Strongly topologically orderable gyrogroups with a suitable set

TL;DR

This paper investigates the existence of suitable sets in strongly topologically orderable gyrogroups, extending previous results and establishing their existence in locally compact and non-totally disconnected cases.

Contribution

It extends the theory of suitable sets to strongly topologically orderable gyrogroups, including cases that are locally compact or not totally disconnected.

Findings

Suitable sets exist in all locally compact strongly topologically orderable gyrogroups.

Suitable sets also exist in strongly topologically orderable gyrogroups that are not totally disconnected.

The results generalize previous findings in the literature.

Abstract

A discrete subset $S$ of a topologically gyrogroup $G$ is called a {\it suitable set} for $G$ if $S\cup \{1\}$ is closed and the subgyrogroup generated by $S$ is dense in $G$, where $1$ is the identity element of $G$. In this paper, we mainly study the existence of suitable set of strongly topologically orderable gyrogroups, which extends some result in some papers in the literature. In particular, the existences of suitable set of each locally compact or not totally disconnected strongly topologically orderable gyrogroup are affirmative.