Endomorphisms of free Steiner quasigroups

TL;DR

This paper characterizes homomorphisms between substructures of free Steiner quasigroups, focusing on how replacing a generator with a term affects the structure, based on syntactic properties.

Contribution

It provides a new characterization of homomorphisms in free Steiner quasigroups involving generator replacement and syntactic term properties.

Findings

Homomorphisms depend on syntactic properties of terms.

Characterization applies to substructure homomorphisms.

Results enhance understanding of free Steiner quasigroup structure.

Abstract

A free Steiner quasigroup is a free object in the variety of Steiner quasigroups. Free Steiner quasigroups are characterised by the existence of a levelled construction that starts with a free base - that is, a set of elements none of which is a product of the others, and which generate the quasigroup. Then each element in a free Steiner quasigroup $M$ can be obtained as a term on the free base. We characterise homomorphisms between substructures of a free Steiner quasigroup where one generator is replaced by a term in the original generators. The characterisation depends on certain synctactic properties of the term in question.