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.

Let m and n be cardinals with 3 ≤ m, n ≤ ω. 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.

It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be jCRL ∩ mCRL. We provewhere the marked inclusions are proper.Let L be a DL. Then L ∈ mCRL iff L has a distinguishing set of complete, prime filters. Similarly, L ∈ jCRL iff L has a distinguishing set of completely prime filters, and L ∈ CRL iff L has a distinguishing set of complete, completely prime filters.Each of the classes above is shown to be pseudo-elementary hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary.

Given a poset P and a standard closure operator Γ : ℘(P ) → ℘(P ) we give a necessary and sufficient condition for the lattice of Γ-closed sets of ℘(P ) to be a frame in terms of the recursive construction of the Γ-closure of sets. We use this condition to show that given a set U of distinguished joins from P , the lattice of U-ideals of P fails to be a frame if and only if it fails to be σ-distributive, with σ depending on the cardinalities of sets in U. From this we deduce that if a poset has the property that whenever a ∧ (b ∨ c) is defined for a, b, c ∈ P it is necessarily equal to (a ∧ b) ∨ (a ∧ c), then it has an (ω, 3)-representation. This answers a question from the literature.

We define a fragment of monadic infinitary second-order logic corresponding to a kind of abstract separation property. We use this to define certain subclasses of elementary classes as separation subclasses. We use model theoretic techniques and games to show that separation subclasses which are, in a sense, recursively enumerable in our second-order fragment can also be recursively axiomatized in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. As applications, we use simple characterizations as separation subclasses to obtain axiomatizability results related to graph colourings and partial algebras.2010 Mathematics Subject Classification. Primary 03C98. Secondary 03B15, 03B70, 05C15, 08A55.