Let V(G) be the set of vertices of a simple connected graph G. The set 1(G) consisting of empty set, V(G), and all neighborhoods N(v) of vertices v is an element of V(G) is a subposet of the complete lattice L(G) (under inclusion) of all intersections of elements in L-1(G). In this paper, it is shown that L-1(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 L-1(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. (C) 2016 Elsevier B.V. All rights reserved.
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