Rob Egrot

Axiomatizing representation classes for posets in first order logic

We present some recent results on first order axiomatizations for classes of posets embeddable into fields of sets. The extent to which such axiomatizations are possible depends on how much of the existing lattice structure we wish to preserve. We say a poset is representable if an embedding exists that preserves all existing finite meets and joins. More generally, for cardinals α and β a poset is (α,β)-representable if an embedding exists that preserves meets of cardinality < α, and joins of cardinality < β. We say a poset is e.g. (C, β)-representable if there is an embedding that preserves all existing meets and all joins of cardinality < β. Posets that are (C, C)-representable we call completely representable. In this presentation we discuss for which choices of α, β the resulting representation class is elementary (or pseudoelementary). We compare these results to what is known about the special cases of Boolean algebras, lattices and semilattices.