a note on the zero divisor graph of a lattice

abstract. let $l$ be a lattice with the least element $0$. an element $xin l$ is a zero divisor if $xwedge y=0$ for some $yin l^*=lsetminus left{0right}$. the set of all zero divisors is denoted by $z(l)$. we associate a simple graph $gamma(l)$ to $l$ with vertex set $z(l)^*=z(l)setminus left{0right}$, the set of non-zero zero divisors of $l$ and distinct $x,yin z(l)^*$ are adjacent if and only if $xwedge y=0$. in this paper, we obtain certain properties and diameter and girth of the zero divisor graph $gamma(l)$. also we find a dominating set and the domination number of the zero divisor graph $gamma(l)$