Discrete equational theories

TL;DR

This paper introduces discrete equational theories with operations of discrete arities, characterizes the associated monads, and extends Birkhoff theorems from metric spaces to broader categories.

Contribution

It defines discrete equational theories, characterizes their monads, and generalizes Birkhoff theorems to symmetric monoidal closed categories.

Findings

Monads preserve surjections.

Birkhoff theorems extended to new categorical contexts.

Characterization of discrete equational theories.

Abstract

We introduce discrete equational theories where operations are induced by those having discrete arities. We characterize the corresponding monads as monads preserving surjections. Using it, we prove Birkhoff type theorems for categories of algebras of discrete theories. This extends known results from metric spaces to general symmetric monoidal closed categories.