Equational Axiomatization of Algebras with Structure

TL;DR

This paper introduces a category theoretic framework for axiomatizing classes of algebras with additional structures, providing a unified approach that covers various algebraic systems and their equational theories.

Contribution

It presents a generic HSP theorem and a sound and complete equational logic applicable to a wide range of structured algebras, extending existing theories.

Findings

Unified categorical framework for structured algebras

Generic HSP theorem applicable to diverse algebraic classes

Sound and complete equational logic for structured algebras

Abstract

This paper proposes a new category theoretic account of equationally axiomatizable classes of algebras. Our approach is well-suited for the treatment of algebras equipped with additional computationally relevant structure, such as ordered algebras, continuous algebras, quantitative algebras, nominal algebras, or profinite algebras. Our main contributions are a generic HSP theorem and a sound and complete equational logic, which are shown to encompass numerous flavors of equational axiomizations studied in the literature.