$\downarrow$-posets

TL;DR

This paper studies a special class of posets called $ warrow$-posets, generated by semilattice actions, providing a second order characterization and proving the absence of a first order one.

Contribution

It introduces $ warrow$-posets from semilattice actions and characterizes them with second order logic, showing no first order characterization exists.

Findings

Second order characterization of $ warrow$-posets

No first order characterization exists for these posets

Provides a logical framework for understanding semilattice-induced posets

Abstract

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.