A Variety Theorem for Relational Universal Algebra

TL;DR

This paper introduces a relational analogue of universal algebra with string-diagrammatic syntax, establishing a variety theorem that characterizes categories of models as definable categories, extending classical algebraic theory concepts.

Contribution

It develops a new relational algebraic framework with a variety theorem, linking model categories to definable categories, and extends traditional algebraic syntax to a string-diagrammatic form.

Findings

Categories of models are exactly the definable categories.

Relational algebraic theories extend classical algebraic theories.

String-diagrammatic syntax enhances the expressiveness of relational theories.

Abstract

We develop an analogue of universal algebra in which generating symbols are interpreted as relations. We prove a variety theorem for these relational algebraic theories, in which we find that their categories of models are precisely the definable categories. The syntax of our relational algebraic theories is string-diagrammatic, and can be seen as an extension of the usual term syntax for algebraic theories.