Microsoft Word - 160-170 Mathematics | 160 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2009 Vol. 31 (3) 2018 Study of Two Types Finite Graphs in KU-Semigroups Elaf R. Hasan math88012@gmail.com Fatema F. Kareem fa_sa20072000@yahoo.com Department of Mathematics, College of Education for Pure Science Ibn Al-Haitham, University of Baghdad, Baghdad, Iraq. Article history: Received 25 June 2018, Accepted 18 July 2018, Published December 2018 Abstract In this ˑresearch, we present theˑ notion of the ˑgraph for a KU-semigroup X as theˑ undirected simple graphˑ with the vertices are the elementsˑ of X and weˑˑstudy the ˑgraph ofˑ equivalence classes ofˑ X which is determinedˑ by theˑ definition equivalenceˑ relation ofˑ these verticesˑ, andˑ then some related ˑproperties areˑ given. Several examples are presented and some theorems are proved. Byˑ usingˑ the definitionˑ ofˑ isomorphicˑ graph, ˑwe showˑ thatˑ the graphˑ of equivalence ˑclasses ˑand the ˑgraphˑofˑa KU-semigroup ˑ areˑ theˑ sameˑ, in special cases. Key words: KU-algebra, KU-semigroup, graph, annihilator. 1.Introduction Mathematicians Prabpayak andˑ Leerawat [1, ˑ2] constructed algebraicˑ structure whichˑ isˑ calledˑˑKU-algebra and they ˑintroduced the conceptˑ of ˑa homomorphism of ˑKU-algebra. Kareem and ˑHasan [3] introduced a new class of algebras related to KU-algebras and semigroups, called a KU-semigroup. They defined some types of ideals and discussed few properties. The study of graph theory and its properties are topics of interest in algebraic structures. Beck in [4] introduced the graph of commutative ring by studied the ˑzerodivisor graphs of thisˑ ring. Many mathematicians studied a graph of a commutative ring by different ways; see [5- 10]. In [11], Jun and Leeˑ introduced theˑ concept of theˑ associated graphˑ of ˑBCK/BCI-algebra andˑ they provedˑ that: if X ˑis a BCK-ˑalgebra, ˑthen theˑ associatedˑ graph ofˑ X isˑ connected but ˑif X is a BCI-algebra, then it's not connected. Zahiri and Borzooei [12] introduced aˑnewˑgraph of a BCIˑ-algebraˑ X andˑ they definedˑ the concept ofˑ a-ˑdivisor of BCI-ˑalgebraˑ X . ˑMostafa and Kareem [13] introduced the graphˑ of aˑ commutative ISˑ-algebra ˑX, denotedˑ by (X) and studiedˑ the graph of ˑequivalenceˑ classes of X. In thisˑ research, we introduceˑ the idea ofˑ graph for a KU-ˑsemigroup. We defineˑ the graph as the ˑundirected graph with ˑthe vertices are the elements ˑin KU-ˑsemigroupˑ X ˑ and forˑ distinct vertices ˑ x andˑ y ˑare adjacentˑˑif and onlyˑ ifˑ }0{}),({}),({  yxLyxR ˑ.ˑMoreoverˑ,ˑ weˑ studyˑ the other graph namely, graph ofˑˑequivalenceˑ classesˑ of X byˑ definition of equivalenceˑ relation ofˑ these verticesˑ andˑ then some related ˑproperties areˑ given. 2.Preliminaries Inˑ this section, we present some definitions and background about a KU-algebra and KU- semigroup. Definition 1[1-2]. Algebra )0,,( X is called a KU-algebra if it satisfies the following axioms: Mathematics | 161 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2009 Vol. 31 (3) 2018 ( 1ku ) 0)]())[()(  zxzyyx , ( 2ku ) 00 x , ( 3ku ) xx 0 , ( 4ku ) 0 yx and 0 xy implies yx  , ( 5ku ) 0 xx . On aˑ KU-algebraˑ X , we can defineˑ a binary relationˑ  by ˑputting 0 xyyx . ˑThen ),( X is a partially ˑordered set ˑand 0 is ˑits smallestˑ element. Thusˑ )0,,( X satisfies the followingˑ conditions. For allˑ Xzyx ,, , weˑ that ( \1ku ) )()()( yxzxzy  , ( \2ku ) x0 , ( \3ku ) xyyx  , implies yx  , ( \4ku ) xxy  .ˑ ˑTheoremˑ2ˑ[14].ˑIn a KUˑ-algebra X . ˑTheˑ followingˑ axiomsˑ are satisfied, for all Xzyx ,, (1) yx  ˑimplyˑ zxzy  , (2) )()( zxyzyx  ,ˑfor allˑ Xzyx ,, , (3) yxxy  ))(( ˑ.ˑ Definition 3 [1-2]. ˑA non-ˑempty subsetˑ I of a KU-algebra )0,,( X is calledˑˑan ˑideal of X if ˑfor anyˑ Xyx , , then (i) I0 and (ii) Ixyx  , imply that Iy  .ˑˑ Definition 4 [1-2].ˑLet I be a nonempty subset of a KU-algebra X . Then I is said to be a KU-ideal of X , if )( 1I I0 and )( 2I Xzyx  ,, , Izyx  )( and Iy  imply that Izx  . ˑˑDefinition 5[15].ˑA KU-ˑalgebraˑ )0,,( X ˑis saidˑˑto beˑa commutativeˑifˑitˑsatisfiesˑ: for all ˑ yx, in X , yyxxxy  )()( , where xxyyx  )( , i.e. xyyx  .ˑˑ Lemma 6ˑ[15]. If X is a commutative KU-algebra, then )()()( zxyxzyx  . Example 7[15]. Let e}d,c,b,a,0,{X ˑbe a set, with the operation defined by the following table: Mathematics | 162 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2009 Vol. 31 (3) 2018 Thenˑ )0,,( X is aˑKU-algebra andˑKU-commutative. Definition 8 [3]. A KU-semigroup is a nonempty set X with twoˑbinary operations , ˑand constantˑ 0 satisfying ˑtheˑfollowing axiomsˑ (I) )0,,( X isˑˑa KU-ˑalgebra, (II) ),( X is aˑˑsemigroup, (III)The ˑoperation  is ˑdistributiveˑ(on bothˑ sides) over theˑˑoperation , i.e. )()()()()()( zyzxzyxandzxyxzyx   , for all Xzyx ,, . Example 9[3]. Let }3,2,1,0{X beˑa set. Define  -ˑoperation and  -ˑoperation by the following tables Then, )0,,,( X is a KU-semigroup. Proposition 10[3]. Let )0,,,( X be a KU-semigroup. The followingˑaxioms areˑsatisfied. For all Xzyx ,, , (1) 00 x and 00 x , (2) If yx  implyˑ yzxz   and zyzx   , (3) )()()()()()( zyzxzyxandzxyxzyx   . 3.A graph ofˑKU-semigroups In thisˑ part, we introduceˑ the conceptsˑ of graph KU-semigroupsˑ X and theˑ graph of equivalence ˑclasses of X . We ˑrecall some definitions ˑand basic ˑfacts. For aˑ graph G ˑ, we ˑdenotedˑ theˑ set of ˑvertices ˑof G ˑasˑ )(GV and ˑthe ˑset ˑof edgesˑ as )(GE . An edge to be  0 aˑ b c d e 0 0 a bˑ c d e a 0 0 b c bˑ c b 0 a 0ˑ b aˑ d c 0 aˑ 0 0 a a d 0 0ˑ 0 b 0 b e 0 0 0 0 0 0 * 0 1 2 3 0 0 1 2 3 1 0 0 0 2 2 0 2 0 1 3 0 0 0 0  0 1 2 3 0 0 0 0 0 1 0 1 0 1 2 0 0 2 2 3 0 1 2 3 Mathematics | 163 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2009 Vol. 31 (3) 2018 associated with a vertex pair ji xx  ; such an edge having the same vertex as end vertices are called a loop. Also if more than one edge to be associated with a given pair of vertices, then edges referred to as parallel edges.A simple graph is a graph that has neither loops nor parallel edges. A graphˑ G is ˑsaid to be complete ˑ if everyˑ two distinct ˑvertices areˑ joined byˑ exactly one ˑedge. ˑA ˑgraph ˑ G is said ˑto be ˑbipartite graph ˑif its vertex ˑset )(GV can ˑbe partitionedˑ into disjointˑ subsets 1V andˑ 2V such thatˑ, every edge of ˑ G joinsˑ a vertexˑ of 1V withˑ aˑ vertex ofˑ 2V . Soˑ, G ˑ is calledˑ a completeˑ bipartiteˑˑ graphˑ if every vertexˑ in ˑone of theˑˑbipartition subsetˑ isˑ joined to ˑevery vertexˑ inˑ theˑ otherˑ bipartition ˑˑsubset. If 1V andˑ 2V haveˑ m and n ˑvertices ˑrespectivelyˑ, then a completeˑˑ bipartite ˑgraphˑ will be denotedˑ by nmK , . Consequently, a star ˑgraph is aˑ complete bipartiteˑ graph ofˑ the form nK ,1 .ˑ A ˑgraph ˑGˑ is saidˑ to beˑ a connected if ˑthere is ˑa pathˑˑ between anyˑ given pairs ofˑ vertices, otherwiseˑ the graph is a ˑdisconnected. The neighborsˑ of a vertex x inˑ a graphˑ G , denoted ˑby )(xN ˑare the set ofˑ vertices that are ˑadjacent to x . A graph ˑH is called a subgraphˑ of G ˑ if )()( GVHV  ˑand )()( GEHE  .ˑ Two graphs 1G andˑ 2G are saidˑ to be isomorphicˑ if there exists ˑa bijective mappingˑ )()(: 21 GVGVf  such thatˑ )( 1GEyx  then ˑ )()()( 2GEyfxf  . For more ˑdetails we refer ˑto [16] ˑ. Definitionˑ11. Letˑ A be aˑ subset of a KU- semigroup ˑ )0,,,( X , then ˑwe define the followingˑ },0,0:{)( AaaxaxXxAL   ,ˑ ˑ },0,0:{)( AaxaxaXxAR   . If ˑ }{aA  ˑ, then weˑ write )(aL and )(aR ˑinstead of })({aL ˑand })({aR ,ˑrespectively. Proposition12.ˑ Let A ˑand B be non-empty subsetsˑ in )0,,,( X , then the ˑfollowing statements areˑ true: (1) )]([ ALRA  andˑ )]([ ARLA  , (2) ˑIf BA , thenˑ )()( ARBR  and ˑ )()( ALBL  , (3) ˑ )]]([[)( ARLRAR  ˑˑ andˑ )]]([[)( ALRLAL  . (4) ˑ )()()( BRARBAR   ˑˑandˑ )()()( BLALBAL   (5) ˑ )()()( BARBRAR   ˑand )()()( BALBLAL   Proof. (1) Let ˑ Aa  ˑand )( ALx  , ˑthen 0,0  axax  . It follows that )]([ ALRa  , hence )]([ ALRA  . Similarlyˑ, )]([ ARLA  . (2) Suppose thatˑ BA  and )(BRx  , ˑthen 0,0  xbxb  for all Bb  . AbxbxbtherforeBAbut  0,0,  . So )( ARx  ,ˑ hence )()( ARBR  . Similarly,ˑ )()( ALBL  . (3) by usingˑ (1) andˑ (2) we have ˑ )]([ ALRA  ˑand )]([ ARLA  ˑimpliesˑˑ that byˑ (2) )()]]([[ ALALRL  and )()]]([[ ARARLR  . If ˑˑweˑ applyˑ (1) to )( AL and ˑ )( AR , then Mathematics | 164 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2009 Vol. 31 (3) 2018 )]]([[)( ALRLAL  andˑ )]]([[)( ARLRAR  . Henceˑˑ )]]([[)( ARLRAR  andˑ )]]([[)( ALRLAL  . (4) Sinceˑ BAA  and BAB  , weˑ have by ˑpart (2) of ˑProposition 3.2 thatˑ, )()( ARBAR  and )()( BRBAR  , hence )()()()( IBRARBAR   Converselyˑ, ifˑ )()( BRARx  , ˑthen )()( BRxandARx  ,ˑ therefore Aaxaxa  ,0,0  andˑ Bbxbxb  ,0,0  . Butˑ ˑifˑ  BAc  , ˑthen )(,0,0 BAcxcxc   weˑ haveˑ )( BARx  , henceˑ )()()()( IIBARBRAR   Fromˑ (I)ˑ and (II), ˑwe have )()()( BRARBAR   .ˑ Similarly, )()()( BLALBAL   . (5) we ˑhave BAA  , BAB  ˑ from (2) )()( BARAR  and )()( BARBR  which impliesˑ that )()()( BARBRAR   . Similarlyˑ, )()()( BALBLAL   .ˑ Example ˑ13.ˑ Letˑ }3,2,1,0{X beˑa ˑsetˑ. Define  -ˑoperation and  -ˑoperation by the following tables Then )0,,,( X is a KU-semigroup. We have the following }3,0{})2,1({})2,0({})1,0({  RRR , }2,0{})3,0({ R and }0{})3,2({})3,1({  RR . Also, }2,1,0{})3,0({},0{})2,1({},3,0{})2,0({},0{})1,0({  LLLL and }0{})3,2({})3,1({  LL . Remark 14. If )0,,,( X is a KU-semigroup, then (i) XRL  })0({})0({ , (ii) }0{})({})({  XRXL . Proof. ˑClear. Lemma ˑ15.ˑ Let )0,,,( X ˑbe a KU-ˑsemigroup and aˑ KU-algebra beˑ a commutative, then forˑ any elements ˑ a and b of X . If 0ba , ˑthen })({})({ bLaL  . Proof.ˑˑSupposeˑ ˑthat 0ba . Let 0,0})({  axaxaLx  , then by lemma 6 )()(0)()()(0 bxbxbxaxbax  and * 0 1 2 3 0 0 1 2 3 1 0 0 1 3 2 0 0 0 3 3 0 1 2 0  0 1 2 3 0 0 0 0 0 1 0 1 0 0 2 0 0 2 0 3 0 3 0 0 Mathematics | 165 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2009 Vol. 31 (3) 2018 )()(0)()()(0 bxbxbxaxbax   . It follows that })({bLx  , hence })({})({ bLaL  . Lemma16. If A is a subset ofˑ a KU-ˑsemigroup )0,,,( X and a KU-ˑalgebra is aˑ commutative, then )( AR is anˑ ideal of X . Proof. For any Aa  , we have 0)0(0  aaaaa and 00 a . Hence )(0 AR . Let )(),( ARxARyx  , then 0)(  yxa , 0)(  yxa  which implies that by lemma6 0)()(  yaxa and since )( ARx  , then 0)(0  ya , hence 0 ya . Also, 0)()(  yaxa  and since )( ARx  , 0)(0  ya  , hence 0ya  , i.e. )( ARy  . Which implies that )( AR is ˑan ideal ˑofˑ X . Definitionˑ 17.ˑ A graph of aˑ KU-semigroup X , ˑdenoted by ˑΩ 𝑋 is ˑ a simple graph whoseˑ vertices areˑ the elements ˑof X and ˑtwo distinctˑ elements Xyx , areˑ adjacentˑ ifˑ andˑ onlyˑ ifˑˑ }0{}),({}),({  yxLyxR . Exampleˑ 18.ˑ Letˑ },,,,0{ dcbaX  ˑbeˑaˑ setˑ.ˑ Define  -ˑoperation andˑ  -ˑoperation by ˑthe followingˑ tables Thenˑ )0,,,( X is a KU-semigroup. So ˑ },,,0{}),0({ dcaR  },0{}),({}),0({ cbaRbR  },,0{}),0({ bacR  , }0{}),({}),({}),({}),({}),({}),0({  dcRdbRcbRdaRcaRdR . Also, }0{}),({}),({}),({}),({}),({}),0({  dcLdbLcbLdaLcaLdL . Byˑ definition17, we determineˑ the graphˑ of ˑ X asˑ followsˑ: ˑThe ˑset ˑofˑ vertices ˑis ˑ },,,,0{)( dcbaXV  ˑˑandˑˑˑ theˑ set ˑof ˑedgesˑˑis ˑ },,,,,0{)( dcdbcbdacadXE  . Theˑ figure (1) ˑshowsˑ the graphˑ Ω 𝑋 . ˑ  0 a b c d 0ˑ 0 ˑ a b ˑ c d a ˑ 0 0 ˑ a ˑ c d ˑ b 0 0 0 c d ˑ c 0 a ˑ b 0 ˑ d ˑ d 0 a ˑ b c 0  0 a b c d 0 ˑ 0 0 ˑ 0 0 ˑ 0 a 0 0 0 0 a b ˑ 0 0 ˑ 0 0 b c 0 0 0 b c d 0 ˑ a b ˑ c d Mathematics | 166 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2009 Vol. 31 (3) 2018 Figure 1.The GraphˑΩ 𝑋 Example 19. ˑLet }3,2,1,0{X beˑaˑ set in example13. ˑThe set of vertices ˑis }3,2,1,0{ and the set of edges is 1 3 𝑎𝑛𝑑 2 3 . Theˑ figure (2) showsˑ the ˑ graph ˑΩ 𝑋 . Figure 2. The GraphˑΩ 𝑋 Theorem 20. A disconnected graph cannot be a graph of any KU-semigroups 𝑋. Proof. Suppose G is a disconnected graph with components 1G and 2G . Let ˑ𝐺 Ω 𝑋 ˑ beˑ a graphˑ of ˑ KU-ˑsemigroups𝑋. Then, there exist vertices 1Gx  and 2Gy  such that there is no path between 𝑥 and 𝑦. Let 1Ga  and 2Gb be vertices adjacent to 1Gx  and 2Gy  , respectively. Then 𝑥  𝑎 𝑎𝑥 0, 𝑥a a𝑥 0 and 𝑦  𝑏 𝑏𝑦 0, 𝑦b b𝑦 0. If  𝑏 𝑏𝑎 𝑧, 𝑎b b𝑎 z , for ˑsomeˑz 𝑋. Then 𝑥 𝑧 𝑧𝑥 0, 𝑦 𝑧 𝑧𝑦 0, 𝑥z z𝑥 0, yz z𝑦 0 . Thus z is a common neighbor of 𝑥 and 𝑦 , that is a contradiction. Hence, a disconnected graph cannot be a graph of any KU-semigroups. Definition21. Let X be a KU-semigroup. Forˑ any Xx , we have }}0{}),({}),({:{)(  yxLyxRXyxann ˑis called ˑthe set of annihilator of x . Lemma 22. Let X be a KU-semigroup and )( xann ˑbe ˑthe setˑ of annihilator of x . Then (i) Xxxannx  )( , (ii) Thereˑ is anˑ edgeˑ connectingˑ x and y if ˑand onlyˑ if ˑ )( yannx  ˑand ˑ )( xanny  . Mathematics | 167 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2009 Vol. 31 (3) 2018 Proof. (i) Clear by definition 17. (ii): Suppose that )( yannx  or )( xanny  , then }0{}),({}),({  yxLyxR . It implies thatˑ thereˑ is no ˑedge connectingˑ x andˑ y , thisˑ is aˑ contradiction. Thus )( yannx  and )( xanny  . Conversely, suppose that )( yannx  and )( xanny  , then }0{}),({}),({  yxLyxR . It implies that ˑthere is ˑan edgeˑ connectingˑ x andˑ y . Definition 23.ˑ Define aˑ relationˑ on a KUˑ-semigroupˑ X as followsˑ: x1 x2 ifˑ and only ˑif ˑ Xxxxannxann  2121 ,),()( . Lemma 24. ˑThe relation ˑ (from definition23) is ˑan equivalenceˑ relation ˑon X . Proof.ˑ Clear. 4. A graphˑ of EquivalenceˑClasses ofˑ KU-ˑSemigroups Nowˑ, we introduceˑ ˑthe graphˑ of ˑequivalence classesˑ of aˑ KUˑ-ˑsemigroup X , ˑwhich is constructedˑ fromˑ classes ofˑˑ equivalence relationˑ ~ ˑin ˑdefinitionˑ 23. For Xyx , , ˑwe say thatˑ x ~ y if andˑ only ifˑ )()( yannxann  . As denoted ˑin (lemma 4), ~ is anˑ equivalenceˑ relation. Furthermoreˑ, ifˑ ][ x denotes theˑclass of x, thenˑ the product ˑ ][][][ yxyx   andˑ ][][][ yxyx  . Definition25.ˑThe graphˑ of equivalence ˑclasses of ˑa KU-semigroupˑ X , denoted by ˑˑ X is the undirected simpleˑˑgraph whoseˑ vertices ˑareˑ theˑ set of ˑequivalenceˑˑclasses ˑ  Xxx ];[ ˑ and twoˑˑdistinctˑ classesˑˑ ][],[ yx are ˑadjacentˑ in ˑ  X ˑ if ˑandˑ only ifˑ }0{][][ yx  and }.0{][][  yx Example 26.ˑ Let },,,0{ cbaX  beˑˑa set. Define  -ˑˑoperation and  -ˑˑoperation by the following tables Thenˑ )0,,,( X is aˑ KU-ˑsemigroup. We haveˑ the set of ˑvertices is ˑV Ω 𝑋 0, 𝑎, 𝑏, 𝑐 ˑ and ˑthe ˑset of edges ˑis ˑE Ω 𝑋 0 𝑎,0 𝑏,𝑎 𝑏, ˑ𝑎 𝑐,𝑏 𝑐 .ˑ Soˑ, the set ˑof vertices ˑof ˑ 𝑋 is ]}[],[],0{[ ba ˑsince },,{)0( baann  },,0{)( cbaann  , },,0{)( cabann  ˑand },{)( bacann  ,ˑ then 𝐸  𝑋 0 𝑎 , 0 𝑏 , 𝑎 𝑏 . ˑTheˑ followingˑ Figure showsˑ the graph Ω 𝑋 ˑ and  𝑋 .  0ˑ aˑ bˑ c 0ˑ 0 aˑˑ b cˑ aˑ 0 0ˑˑ a cˑ bˑ 0 0ˑˑ 0 c cˑ 0 aˑ bˑ 0  0 aˑ b c 0 0 0ˑ 0 0 a 0 0ˑ 0 a b 0 0ˑ 0 b c 0 a b c Mathematics | 168 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2009 Vol. 31 (3) 2018 Figure 3. The GraphsˑΩ 𝑋 and 𝑋 ˑ Example ˑ27ˑ.ˑLet ˑ },,,,0{ dcbaX  beˑaˑ setˑ in example 18. Thenˑ theˑ set ofˑˑ vertices ˑof ˑ 𝑋 ˑ is ˑ ]}[],[],[],0{[ dca ˑand ˑthe set of edges is }],[][],[][],[][],[]0{[ dcdacad  The figure (4) shows ˑthe graph of ˑequivalence classes ˑ 𝑋 . Figure 4. The Graph 𝑋 Lemma 28. With ˑnotations asˑ before. 1) ˑ 𝑋 is ˑa sub graphˑ of Ω 𝑋 ; 2) For all ˑ Xx  , weˑ have )()( xannxN  . Proof.ˑ Straightforward. Theorem29.ˑ Let  𝑋 be theˑ graphˑ ofˑ equivalence ˑclassesˑˑ of X .ˑ For ˑany distinct ˑvertices 𝑥ˑ , 𝑦 ˑ 𝑋 ˑ, if ˑ ][][ yandx ˑˑconnected ˑby ˑanˑ edgeˑ,ˑ thenˑ )()( yannxann  . Proof.ˑ Suppose that ˑ )()( yannxann  , thenˑ yx ~ . Henceˑ ][][ yx  this isˑ a contradictionˑ. Thereforeˑ )()( yannxann  . The ˑconverse of thisˑ theorem is ˑnot true. In exampleˑ27, we haveˑ the verticesˑ ][,]0[ c and ˑ )()0( cannann  but noˑ edge jointˑ between themˑ. Theorem30. ˑLet X as ˑmentioned above. If Ω 𝑋 is a ˑcomplete graph, ˑthen Ω 𝑋  𝑋 . But theˑ convers is notˑ true. Proof.ˑ Suppose that ˑV Ω 𝑋 𝑥 , 𝑥 , … , 𝑥 . If Ω 𝑋 ˑ is the completeˑ graph, then ˑevery pair of ˑits verticesˑ are adjacent.ˑ Thus nixxxxN i ,...,2},,...,,{)( 321  nixxxxN i ,...,3,1},,...,,{)( 312  ,…, },...,,{)( 121  nn xxxxN . Then, )()(),...,()(),()( 2211 nn xNxannxNxannxNxann  ,Thus )(...)()( 21 nxannxannxann  Mathematics | 169 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2009 Vol. 31 (3) 2018 thereforeˑˑ everyˑ vertex of ˑΩ 𝑋 isˑˑ a equivalenceˑ classˑ of ˑˑ 𝑋 , thusˑ theˑ vertices of ˑˑ 𝑋 ˑ are distinctˑ and ˑthe sameˑ number of ˑvertices of ˑ Ω 𝑋 ˑ , thenˑ there existˑ an isomorphic ˑ 𝑓: Ω 𝑋 →  𝑋 ˑ such thatˑ ][)( ii xxf  for ˑeachˑˑ },...,2,1{ ni  ˑand theˑˑ mapping of ˑedges ˑ𝑓: 𝐸 Ω 𝑋 → 𝐸  𝑋 , ˑwhich sendsˑ theˑ edge ji xx  in ˑ𝐸 Ω 𝑋 ˑ toˑ the edge ˑ ][][ ji xx  inˑ 𝐸  𝑋 is ˑa well-ˑˑdefined ˑbijection. The converseˑ of this theoremˑ is false asˑ illustrated inˑ example31, ˑwe have Exampleˑˑ31.ˑˑLet },,,0{ cbaX  beˑˑaˑ set. Define  -ˑoperation and  -ˑoperationˑ by ˑthe ˑfollowing ˑˑtablesˑ Thenˑˑ )0,,,( X is a ˑKU-semigroup. ˑ We ˑdetermine the ˑgraph Ω 𝑋 ˑ as follows: ˑV Ω 𝑋 0, 𝑎, 𝑏, 𝑐 and ˑˑ E Ω 𝑋 0 𝑎, 𝑎 𝑏, 𝑎 𝑐, 𝑏 𝑐 . ˑThe set ˑof verticesˑ of  𝑋 is ]}[],[],[],0{[ cba , sinceˑ },{)0( aann  },,0{)( cbaann  , },{)( cabann  and ˑ },{)( bacann  ,ˑ then Eˑ  𝑋 0 𝑎 , 𝑎 𝑏 , 𝑎 ˑ 𝑐 , 𝑏 ˑ 𝑐 . ˑ Theˑ followingˑ Figure showsˑ the graph Ω 𝑋 ˑ and  𝑋 . Figure 5. The graphsˑΩ 𝑋 and 𝑋 ˑIn above figure Ω 𝑋  𝑋 ˑ, butˑ Ω 𝑋 is not aˑ complete graphˑ. Theorem 32.ˑIf Ω 𝑋 is aˑ star graph,ˑ then  𝑋 is ˑan edge. Proof.ˑ Suppose that ˑΩ 𝑋 is a starˑ graph with vertexˑ set V Ω 𝑋 𝑥 , 𝑥 , … , 𝑥 . Thisˑ set can be splitˑ into two ˑsets }{ 11 xV  andˑ },...,{ 22 nxxV  such thatˑ the vertex ˑof 1V is joinedˑ to each vertex ˑof 2V by exactlyˑ one edge. Thusˑ, the set of ˑedges is 𝐸 Ω 𝑋 𝑥 𝑥 , 𝑥 𝑥 , … , 𝑥 𝑥 , so 2321 },...,,{)( VxxxxN n  and 1312 )(...)(}{)( VxNxNxxN n  , ˑthen 21 )( Vxann  and 132 )(...)()( Vxannxannxann n  . ˑThen thereˑ are two distinctˑ equivalence classesˑ ][ 1x and ][ 2x in  𝑋 , whichˑ are adjacent.ˑ Thus  𝑋 is ˑan edge. Lemma 33. Let G and H be two graphs and HG  . ˑIfˑˑ yxf )( ,ˑˑthenˑˑ )())(( yNxNf  ˑfor allˑ )()( HVyandGVx  ˑ.  ˑ 0 aˑ bˑ c 0ˑ 0ˑ aˑ bˑ cˑ a 0ˑ 0ˑ aˑ c b 0ˑ 0ˑ 0ˑ c cˑ 0ˑ aˑ bˑ 0  0 aˑˑ b cˑ 0ˑ 0 0ˑˑ 0 0ˑ a 0ˑ aˑˑ 0ˑ c b 0 0ˑˑ b 0 cˑ 0 cˑ 0ˑ c Mathematics | 170 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2009 Vol. 31 (3) 2018 Proof. Let HGf : be a graph isomorphism, )(GVx  and )()( HVyxf  . Then )()}(:)({)}()(:)({}:)({))(( yNzfyzfzfxfzfzxzfxNf  . Theorem 34. ˑLet X and Y be two KU-semigroups . If ˑΩ 𝑋  Ω 𝑌 ˑ, ˑthen  𝑋  𝑌 . Proof.ˑ Supposeˑ that ˑ𝑉 Ω 𝑋 𝑥 , 𝑥 , … , 𝑥 ˑ and ˑ𝑉 Ω 𝑌 𝑦 , 𝑦 , … , 𝑦 ˑ such thatˑ the isomorphismˑ 𝑓: Ω 𝑋 → Ω 𝑌 ˑ satisfies ˑ ii yxf )( for eachˑ },...,2,1{ ni  . ˑByˑ lemmaˑ 33, )())(( ii yNxNf  for eachˑ i , thenˑ )())(( ii yannxannf  and the mapping ofˑ edges ˑ𝑓: 𝐸  𝑋 → 𝐸  𝑌 , which sendsˑ the edge ˑ ][][ ji xx  in  𝑋 to the edge ][][ ji yy  in  𝑌 is aˑ well-ˑdefined ˑbijection. ˑThus  𝑋 ˑ 𝑌 . Theˑ converseˑ of this theoremˑ is not true. Inˑ Examples 27 and 31, we have  𝑋  𝑌 ˑ but Ω 𝑋 Ω 𝑌 . References 1. Prabpayak. C.; Leerawat. U. On ideals and congruence in KU-algebras. Scientia Magna Journal. 2009, 5, 1, 54-57. 2. Prabpayak. C.; Leerawat. U. On isomorphisms of KU-algebras. Scientiamagna journal. 2009, 5, 3, 25-31. 3. Fatema. F. Kareem; Elaf. R. H. On KU-semigroups. Int. J.Sc. Nat. 2017, 8(4), 1- 4. Beck, I. Coloring of commutative rings. J. Algebra.1988, 116, 208-226. 5. Akbari. S.; Kiani. D.; Mohammadi. F.; Moradi. S. 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