Subdirectly irreducible medial quandles

TL;DR

This paper classifies all subdirectly irreducible medial quandles, showing they belong to four classes, and provides representations for the non-connected cases, advancing the understanding of their structure.

Contribution

It offers a complete classification of subdirectly irreducible medial quandles and introduces a representation for non-connected examples.

Findings

Subdirectly irreducible medial quandles fall into four classes.

Finite subdirectly irreducible medial quandles are either connected or reductive.

A representation for non-connected subdirectly irreducible medial quandles is provided.

Abstract

We describe all subdirectly irreducible medial quandles. We show that they fall within one of four disjoint classes. In particular, in the finite case they are either connected (and therefore Alexander quandles) or reductive. Moreover, we provide a representation of all non-connected subdirectly irreducible medial quandles.