Some observations on the substructure lattice of a Δ1 ultrapower

Given a (nontrivial) Δ1 ultrapower F/U, let L︁U denote the set of all Π2‐correct substructures of F/U; i.e., L︁U is the collection of all those subsets of |F/U| that are closed under computable (in the sense of F/U) functions. Defining in the obvious way the lattice L︁(F/U)) with domain L︁U, we obtain some preliminary results about lattice embeddings into – or realization as – an L︁(F/U). The basis for these results, as far as we take the matter, consists of (1) the well‐known class of (non‐trivial) minimal F/U's, which function as atoms, and (2) the class of minimalfree F/U'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 L︁(F/U), and that the initial segment lattices {0, 1,..., n } are not just embeddable in (as is trivial), but in fact realizable as, lattices L︁(F/U). 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)