Superconnected left quasigroups and involutory quandles

TL;DR

This paper explores the properties of superconnected and superfaithful left quasigroups, with a focus on involutory quandles, extending known results to infinite cases and providing new characterizations based on displacement groups.

Contribution

It extends the classification of involutory quandles to infinite cases and offers new characterizations using the properties of their displacement groups.

Findings

Extended involutory quandle classification to infinite cases

Provided new characterizations based on displacement group properties

Improved upon previous main results in the field

Abstract

In this paper we study the classes of superconnected and superfaithful left quasigroups, that are relevant in the study of Mal'cev varieties of left quasigroups \cite{Maltsev_paper}. Then we focus on quandles and in particular to the involutory ones. We extend the main result of \cite{involutive_quandles_russo} to the infinite case and we offer a characterization of several classes of involutory quandles in terms of the properties of the canonical generators of the displacement group, improving the main results of \cite{Nobu}.