( Θ ) s closed Cosder α such that : S α whch s compatble wth the semgroup operato Θ: as follow: For all s t S sα ( tα ) ( sθ t) α Wth the aboe mplemetatos we hae the Q S V CONCLUSION ts Fuzzy Autocatalytc Set ca be composed to Omega Algebra s establshed The the structure of s further eteded to trasformato semgroup of ACKNOWLEDGMENT Ths research s partly supported by Fudametal Research Grat Scheme (FRGS) ot o 785 awarded by MOHE X REFERENCES [] Kauffma S A (97) Cellular Homeostass Epgeesst Replcato Romly Aggregated Macromolecular Systems Joural of Cyberetcs :7-96 [] Rossler O E (97) A System Theoretc Model of Bogeess Z Naturforchug 6b: [] Ja S Krsha S (998) Autocatalytc Sets the Growth of Complety a Eolutoary model Physcal Reew Letters [4] Tahr A Sabarah B Kharl A A (006) Fuzzy Autocatalytc Set I Modelg A Icerato Process Joural Fuzzy Set System (submtted; Mauscrpt Id No:FSS-D ) [5] Sabarah B (006) Modelg of Clcal Waste Icerator Process Usg Noel Fuzzy Autocatalytc Set: Mafestato of Mathematcal Thkg Uerst Tekolog Malaysa Upublshed PhD Thess [6] Plotk B I Greeglaz L Ja Garama A A (99) Algebrac Structures Automata Databases Theory Sgapore: World Scetfc [7] Ja S Krsha S (999) Emergece the Growth of Complety Networks a Adapte Systems Computer Physcs Commucatos -: 6- [8] Dg C He X Xog H Peg H Holbrook S R (006) Traste Closure Metrc Iequalty of Weghted Graphs: Detectg Prote Iteracto Modules Usg Clques It J Data Mg Boformatcs (): 6-77 [9] Eleberg S (976) Automata laguages maches ol B New York: Academc Press Iteratoal Scholarly Scetfc Research & Ioato 4() 00 5 scholarwasetorg/9997/4570