Annihilators in zero-divisor graphs of semilattices and reduced commutative semigroups

Abstract

Let V(G) be the set of vertices of a simple connected graph G. The set L1(G) consisting of ∅, V(G), and all neighborhoods N(v) of vertices v∈V(G) is a subposet of the complete lattice L(G) (under inclusion) of all intersections of elements in L1(G). In this paper, it is shown that L1(G) is a join-semilattice and L(G) is a Boolean algebra if and only if G is realizable as the zero-divisor graph of a meet-semilattice with 0. Also, if L1(G) is a meet-semilattice and L(G) is a Boolean algebra, then G is realizable as the zero-divisor graph of a join-semilattice with 0. As a corollary, graphs that are realizable as zero-divisor graphs of commutative semigroups with 0 that do not have any nonzero nilpotent elements are classified.