SOME PROPERTIES OF CAYLEY GRAPHS OF CANCELLATIVE SEMIGROUPS

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1 THE PUBLISHING HOUSE PROEEDINGS OF THE ROMANIAN AADEMY, Serie A, OF THE ROMANIAN AADEMY Volume 7, Number /06, pp 0 MATHEMATIS SOME PROPERTIES OF AYLEY GRAPHS OF ANELLATIVE SEMIGROUPS Bahman KHOSRAVI Qom unierity of Technology, Faculty of Science, Department of Mathematic, Qom, Iran khoraibahman@yahoocom Abtract In [S Panma et al, haracterization of lifford emigroup digraph, Dicrete Math 06 () (006), 47 5] a characterization of ayley graph of lifford emigroup i gien In thi paper, firt we characterize ayley graph of cancellatie emigroup, and we gie a criterion to check whether a digraph i a ayley graph of a cancellatie emigroup Alo Kelare and Praeger gae neceary and ufficient condition for ayley graph of emigroup to be ertex-tranitie Then, ome author gae decription for all ertex-tranitie ayley graph of ome pecial clae of emigroup In thi note imilar decription for all ertex-tranitie ayley graph of cancellatie emigroup are gien Key word: ayley graph, ertex-tranitie graph, cancellatie emigroup INTRODUTION Let S be a emigroup and S Recall that the ayley graph ay ( of S with the connection et i defined a the digraph with ertex et S and edge et E( ay( ) = { (, c): c } ayley graph of group hae been exteniely tudied and ome intereting reult hae been obtained (ee for example, []) Alo, the ayley graph of emigroup hae been conidered by ome author (ee for example, [ 4, 6 0, ]) One of the intereting ubject in the tudy of ayley graph of emigroup i conidering how the reult obtained for the ayley graph of group work in cae of emigroup It i known that the ayley graph of group are ertex-tranitie; in the ene that for eery two ertice u, there exit a graph automorphim f uch that ( u ) f = [] Alo, Kelare and Praeger in [] characterized ertex- tranitie ayley graph of emigroup S for which all principal left ideal of the ubemigroup generated by the connection et are finite In fact Kelare and Praeger gae neceary and ufficient condition for ayley graph of emigroup to be ertex-tranitie Then, ome author gae decription for all ertex-tranitie ayley graph of ome pecial clae of emigroup In thi note imilar decription for all ertex-tranitie ayley graph of cancellatie emigroup are gien In [4] a characterization of ayley graph of group wa preented by Sabidui A characterization of ayley graph of lifford emigroup wa preented by Panma et al [] Alo ome combinatorial propertie of ome clae of emigroup were conidered [ 4, 6 ] In thi note, we extend Sabidui Theorem and we preent a characterization of ayley graph of cancellatie emigroup PRELIMINARIES In thi ection, we gie ome preliminarie needed in the equel on graph, emigroup, and ayley graph of emigroup For more information on graph, we refer to [], and for emigroup ee [5] Recall that a digraph (directed graph) = ( V, E) i a non-empty et V = V () of ertice, together with a binary relation E = E() on V For eery ertex u in, let N = { : ( u, ) )},

2 4 Bahman KHOSRAVI N = { : (, u) )}, d = N and d = N The number d (u) and d (u) are called in-degree and out-degree of u, repectiely Throughout thi paper, by a graph we mean a digraph without multiple edge (poibly with loop) By the underlying undirected graph of we mean a graph with the ame et of ertice V and an undirected edge { u, } for each directed edge ( u, ) of The graph i aid to be connected if it underlying undirected graph i connected By a connected component of a digraph we mean any component of the underlying graph of Let ( V, E ) and ( V, E ) be digraph A mapping ϕ : V V i called a (digraph) graph homomorphim if ( u, ) E implie ( ϕ, ( ) ϕ) E, and i called a (digraph) graph iomorphim if it i bijectie and both of ϕ and ϕ are graph homomorphim Alo it i called a graph monomorphim if it i one-to-one A graph homomorphim ϕ :( V, E) ( V, E) i called an endomorphim, and a graph iomorphim on = ( V, E) i aid to be an automorphim We denote the et of all endomorphim on the graph by End (), the et of all monomorphim on the graph by Mon (), and the et of all automorphim on by Aut () Let S be a emigroup, and be a ubet of S The ayley graph ay ( of S relatie to i defined a the graph with ertex et S and edge et E( ay( ) coniting of thoe ordered pair (, t) uch that c = t for ome c The et i called the connection et of ay ( (ee [7]) The following propoition, known a Sabidui' Theorem, gie a criterion for a digraph to be a ayley graph of a group PROPOSITION [4] A finite digraph = ( V, E) i a ayley graph of a group G if and only if the automorphim group Aut() contain a ubgroup Δ iomorphic to G uch that for eery two ertice u, V there exit a unique Δ uch that ( u ) = For a ayley graph ay (, we denote End ( ay( ) by End, and Aut ( ay( ) by Aut An element f End i called a color-preering endomorphim if cx = y implie c ( x) f = ( y) f for eery x, y S and c The et of all color-preering endomorphim of ay( i denoted by olend, and the et of all color-preering automorphim of ay( by olaut The ayley graph ay ( i aid to be ertex-tranitie (automorphim-ertex tranitie) or Aut -ertex-tranitie if, for eery two ertice x, y S, there exit f Aut uch that ( x ) f = y The notion of olaut -ertex-tranitie, olend -ertex-tranitie, and End - ertex-tranitie for ayley graph are defined imilarly A a corollary of Propoition, we hae that eery ayley graph of a group i ertex tranitie Alo in the equel we apply the following lemma LEMMA [, Lemma 6] Let S be a emigroup, and be a ubet of S i If ay ( i End -ertex-tranitie, then S = S ; ii If ay( i olend -ertex-tranitie, then cs=s for each c The following propoition decribe emigroup S and ubet of S, atifying a certain finitene condition, uch that the ayley graph ay ( i olaut -ertex-tranitie ( Aut -ertextranitie)

3 Some propertie of ayley graph of cancellatie emigroup 5 PROPPOSITION [, Theorem ] Let S be a emigroup, and be a ubet of S which generate a ubemigroup uch that all principal left ideal of the ubemigroup are finite Then, the ayley graph ay( i olaut -ertex-tranitie if and only if the following condition hold: i cs = S, for all c ; ii i iomorphic to a direct product of a right zero emigroup and a group; iii i independent of the choice of S PROPOSITION 4 [, Theorem ] Let S be a emigroup, and be a ubet of S uch that all principal left ideal of the ubemigroup are finite Then, the ayley graph ay ( i Aut - ertex-tranitie if and only if the following condition hold: i S = S ; ii i a completely imple emigroup; iii The ayley graph ay (, i Aut ( ) -ertex-tranitie; i i independent of the choice of S Recall that a right zero emigroup (left zero emigroup) i a emigroup S atifying the identity xy = y (repectiely, xy = x ), for all x, y S An element in emigroup S i called left cancellable (right cancellable) if r = t ( r = t ), for r, t S, implie r = t, and i called cancellable if S i left cancellable and right cancellable The emigroup S i called left cancellatie, right cancellatie or cancellatie if all element of S are left cancellable, right cancellable or cancellable, repectiely Alo, recall that a emigroup i aid to be left imple (right imple) if it ha no proper left (right) ideal A emigroup i called a left group (right group) if it i left (right) imple and right (left) cancellatie It i known that a emigroup i a right (left) group if and only if it i iomorphic to the direct product of a group and a right (left) zero emigroup (ee [6]) A emigroup i completely imple if it ha no proper ideal and ha an idempotent element which i minimal with repect to the partial order e f e = ef = fe on idempotent element Let S be a monoid and A be a et If we hae a mapping μ : A S A, by μ ( a, ) = a uch that a) a = a, b) a ( t) = ( a) t, for a A,, t S, we call A a right S -act, which i denoted by A S, and ay S act on A Alo we ay S act trongly faithfully on A if for, t S the equality a = at, for ome a A, implie that = t We note that ome author call thi property a free action of the emigroup S LEMMA 5 [, Lemma 5] Let S be a emigroup with a ubet, let g S and let g be the et of all ertice of the ayley graph ay ( uch that there exit a directed path from g to Then g i equal to the right coet g LEMMA 6 [, Lemma 5, orollary 5] Let S be a emigroup with a ubet uch that i completely imple, and S = S Then, eery connected component of the ayley graph ay ( i trongly connected, and for eery S, the component containing i equal to Alo, if i iomorphic to a right group, then the right -coet are the connected component of ay (

4 6 Bahman KHOSRAVI 4 HARATERIZATION OF AYLEY GRAPHS OF ANELLATIVE MONOIDS Sabidui in [0] preented a characterization of ayley graph of group We note that eery finite cancellatie emigroup i a group In thi ection, we characterize ayley graph of cancellatie monoid LEMMA Let be a digraph If Mon () ha a ubemigroup S uch that it act trongly faithfully on V (), then S i a cancellatie emigroup Proof To proe S i a cancellatie emigroup, let,, S uch that = So, for eery V (), ( ) = ( ) Let u = ( ) Thu ( u ) = Since S act trongly faithfully on V (), we get that = Therefore S i left cancellatie Now we proe that S i right cancellatie For thi purpoe we aume that = Similarly to the aboe we get that ( ) = ( ), for eery V () Since i one-to-one, we conclude that ( ) = ( ), for eery V () Hence = Therefore S i cancellatie THEOREM Let be a digraph uch that the out-degree of all ertice of are finite and equal to each other Then i iomorphic to a ayley graph of a cancellatie monoid if and only if Mon () ha a ubmonoid T uch that T act trongly faithfully on V (), and there exit u V () uch that ut = u = T = V ( T { } ) Proof ( ) Let = ay(, where S i a cancellatie monoid and S Firt we note that eery ertex of ay ( i joined to exactly ertice of which are { c c }, ince S i a cancellatie emigroup Becaue the out-degree of all ertice of are finite and equal to each other, we get that < Now we take T = { S}, where : S S i defined by ( x ) = x, for x S Let S We know that S i a cancellatie emigroup and o x = y if and only if ( x) = x = y = ( y) Thu i one-to-one Now we proe that i a graph homomorphim For thi purpoe, we conider an arbitrary edge ( x, y) in E ( ay( ) Hence there exit c uch that y = cx Therefore y = cx Thu ( y) = c( x) Hence (( x),( y) ) ay( ) and o i a graph homomorphim By the definition of where S, we get that = T, for S, So T i a emigroup Since S i a monoid, S ha an identity Hence i the identity of T and o T i a monoid Now we how that T act trongly faithfully on V () To proe it, let = ', for, ' S and u V () So u = u' Since S i cancellatie, = ' Therefore T act trongly faithfully on V () To complete the proof of the neceary part, we take u =, where i the identity element of S Then for eery V ( ) = S, ince ( ) =, we conclude that T = {() T} = V ( ) ( ) Suppoe that there exit T Mon() and u V () uch that ut = V () So, for eery V (), there exit T uch that = Now we proe that for each V (), there exit a unique uch that ( u ) = Let, T and ( u ) = = Hence trongly faithfully property of T implie that = So T i unique By Lemma, we conclude that T i a cancellatie monoid Let = { T =, N } We claim that ay( T, ) For thi purpoe we define ψ : ay( T, by ( )ψ =, where V () By the aboe dicuion ψ i

5 5 Some propertie of ayley graph of cancellatie emigroup 7 well-defined We claim that ψ i a graph iomorphim Since ( ) ψ = = ' = ( ' ) ψ implie that = = ' = ', we conclude that = and o ψ i one-to-one To proe ψ i onto, we conider T We know that V ( ) Thu there exit V () uch that = (u) By the aboe dicuion we know that = Thu ( ) ψ = ' ' = Therefore ψ i onto Finally we proe that ψ i a graph iomorphim To proe ψ preere adjacency, we conider ) T Mon() and ( u, x) ), for each x N (u), then {(,( ) ) x x N } E( ) We claim that, for eery y N (), there exit x N (u) uch that y = ( x) We know that {(,( x) ) x N } E( ), ince ( u, x) ) Becaue i one-to-one, we get that if x, x N, and x x, then ( x ) ( x ) Alo we know that N = N ( ) < Hence N ( ) = {( x) x N } Thu ince (, ) ), = ( x 0 ), for ome x0 N Alo we know that x0 = x Therefore '= 0 x 0 Hence ' = x 0 Since T act trongly faithfully on V (), we conclude that ' = x 0 Hence (, ) ay( T, ) Therefore (( ) ψ,( ) ψ ) ay( T, ) Thu ψ preere adjacency Now we proe that ψ preere nonadjacency For thi purpoe we conider an arbitrary edge (( ) ψ,( ) ψ ) = (, ) ay( T, ) Hence there exit '' uch that = By definition of, we get that N (u) Thu = So = ( ) Since ( u, ) ) and Mon(), we conclude that (, ( ) ) ) Thu (, ) ) Therefore ψ preere non-adjacency So ψ i a graph iomorphim Hence ay( T, Therefore i iomorphic to a ayley graph of a cancellatie emigroup Uing the aboe theorem, Sabidui Theorem and the fact that eery finite cancellatie emigroup i a group, we hae the following corollary OROLLARY Let be a finite graph Then the monomorphim et of ha a ubmonoid T uch that T act trongly faithfully on V () and there exit u T uch that ut = { u = T} = V ( ) if and only if the automorphim group Aut () contain a ubgroup Δ uch that Δ = V ( ) and for eery two ertice u, V ( ) there exit a unique Δ uch that ( u ) = 4 OLOR-PRESERVING AUTOMORPHISM VERTEX TRANSITIVITY Kelare and Praeger in [] decribed all emigroup S and all ubet of S, atifying a certain finitene condition, all principal left ideal of are finite, uch that the ayley graph ay ( i olaut ertex-tranitie Under their condition, eery component of ay ( i finite In thi ection it i hown that if S i a cancellatie emigroup, then ome of the condition of Propoition will be atified LEMMA 4 Let S be a cancellatie emigroup, and uppoe that S ha no identity Then there i no pair of element e, a S uch that ea = a or ae = a

6 8 Bahman KHOSRAVI 6 Proof On the contrary uppoe that there exit a pair e, a S uch that ea = a Then aea = a So ae = a, ince S i cancellatie Now for eery element b bea = ba Hence be = b Thu beb = b So eb = b Therefore S ha an identity which i a contradiction LEMMA 4 Let S be a cancellatie emigroup and there exit a monoid c S uch that cs = S Then S i Proof On the contrary uppoe that S i not a monoid Since cs = S and c S, we conclude that there exit t S uch that ct = c which i a contradiction by Lemma 4 LEMMA 4 Let S be a cancellatie emigroup and there exit i a monoid and c ha an inere in S c S uch that cs = S Then S Proof Since cs= by Lemma 4, we conclude that S i a monoid Since cs = S, there exit c' S uch that = cc' So cc ' c = c Hence c ' c = ince S i a cancellatie emigroup Therefore c ' i the inere of c in S OROLLARY 44 Let S be a cancellatie emigroup, S and ay( olend -ertex-tranitie Then S i a monoid and eery element of ha an inere in S be Proof By Lemma and 4, we get the reult THEOREM 45 Let S be a cancellatie emigroup and let be a nonempty ubet of S uch that = Then the following tatement are equialent i ay ( i olaut -ertex-tranitie; ii ay ( i olend -ertex-tranitie; iii cs = for all c, and i a group Proof (i) (ii) It i obiou (ii) (iii) By Lemma, we conclude that cs = S, for all c Since i a group =, we conclude that (iii) (i) Since cs = S, S = S Alo by hypothei we know that i a group which i a completely imple emigroup Hence by Lemma 6, we conclude that the right -coet are the connected component of ay ( hooe ditinct element, ' S We will define a mapping ϕ : S S uch that ( ) ϕ = ', and we how that ϕ olaut If = ', then we define a mapping λ : ' by α', if x = α, for ome α ; ( x) λ = x, if x, which i a graph automorphim If ', then for eery x we define α', if x = α, for ome α ( x) ϕ = α, if x = α', for ome α x, if x '

7 7 Some propertie of ayley graph of cancellatie emigroup 9 Now we note that for, if α = = = α (or α ' = = = α ' ), then α = α and o α ' = α ' (or α = α ) Alo if = ', then it i obiou that ( ) ϕ = ( ) ϕ Therefore we conclude that ϕ i well-defined and one-to-one By the definition of ϕ, we get that ϕ i onto Obiouly ( ) ϕ = Now we how that ϕ preere adjacency and non-adjacency For thi purpoe take an arbitrary edge ( t, ct) ay( ), for t S and c If t = α where α, then ct = cα So ( ct) ϕ = ( cα) ϕ = cα' Alo we know that ( t) ϕ = α' Therefore ( ct ) ϕ = ( cα) ϕ = cα' = c( t) ϕ and o (( t) ϕ,( ct) ϕ) ay( ) Similarly if t = α ' ', we get that (( t) ϕ,( ct) ϕ) ay( ) Alo if x ', then (( t) ϕ, c( t) ϕ) = ( t, ct) ay( ) Hence ϕ preere adjacency Similarly we can conclude that ϕ preere non-adjacency Alo by the aboe argument we can conclude that ϕ preere color, too Thu ϕ olaut Therefore ay ( i olaut -ertex-tranitie THEOREM 46 Let S be a cancellatie emigroup, and let finite Then the following tatement are equialent i ay ( i Aut ertex-tranitie; ii ay ( i End ertex-tranitie; iii S = S be a ubet uch that i Proof (i) (ii) It i obiou (ii) (iii) By Lemma, we get the reult (iii) (i) Since i a finite cancellatie emigroup, we get that i a group So i independent of the choice of S and i a completely imple emigroup Alo ince i a group by Propoition we know that eery ayley graph of a group i ertex-tranitie and o ay (, i Aut ( ) ertex-tranitie Therefore by Propoition 4, we conclude that ay ( i Aut ertex-tranitie About trongly connected we hae the following reult LEMMA 47 Let S be a emigroup and S If ay ( i trongly connected, then S = Proof We know that S Let c Since ay ( i trongly connected, for eery ertex there exit a path between c and So there exit c,, c t uch that = c ctc Hence Therefore S Thu S = THEOREM 48 Let S be a cancellatie emigroup and be a ubet of S If ay ( i trongly connected, then S i a group Poof By Lemma 47, we conclude that S = Now we claim that S ha an identity On the contrary uppoe S doe not hae any identity By Lemma 5 we conclude that, the et of all ertice of ay ( uch that there exit a directed path from to, i equal to On the other hand ince ay ( i trongly connected, = S = Hence there exit α uch that α =

8 0 Bahman KHOSRAVI 8 which i a contradiction by Lemma 4 Therefore S ha an identity e Now we note that = =, for eery S = Hence for eery S there exit a ' uch that ' = e Similarly there exit " uch that " ' = e Hence ' = e' = " ' ' = " e' = " ' = e Thu eery element of S ha an inere Therefore S i a group AKNOWLEDGMENTS The author would like to thank the referee for ery helpful uggetion and ueful comment which improed the manucript REFERENES J A Bondy and U S R Murty, Graph Theory with Application, American Eleier Publihing o, Inc, New York, 976 J Dene, onnection between tranformation emigroup and graph, in Theory of Graph (Internat Sympo, Rome, 966), Gordon and Breach, New York, Dunod, Pari, 967, pp 9 0 S Fan and Y Zeng, On ayley graph of band, Semigroup Forum, 74,, pp 99 05, Y Hao, X Gao and Y Luo, On the ayley graph of Brandt emigroup, omm Algebra, 9, 8, pp , 0 5 J M Howie, Fundamental of Semigroup Theory, larendon Pre, Oxford, Z Jiang, An anwer to a quetion of Kelare and Praeger on ayley graph of emigroup, Semigroup Forum, 69,, pp , A V Kelare, On undirected ayley graph, Autralaian J ombin, 5, pp 7 78, 00 8 A V Kelare, Graph Algebra and Automata, Marcel Dekker, New York, 00 9 A V Kelare, Labelled ayley graph and minimal automata, Autralaian J ombin, 0, pp 95 0, A V Kelare, On ayley graph of inere emigroup, Semigroup Forum, 7,, pp 4 48, 006 A V Kelare and E Praeger, On tranitie ayley graph of group and emigroup, European J ombin, 4,, pp 59 7, 00 A V Kelare and S J Quinn, Directed graph and combinatorial propertie of emigroup, J Algebra, 5,, 6 6, 00 A V Kelare and S J Quinn, A combinatorial property and ayley graph of emigroup, Semigroup Forum, 66,, 89 96, 00 4 A V Kelare, Ra and S Zhou, Ditance labelling of ayley graph of emigroup, Semigroup Forum DOI: 0007/ A V Kelare, J Ryan and J L Yearwood, ayley graph a claifier for data mining: the influence of aymmetrie, Dicrete Math 09, 7, pp , B Khorai, On ayley graph of left group, Houton J Math, 5,, pp , B Khorai and M Mahmoudi, On ayley graph of rectangular group, Dicrete Math, 0, 4, pp 804 8, 00 8 B Khorai and B Khorai, A haracterization of ayley Graph of Brandt Semigroup, Bull Malayian Math Sci Soc, 5,, 99 40, 0 9 B Khorai and B Khorai, On ayley graph of emilattice of emigroup, Semigroup Forum, 86,, pp 4, 0 0 B Khorai, B Khorai and B Khorai, On color-automorphim ertex tranitiity of emigroup, European J ombin, 40, 55 64, 04 U Knauer, Algebraic Graph Theory: Morphim, Monoid and Matrice, de Gruyter Studie in Mathematic, Vol 4, Walter de Gruyter & o, Berlin, 0 S Panma, N Na hiangmai, U Knauer and Sr Arworn, haracterization of lifford emigroup digraph, Dicrete Math 06,, 47 5, 006 S Panma, U Knauer and Sr Arworn, On tranitie ayley graph of trong emilattice of right (left) group, Dicrete Math, 09, , G Sabidui, On a cla of fixed-point-free graph, Proc Amer Math Soc, 9, pp , 958 Receied April, 0