Nondiversity in substructures

For a model Jt of Peano Arithmetic, let Lt{Jf) be the lattice of its elementary substruc tures, and let Lt+{Jf) be the equivalenced lattice (Lt(^#), =?#), where ^^ is the equivalence relation of isomorphism on Lt(^#). It is known that Lt+(^#) is always a reasonable equivalenced lattice. Theorem. Let L be a finite distributive lattice and let (L, E) be reasonable. If J?q is a nonstandard prime model of PA, then Jt? has a cofinal extension Jt such that Lt+ {Jt) = (L, E). A general method for proving such theorems is developed which, hopefully, will be able to be applied to some nondistributive lattices. Let JV = (N, +, x, 0,1, <) be a model of Peano Arithmetic. Then Lt(jV) is the lattice of its elementary substructures. More generally, if Jt -< JV, then \x(JV ?Jt) is the sublattice of Lt(jV) that is the principal filter determined by J[\ that is, \X(JV ?Jt) = { X : Jt ^ Jt ^ JV }. We say that Jt is diverse if no two of its elementary substructures are isomorphic. The extension Jt -< JV is diverse if no two models in \A.(jV ?Jt) are isomorphic over Jt. Thus, Jt is diverse if and only if it is a diverse extension of its prime elementary submodel. The following is Theorem 1.2 of [11]. Theorem 1. Suppose that Jt is a countable nonstandard model of PA and that L is a finite lattice. If Jt has an extension JV such that \A,(JV ? Jt) = L, then Jt has a cofinal diverse extension JV such that Lt(jV/Jt) = L. It is known that for every finite distributive L, and also for many other finite lattices, such as M3 and N5, every countable nonstandard model M of PA has a cofinal extension JV such that \A(jV?Jt) = L. However, it is an open question whether the same conclusion holds for every finite lattice L. Consult [7] for more on this question. Getting nondiverse extensions is a more difficult matter. Theorem 2, the principal new result of this paper, implies that for finite distributive L, nondiverse extensions are always possible (unless L is a chain). Recall that a lattice is distributive if and only if it does not embed either M3 or N5. It is open whether or not there is a nondiverse cofinal extension Jt -< JV for which \A.(JV?Jt) is isomorphic to either M3 orN5. If L is a lattice and E is an equivalence relation on L, then (L, E) is an equivalenced lattice. If JV \= PA, then Lt+(jV) is the equivalenced lattice (Lt(jV),E), where E is the equivalence relation of isomorphism. More generally, if Jt -< JV, then Received October 9, 2006. ? 2008. Association for Symbolic Logic 0022-4812/08/7301-0008/S2.90