Congruence relations, filters, ideals, and definability in lattices of α-recursively enumerable sets

Abstract

Throughout this paper, α will denote an admissible ordinal. Let (α) denote the lattice of α-r.e. sets, i.e., the lattice whose elements are the α-r.e. sets, and whose ordering is given by set inclusion. Call a set A ∈ (α)α*-finite if it is α-finite and has ordertype < α* (the Σ1-projectum of α). The α*-finite sets form an ideal of (α), and factoring (α) by this ideal, we obtain the quotient lattice *(α).We will fix a language ℒ suitable for lattice theory, and discuss decidability in terms of this language. Two approaches have succeeded in making some progress towards determining the decidability of the elementary theory of (α). Each approach was first used by Lachlan for α = ω. The first approach is to relate the decidability of the elementary theory of (α) to that of a suitable quotient lattice of (α) by a congruence relation definable in ℒ. This technique was used by Lachlan [4, §1] to obtain the equidecidability of the elementary theories of (ω) and *(ω), and was generalized by us [6, Corollary 1.2] to yield the equidecidability of the elementary theories of (α) and *(α) for all α. Lachlan [3] then adopted a different approach.