Graphs and Their Associated Inverse Semigroups

TL;DR

This paper explores four inverse semigroups constructed from undirected graphs, characterizing their algebraic structures and proving each graph's unique associated inverse semigroup up to isomorphism.

Contribution

It introduces four new inverse semigroups based on undirected graphs and provides their structural characterizations and a uniqueness theorem.

Findings

Characterization of the semilattice of idempotents

Description of the lattice of ideals

Uniqueness of the inverse semigroup for each graph

Abstract

Directed graphs have long been used to gain understanding of the structure of semigroups, and recently the structure of directed graph semigroups has been investigated resulting in a characterization theorem and an analog of Fruct's Theorem. We investigate four inverse semigroups defined over undirected graphs constructed from the notions of subgraph, vertex induced subgraph, rooted tree induced subgraph, and rooted path induced subgraph. We characterize the structure of the semilattice of idempotents and lattice of ideals of these four inverse semigroups. Finally, we prove a characterization theorem that states that every graph has a unique associated inverse semigroup up to isomorphism.