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 a and 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 a and joins of sets smaller than beta. We show using an ultraproduct/ultraroot argument that when 2 <= alpha, beta <= 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. (C) 2016 Elsevier B.V. All rights reserved.