The Graph of Equivalence Classes of Zero Divisors

Abstract

We introduce a graph GE(L) of equivalence classes of zero divisors of a meet semilattice L with 0. The set of vertices of GE(L) are the equivalence classes of nonzero zero divisors of L and two vertices [x] and [y] are adjacent if and only if [x]∧[y]=[0]. It is proved that GE(L) is connected and either it contains a cycle of length 3 or GE(L)≅K2. It is known that two Boolean lattices L1 and L2 have isomorphic zero divisor graphs if and only if L1≅L2. This result is extended to the class of SSC meet semilattices. Finally, we show that Beck's Conjecture is true for GE(L) .