The existence of suitable sets in locally compact strongly topological gyrogroups

A subset $S$ of a topological gyrogroup $G$ is said to be a {\it suitable set} for $G$ if $S$ is discrete, the gyrogroup generated by $S$ is dense in $G$, and $S\cup {0}$ is closed in $G$, where $0$ is the identity element of $G$. In this paper, it is proved that every locally compact strongly topological gyrogroup has a suitable set, which gives an affirmative answer to a question posed by F. Lin, et al. in \cite{key14}.

💡 Research Summary

The paper investigates the existence of suitable sets in the setting of strongly topological gyrogroups, a non‑associative generalization of topological groups introduced by Ungar. A suitable set S⊂G is defined by three conditions: (i) S is discrete, (ii) the gyrogroup generated by S is dense in G, and (iii) S∪{0} is closed (0 being the identity). The question posed by Lin et al. (2020) asked whether every locally compact strongly topological gyrogroup possesses such a set. The authors answer this affirmatively.

The authors begin by recalling the algebraic definition of a gyrogroup, the notion of a sub‑gyrogroup, L‑subgyrogroup, and the gyro‑automorphism gyr