Coloring Of Meet-Semilattices

Given a commutative semigroup S with 0, where 0 is the unique singleton ideal, we associate a simple graph Γ(S), whose vertices are labeled with the nonzero elements in S. Two vertices in Γ(S) are adjacent if and only if the corresponding elements multiply to 0. The inverse problem, i.e., given an arbitrary simple graph, whether or not it can be associated to some commutative semigroup, has proved to be a difficult one. In this paper, we extend results by DeMeyer[3], McKenzie, and Schneider[4] on this problem by studying the complement of graphs. As an application and an extension of work in [3] we prove that every compact connected 2-manifold admits an Eulerian triangulation that can be associated to a zero divisor semigroup graph.