Rob EgrotMahidol University2018-12-112019-03-142018-12-112019-03-142016-01-13Journal of Applied Logic. Vol.16, (2016), 60-71157086832-s2.0-84992304894https://repository.li.mahidol.ac.th/handle/20.500.14594/40954© 2016 Elsevier B.V. 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≤α,β≤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.Mahidol UniversityMathematicsArticleSCOPUS10.1016/j.jal.2016.03.003