No finite axiomatizations for posets embeddable into distributive lattices

TL;DR

This paper proves that for certain classes of posets embeddable into distributive lattices with meet and join preservation constraints, no finite set of axioms can fully characterize them.

Contribution

It establishes the non-finite axiomatizability of classes of posets embeddable into distributive lattices under specific meet and join preservation cardinality constraints.

Findings

Classes cannot be finitely axiomatized

Embeddability depends on infinite axiomatizations

Results hold for all cardinals between 3 and infinity

Abstract

Let $m$ and $n$ be cardinals with $3\leq m,n\leq\omega$. We show that the class of posets that can be embedded into a distributive lattice via a map preserving all existing meets and joins with cardinalities strictly less than $m$ and $n$ respectively cannot be finitely axiomatized.