A NOTE ON THE ZERO DIVISOR GRAPH OF A LATTICE

Let L be a lattice with the least element 0. An element x 2 L is a zero divisor if x^y = 0 for some y 2 L = L \ {0}. The set of all zero divisors is denoted by Z(L). We associate a simple graph 􀀀(L) to L with vertex set Z(L) = Z(L) \ {0}, the set of non-zero zero divisors of L and distinct x, y 2 Z(L) are adjacent if and only if x ^ y = 0. In this paper, we obtain certain properties and diameter and girth of the zero divisor graph 􀀀(L). Also we find a dominating set and the domination number of the zero divisor graph 􀀀(L).