Uniform Birkhoff

TL;DR

This paper introduces a uniform structure framework to characterize pseudovarieties of finitely generated algebras, unifying previous results and providing a new perspective on algebraic classes via uniform continuity.

Contribution

It establishes a novel uniformity-based characterization of pseudovarieties of finitely generated algebras, linking algebraic properties with topological uniform structures.

Findings

Characterizes pseudovarieties using uniform structures on free algebras.

Unifies earlier theorems on finite algebra pseudovarieties.

Connects algebraic classes with uniform continuity concepts.

Abstract

We show that pseudovarieties of finitely generated algebras, i.e., classes $C$ of finitely generated algebras closed under finite products, homomorphic images, and subalgebras, can be described via a uniform structure $U$ on the free algebra for $C$: the members of $C$ then are precisely those finitely generated algebras $A$ for which the natural mapping from the free algebra onto the term clone of $A$ is well-defined and uniformly continuous with respect to the uniformity $U$ and the uniformity of pointwise convergence on the term clone of $A$, respectively. Our result unifies earlier theorems describing pseudovarieties of finite algebras and the pseudovariety generated by a single oligomorphic algebra.