On Congruence Permutable $G$-sets

TL;DR

This paper characterizes when a $G$-set is congruence permutable by linking it to a specific semigroup structure, providing necessary and sufficient conditions for this property.

Contribution

It introduces a semigroup construction associated with a $G$-set and establishes criteria for congruence permutability based on this structure.

Findings

Provides necessary and sufficient conditions for congruence permutability of $G$-sets.

Defines a semigroup $(G,X,0)$ with a zero element related to the $G$-set.

Connects algebraic properties of $G$-sets with semigroup theory.

Abstract

An algebraic structure is said to be congruence permutable if its arbitrary congruences $\alpha$ and $\beta$ satisfy the equation $\alpha \circ \beta =\beta \circ \alpha$, where $\circ$ denotes the usual composition of binary relations. For an arbitrary $G$-set $X$ with $G\cap X=\emptyset$, we define a semigroup $(G,X,0)$ with a zero $0$ ($0\notin G\cup X$), and give necessary and sufficient conditions for the congruence permutability of the $G$-set $X$ by the help of the semigroup $(G,X,0)$.