Endomorphisms of free Steiner quasigroups
Silvia Barbina, Enrique Casanovas
TL;DR
The paper develops a syntactic framework for free Steiner quasigroups by introducing reduced terms, term ranks, and the free/ HF-ordering concepts that underpin levelled, stagewise constructions from a free base $A$. It then characterises endomorphisms and automorphisms—showing finitely generated free Steiner quasigroups have tame automorphism groups and that endomorphisms are tightly controlled by the irreducibility of reduced terms. The HF/free extension theory and the independence criteria connect the algebraic structure to model-theoretic properties, including universality, embeddings, and definability. A central conjecture proposes a precise link between dependence on an independent tuple and definability/algebraicity over the generated substructure, with implications for the model theory of free Steiner triple systems.
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.