Finitary Cartesian closed varieties and semigroup actions

TL;DR

This paper explores the relationship between matched pairs of Boolean algebras and monoids, and how their associated categories of sets can be represented through actions of Boolean left restriction monoids, linking algebraic structures to categorical frameworks.

Contribution

It demonstrates that categories of $[B|M]$-sets are equivalent to categories of $S$-actions, providing a new perspective on finitary Cartesian closed varieties via Boolean restriction monoids.

Findings

Categories of $[B|M]$-sets are equivalent to $S$-actions.

Every non-degenerate finitary Cartesian closed variety corresponds to a category of $S$-actions.

The structure of Boolean left restriction monoids underpins these categorical equivalences.

Abstract

We build on some ideas of Richard Garner. Let $M$ be a monoid and $B$ a Boolean algebra. A `matched pair' $[B|M]$ consists of $B$ and $M$ and some mutual interactions. Garner showed that every such matched pair determines (what we shall call) a Boolean left restriction monoid $S = S[B|M]$. In this paper, we show that the data of a $[B|M]$-set (defined later) may be encoded by means of a certain kind of action by $S$. This means that the category $[B|M]$-{\bf sets} is equivalent to a category of {\bf $S$-actions}. We deduce, as a result of Garner's work, that every non-degenerate finitary Cartesian closed variety is equivalent to a special category of $S$-actions where $S$ is a Boolean left restriction monoid.