Infinite substructure lattices of models of Peano Arithmetic

Some new finite lattices (for example, A/4 , M7, and the hexagon lattice) are shown to be isomorphic to the lattice of elementary substructures of a model of Peano Arithmetic. The set of elementary substructures of a model JV of Peano Arithmetic forms a lattice Lt(yf), the substructure lattice of jV. It is unknown whether there are finite lattices that are not isomorphic to any substructure lattice. Indeed, it was conjectured in [8] that for any finite lattice L there is some jV 1= PA for which L = \X(JV). It was proved in [8] that if 2 < n < co, then there is JV such that Lt(^T) = n(«). (The partition lattices U(n) are defined below.) For 3 < n < co, the lattice Mn is the unique lattice that has n + 2 elements and n atoms. Thus, the lattice A/3, which is isomorphic to n(3), is isomorphic to a substructure lattice, although it is known from [2] or [4] that if yV is a model of True Arithmetic and 3 < n < co, then Lt(yT) ^ Mn . Other examples of finite lattices that are not substructure lattices of models of True Arithmetic are given in [8]. The lattices Mn for 4 < n < co and the hexagon lattice were identified in [8] as specific finite lattices that were not known to be isomorphic to substructure lattices. It is a consequence of the theorem in this note that M„ , whenever n = p" + 1 for some prime p, the hexagon lattice, and the lattice Mi are isomorphic to substructure lattices. It is still unknown whether or not Mx! is isomorphic to a substructure lattice. All previously known positive results about finite substructure lattices are contained in [8]. Prior to [8], Paris [4] proved that all finite distributive lattices occur as substructure lattices. (Also see [7].) Wilkie [9] showed that the pentagon lattice is a substructure lattice, and Paris [5] proved that M3 is a sublattice of an infinite substructure lattice. More generally, intermediate structure lattices will be considered. For „# -< yV ^ PA, let Lt(yT/^) = {jf0 e lA(yV): J? < JTq < jV} , regarded as a sublattice of Lt(yF). Notice that if Jf is a minimal model of PA and ^f -< jT , then Lt(^) = \l(j¥JJ?). Given a set A , let n(^) be the set of all partitions of A . If a, b £ A and n £ fl(A), then we write a « b (mod n) if {a, b} C C for some C £ n. Received by the editors February 11, 1991 and, in revised form, July 2, 1991. 1991 Mathematics Subject Classification. Primary 03H15, 03C62; Secondary 06B15. ©1993 American Mathematical Society 0002-9939/93 $1.00 +$.25 per page