Actions arising from intersection and union

An action is a pair of sets, $C$ and $S$, and a function $f\colon C\times S \to C$. Rothschild and Yalcin gave a simple axiomatic characterization of those actions arising from set intersection, i.e. for which the elements of $C$ and $S$ can be identified with sets in such a way that elements of $S$ act on elements of $C$ by intersection. We introduce and axiomatically characterize two natural classes of actions which arise from set intersection and union. In the first class, the $\uparrow\mathrel{\mspace{-2mu}}\downarrow$-actions, each element of $S$ is identified with a pair of sets $(s^\downarrow,s^\uparrow)$, which act on a set $c$ by intersection with $s^\downarrow$ and union with $s^\uparrow$. In the second class, the $\uparrow\mathrel{\mspace{-2mu}}\downarrow$-biactions, each element of $S$ is labeled as an intersection or a union, and acts accordingly on $C$. We describe how these actions might be useful in modeling conversations where there is a shared conversational state consisting of actively entertained possibilities that can be both removed and added. The class of $\uparrow\mathrel{\mspace{-2mu}}\downarrow$-actions is closely related to a class of single-sorted algebras, which was previously treated by Margolis et al., albeit in another guise (hyperplane arrangements), and we note this connection. Along the way, we make some useful, though very general, observations about axiomatization and representation problems for classes of algebras.