Characterizing Finitely Based Abelian Mal'cev Algebras

TL;DR

This paper characterizes when abelian Mal'cev varieties are finitely based, linking finite basis property to finite presentation of associated algebraic structures.

Contribution

It provides a precise characterization of finitely based abelian Mal'cev varieties using finite type and finite presentation conditions.

Findings

Finitely based abelian Mal'cev varieties have finite type.

Their ring of idempotent binary terms is finitely presented.

Their module of unary terms is finitely presented.

Abstract

In this paper, we prove the following characterization: an abelian Mal'cev variety is finitely based if and only it has finite type, its ring of idempotent binary terms is finitely presented, and its module of unary terms is finitely presented.