Representable posets

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 $α$ and $β$ a poset is said to be $(α,β)$-representable if an embedding into a field of sets exists that preserves meets of sets smaller than $α$ and joins of sets smaller than $β$. We show using an ultraproduct/ultraroot argument that when $2\leqα,β\leq ω$ the class of $(α,β)$-representable posets is elementary, but does not have a finite axiomatization in the case where either $α$ or $β=ω$. We also show that the classes of posets with representations preserving either countable or all meets and joins are pseudoelementary.

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