An alternative look at the structure of graph inverse semigroups

TL;DR

This paper investigates the structure of subsemigroups within graph inverse semigroups, revealing isomorphisms to matrix units and conditions for product containment based on graph paths.

Contribution

It characterizes subsemigroups associated with specific classes in graph inverse semigroups and establishes their structural properties and isomorphisms.

Findings

Subsemigroups D^0 are isomorphic to matrix units in acyclic graphs.

Conditions for J-class products based on graph paths.

Structural descriptions of subsemigroups in graph inverse semigroups.

Abstract

For any graph inverse semigroup $G(E)$ we describe subsemigroups $D^0=D\cup\{0\}$ and $J^0=J\cup\{0\}$ of $G(E)$ where $D$ and $J$ are arbitrary $\mathcal{D}$-class and $\mathcal{J}$-class of $G(E)$, respectively. In particular, we prove that for each $\mathcal{D}$-class $D$ of a graph inverse semigroup over an acyclic graph the semigroup $D^0$ is isomorphic to a semigroup of matrix units. Also we show that for any elements $a,b$ of a graph inverse semigroup $G(E)$, $J_a\cdot J_b\cup J_b\cdot J_a\subset J_b^0$ if there exists a path $w$ such that $s(w)\in J_a$ and $r(w)\in J_b$.