The Properties Of Two Graphs On Semigroups

In this dissertation,we give the definitions of the inclusion graph of S-acts on semigroups and the annihilating-ideal graph of commutative semigroups with respect to an ideal.We study the properties of the two graphs.The primary way to study the graph structure is by algebraic theory.The main results are given as follows:In the first chapter,we give the introduction and some basic notions in semi-group theory and graph theory.In the second chapter,we investigate the inclusion graph of S-acts in semi-groups.Let S be a semigroup and M be an S-act.The inclusion graph of M,denoted by G(M),is the undirected simple graph whose vertices are all non-trivial subacts of M and defining two distinct vertices I and J to be adjacent if and only if I(?)J or J(?)I.Some results on completeness,connectivity,diameter,girth,the clique number and the chromatic number of G(M)are given.The main results are as follows:Theorem 2.1.6 Let M be a decomposable S-act.Then G(M)is disconnected if and only if |G(M)|=2.Theorem 2.1.11 Let M be a decomposable S-act.Then G(M)is disconnected if and only if M is a coproduct of two simple S-acts.Theorem 2.2.1 Let M be an indecomposable S-act.Suppose that G(M)is connected,then diam(G(M))?2.Theorem 2.2.3 If there is a cycle in G(M),then g(G(M))=3.Theorem 2.2.4 If G(M)has a cycle of length 4 or 5,then there is a triangle.Theorem 2.3.1 Let S1 and S2 be two semigroups.The inclusion graphs of S1 and S2 are G(S1)and G(S2).Then?(G(S1))+?(G(S2))+1??(G(S1 × S2))??(G(S1))+?(G(S2))+1.Theorem 2.3.4 Let S be a semigroup.Suppose that the number of left ideal of S is finite.Then ?(G(S))=?(G(S)).In the third chapter,we discuss the annihilating-ideal graph of a commutative semigroup with respect to an ideal.Let S be a commutative semigroup.Let I be a proper ideal of S.The annihilating-ideal graph of a commutative semigroup S with respect to I,denoted by AGI(S),is an undirected simple graph.The vertices of AGI(S)are annihilating ideals with respect to I.Two distinct vertices A,B are defined to be adjacent if and only if AB C I.Based on this definition,we will investigate the influence of the annihilator with respect to I and the basic properties of AGI(S),including connectivity,diameter,girth,and cut vertices.The mainly results are given as follows:Theorem 3.1.4 Let S be a commutative semigroup.If AnnI(S)=S,then AGI(S)is a complete graph.Theorem 3.1.8 Let S be a commutative zero-divisor semigroup with respect to I.Then there exists an element A in V(AGI(S))which is adjacent to the other vertex,where I(?)A,if and only if AnnI(?)I or AGI(S)is a star graph with centerTheorem 3.1.13 Let S be a commutative zero-divisor semigroup with respect to I.Suppose that AnnI(S)(?)AI(S),then 2}|AnnI(S)\I|-1??(AGI(S)).Theorem 3.1.14 Let S be a commutative zero-divisor semigroup with respect to I.If every ideal of S is principal ideal which is not contained in I,and from a?b we have aS1?bS1,Then there exists a subgraph of AGI(S)which is isomorphic to ?I(S).Theorem 3.2.3 Let I be a proper ideal of a commutative semigroup S.Then AGI(S)is connected and diam(AGI(S))?3.Moreover,if there exists a cycle of AGI(S),then g(AGI(S)?4.Theorem 3.2.4 Let I be a proper ideal of a commutative semigroup S.If(?)=I,then AGI(S)is a complete graph if and only if AI(S)=Min(NI(S)).Theorem 3.2.6 Let I be a proper ideal of a commutative semigroup S.Let A be a vertex of AGI(S).If I(?)A,then A is not a cut vertex.Theorem 3.3.3 Let I be a proper ideal of a commutative semigroup S.Then 1)g(AGI(S))?{3,4,?}.2)If for all X ?V(AGI(S)),there exists X(?),then g(AGI(S))=? if and only if AGI(S)is a star graph.Theorem 3.3.6 Let I be an ideal of a commutative semigroup S.Suppose that P1 and P2 are two distinct prime ideals of S and I=P1?P2.If |CP1(S)\CP2(S)|?2 and |CP2(S)\CP1(S)|?2,then g(AGI(S))=4.