Some observations on the substructure lattice of a Δ1 ultrapower

Given a (nontrivial) Δ₁ ultrapower ?/??, let ??_U denote the set of all Π₂‐correct substructures of ?/??; i.e., ??_U is the collection of all those subsets of |?/??| that are closed under computable (in the sense of ?/??) functions. Defining in the obvious way the lattice ??(?/??)) with domain ??_U, we obtain some preliminary results about lattice embeddings into – or realization as – an ??(?/??). The basis for these results, as far as we take the matter, consists of (1) the well‐known class of (non‐trivial) minimal ?/??'s, which function as atoms, and (2) the class of minimalfree ?/??'s, to whose nonemptiness a substantial section of the paper is devoted. It is shown that an infinite, convergent monotone sequence together with its limit point is embeddable in an ??(?/??), and that the initial segment lattices {0, 1,…, n } are not just embeddable in (as is trivial), but in fact realizable as, lattices ??(?/??). Finally, the diamond is (easily) embeddable; and if it is not realizable, then either the 1 ‐ 3 ‐ 1 lattice or the pentagon is at least embeddable (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)