On the Structure of Hom Quandles

TL;DR

This paper advances the understanding of Hom quandles by showing they can be characterized using medial source quandles, simplifying their structure analysis especially for 2-reductive targets.

Contribution

It demonstrates that only medial source quandles need to be considered for Hom quandle analysis, extending previous structure theorems to this context.

Findings

Medial source quandles suffice for Hom quandle analysis.

Structure theorem provides a complete characterization.

Simplified form for 2-reductive targets enables easy counting and structure determination.

Abstract

We continue the study of the quandle of homomorphisms into a medial quandle begun in Crans and Nelson. We show that it suffices to consider only medial source quandles, and therefore the structure theorem of Jedlicka et al. provides a characterization of the Hom quandle. In the particular case when the target is 2-reductive this characterization takes on a simple form that makes it easy to count and determine the structure of the Hom quandle.