Definability in well quasi-ordered sets of structures

Abstract

The theorem introduced here provides a sufficient condition for an axiomatic class to be elementary. It resulted from consideration of a definability problem in , but is introduced in a less specific context and accompanied by applications to .As in [La, 1], [La, 2], a reflexive, transitive binary relation R on a nonempty set S will be called a quasi-ordering or q.o. of S. A set of pairwise incomparable elements of S is an antichain and a subset of S linearly ordered by R is a chain. A chain {Si: i ∈ ω} with i > j if and only if (siRsj ∧ ¬ sjRSi) will be called an infinite descending chain. If S has no infinite antichains nor any infinite descending chains, S is well quasi-ordered by R, and is called a well quasi-ordering or w.q.o. Any subset of a w.q.o. is a w.q.o. An element s of a q.o. S is minimal in S if ∀t ∈ S(tRs → sRt). A subset S′ of S is a foundation for S if ∀t ∈ S∃s ∈ S′(sRt).