Representable posets

TL;DR

This paper investigates the conditions under which posets can be embedded into fields of sets while preserving finite meets and joins, establishing elementary class properties and axiomatization limitations.

Contribution

It characterizes when classes of $(eta,eta)$-representable posets are elementary and shows the non-finite axiomatizability for certain cases, also exploring pseudoelementary classes.

Findings

The class of $(eta,eta)$-representable posets is elementary for $2\,\leq\,\beta\leq\omega$.

No finite axiomatization exists when either $eta$ or $eta=\,\omega$.

Classes with countable or all-meets-and-joins-preserving representations are pseudoelementary.

Abstract

A poset is representable if it can be embedded in a field of sets in such a way that existing finite meets and joins become intersections and unions respectively (we say finite meets and joins are preserved). More generally, for cardinals $\alpha$ and $\beta$ a poset is said to be $(\alpha,\beta)$-representable if an embedding into a field of sets exists that preserves meets of sets smaller than $\alpha$ and joins of sets smaller than $\beta$. We show using an ultraproduct/ultraroot argument that when $2\leq\alpha,\beta\leq \omega$ the class of $(\alpha,\beta)$-representable posets is elementary, but does not have a finite axiomatization in the case where either $\alpha$ or $\beta=\omega$. We also show that the classes of posets with representations preserving either countable or all meets and joins are pseudoelementary.