On graphs of bounded semilattices
Parastoo Malakooti Rad, Peyman Nasehpour

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 of a bounded semilattice , 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 is a product of three or more chains, then 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 contains a cycle, then . Finally, we show that if is a dually atomic bounded distributive lattice whose set of dual atoms is nonempty, and the graph of has no isolated vertex, then is connected with .
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