arXiv subjects
Publications and source records attributed to Lawrence Valby.
We investigate a certain class of posets arising from semilattice actions. Let $S$ be a semilattice with identity. Let $S$ act on a set $C$. For $c,d\in C$ put $c\leq d$ iff there is some $s\in S$ with $ds=c$. Then $(C,\leq)$ is a poset. Let's call the posets that arise in this way $\downarrow$-posets. We give a reasonable second order characterization of $\downarrow$-posets and show that there is no first order characterization.
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 give intuitive examples of these actions, one involving conversations and another a university's changing student body. The examples give some motivation for considering these actions, and also help give intuitive readings of the axioms. 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.
First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies the axioms iff it embeds into a first order algebra. Importantly, our argument is modular and also works for, e.g., the positive existential algebras (where we restrict attention to the positive existential formulas) and the quantifier-free algebras. We also explain the relationship to theories, and indicate how to add in function symbols.