THE ANNIHILATING-IDEAL GRAPHS OF MV-ALGEBRAS

In this paper, we introduce and study the annihilating-ideal graph of an MV-algebra (A, circle plus, *, 0). The algebraic structure of MV- algebras (especially Boolean algebras) are described by using the annihilating-ideal graph. The connections between the ideal theory of MV-algebras and graph theory are established, which promote the studying of the coloring of graphs. The annihilatingideal graph AG(A) is a simple graph with the vertex set V (AG(A)) = {I is an element of I(A)\{< 0 >, A} |there exists J is an element of I*(A) such that IJ = < 0 >} and the edge set E(AG(A)) = {I - J | IJ = < 0 >, where I, J is an element of V (AG(A)) and I not equal J}, where I(A) is the set of all ideals of A and I*(A) = I(A)\{< 0 >}. We verify that AG(A) is connected with d(max) (AG(A)) <= 3. And we characterize some MV-algebras with d(max)(AG(A)) = 0 or 1, where d(max)(AG(A)) is the diameter of AG(A). If | A |<= 7, we show that AG(A) is either a null graph, or d(max)(AG(A)) = 1. We restrict MV-algebras to Boolean algebras. The connections between AG(A) and Gamma(A) are studied, where Gamma(A) is the zero-divisor graph of A. We characterize the complete graph AG(A) and the star graph AG(A) by using ann(A\{1}) - {a is an element of A | a circle dot b = 0 for all b is an element of A\{1}}, where ann(A\{1}) is the annihilator of A\{1}. Finally, we study the vertex coloring and girth of AG(A). We give two lower bounds and an upper bound for chi(AG(A)).