Non-elementary classes of representable posets

TL;DR

This paper demonstrates that the class of $( heta,C)$-representable posets cannot be characterized by first-order axioms, extending the result to various $(eta,eta)$-representable classes, highlighting limitations in their logical definability.

Contribution

The paper proves the non-axiomatizability of $( heta,C)$-representable posets in first-order logic and generalizes this to other $(eta,eta)$-representable classes.

Findings

The class of $( heta,C)$-representable posets is not first-order axiomatizable.

Generalization to $(eta,eta)$-representable posets for certain $eta$.

Highlights limitations of first-order logic in characterizing these classes.

Abstract

A poset is $(\omega,C)$-representable if it can be embedded into a field of sets in such a way that all existing joins, and all existing \emph{finite} meets are preserved. We show that the class of $(\omega,C)$-representable posets cannot be axiomatized in first order logic using the standard language of posets. We generalize this result to $(\alpha,\beta)$-representable posets for certain values of $\alpha$ and $\beta$.