Factorizations of surjective maps of connected quandles

TL;DR

This paper characterizes when one surjective homomorphism between connected quandles factors through another, providing a complete solution using existing structure theorems and factorization systems.

Contribution

It offers a comprehensive solution to the factorization problem for surjective connected quandle homomorphisms, extending previous theoretical frameworks.

Findings

Complete characterization of factorization conditions

Application of structure theorem for connected quandles

Utilization of factorization system for surjective homomorphisms

Abstract

We consider the problem of when one quandle homomorphism will factor through another, restricting our attention to the case where all quandles involved are connected. We provide a complete solution to the problem for surjective quandle homomorphisms using the structure theorem for connected quandles of Ehrman et al. (2008) and the factorization system for surjective quandle homomorphsims of Bunch et al. (2010) as our primary tools. The paper contains the substantive results obtained by an REU research group consisting of the first four authors under the mentorship of the fifth, and was supported by National Science Foundation, grant DMS-1659123.