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

The revised version adds a note clearing up a loose end from the background discussion in the introduction