On Graphs of Bounded Semilattices

In this paper, we introduce the graph $G(S)$ of a bounded semilattice $S$, which is a generalization of the intersection graph of the substructures of an algebraic structure. We prove some general theorems about these graphs; as an example, we show that if $S$ is a product of three or more chains, then $G(S)$ is Eulerian if and only if either the length of every chain is even or all the chains are of length one. We also show that if $G(S)$ contains a cycle, then $\hbox{girth}(G(S)) = 3$. Finally, we show that if $(S,+,\cdot,0,1)$ is a dually atomic bounded distributive lattice whose set of dual atoms is nonempty, and the graph $G(S)$ of $S$ has no isolated vertex, then $G(S)$ is connected with $\hbox{diam}(G(S))\leq 4$.