Finitary Cartesian closed varieties and semigroup actions

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.