On graphs of bounded semilattices

TL;DR

This paper introduces the graph of a bounded semilattice, explores its properties, and establishes conditions for Eulerian cycles, girth, and connectivity, extending graph theory to algebraic structures.

Contribution

It generalizes the intersection graph concept to bounded semilattices and proves new properties relating structure and graph characteristics.

Findings

G(S) is Eulerian iff chains have even length or are of length one.

If G(S) contains a cycle, then girth is 3.

G(S) is connected with diameter at most 4 under certain conditions.

Abstract

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 $diam(G(S))\leq 4$.