On profinite polyadic groups

TL;DR

This paper characterizes profinite polyadic groups as those that are compact, Hausdorff, and totally disconnected, and extends the concept to pro-$rak{X}$ groups within a specified class of finite groups.

Contribution

It provides a characterization of profinite polyadic groups and introduces the concept of $rak{X}$-polyadic groups, generalizing the structure theory within a pseudo-variety of finite groups.

Findings

Profinite polyadic groups are exactly the compact, Hausdorff, totally disconnected groups.

A polyadic group is pro-$rak{X}$ if and only if it satisfies specific topological and quotient conditions.

The paper extends the structure theory to $rak{X}$-polyadic groups within a pseudo-variety.

Abstract

We study the structure of profinite polyadic groups and we prove that a polyadic topological group $(G, f)$ is profinite, if and only if, it is compact, Hausdorff, totally disconnected. More generally, for a pseudo-variety (or a formation) of finite groups $\mathfrak{X}$, we define the class of $\mathfrak{X}$-polyadic groups, and we show that a polyadic group $(G, f)$ is pro-$\mathfrak{X}$, if and only if, it is compact, Hausdorff, totally disconnected and for every open congruence $R$, the quotient $(G/R, f_R)$ is $\mathfrak{X}$-polyadic.