On semi-Peano algebras

TL;DR

This paper studies semi-Peano algebras, focusing on finitely presented multi-unary cases, showing they are free products of cyclic algebras with unique defining relations and characterizations.

Contribution

It introduces a classification of finitely presented multi-unary semi-Peano algebras as free products of cyclic algebras with unique relations.

Findings

Finitely presented multi-unary semi-Peano algebras are free products of cyclic algeanoids.

Each cyclic algebra is characterized up to isomorphism by a unique relation.

The case of single-operation semi-Peano algebras is straightforward, generalizing to multiple unary operations.

Abstract

A semi-Peano algebra is an algebra for which each operation is injective, and the images of the operations are pairwise disjoint. The most straightforward non-trivial kind of finitely presented semi-Peano algebra are algebras with a single unary operation. There are two possible directions of generalization: algebras with a single operation of any arity, and algebras with several unary operations. The former can be solved easily by adapting results on equidecomposable groupoids from [2]. However, the second way is somewhat different. We will show that a finitely presented multi-unary semi-Peano algebra is the free product of cyclic semi-Peano algebras and that a unique relation defines such cyclic algebras. In addition, we will characterize each cyclic algebra up to isomorphism.