Pseudo-finite homogeneity and saturation

When analyzing database query languages a roperty, of theories, the pseudo-finite homogeneity property, has been introduced and applied (cf. [3]). We show that a stable theory has the pseudo-finite homogeneity property just in case its expressive power for finite states is bounded. Moreover, we introduce the corresponding pseudo-finite saturation property and show that a theory fails to have the finite cover property if and only if it has the pseudo-finite saturation property. ?1. Pseudo-finite homogeneity. Throughout let T be a complete first-order theory in a countable language L with infinite models. Suppose that p is a finite non-empty set of relation symbols not contained in L. Set L(p) := L U p. If M is a model of T, and (M, P) is an L(p)-structure with, say, P = PI ... Pr, then P is a (p)-state in M. fld(P), the field or active domain of the state P, is the set fld(PI) U ... U fld(P,.), where fld(Pj) is the field of the relation P1. P is a finite state, if every fld(Pj) is finite and non-empty. In the following we will denote finite states by s,' ..... A state P in M is pseudo-finite, if (M, P) is a model of F(T, p), the theory of all finite states, i.e., F (T, p) := Th({(N, s) I N l= T, (N, s) an L(p)-structure, s a finite state}). In general, we use r, F', t, . . . to denote pseudo-finite states. EXAMPLE 1.1. For L :_ 0, T the L-theory of infinite sets, and p := {P} with unary P, a subset r of a model M of T is pseudo-finite if and only if M \ r is infinite. In fact, for every k, the complement of every finite subset contains at least k elements, hence the same holds for a pseudo-finite subset. If M \ r and r are infinite and 1 > 1, then (M, r) satisfies the same sentences of quantifier rank 1I of L-sentences such that (p E F(T, p) X for all n > 1, T = pn . (b) Let r be a pseudo-finite subset of M (i.e., p = {P} with unary P). If s C M is finite, then r U s is pseudo-finite. If s is definable (in particular,finite) then r \ s is pseudo-finite. Received December 1, 1997; revised June 23, 1998. ? 1999, Association for Symbolic Logic 0022-48 12/99/6404-0023/$2. 1 0