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 \(girth(G(S)) = 3\). 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 \(diam(G(S))\leq 4\).