Definition 1a

Let M be a model of T and an infinite set I be an indiscernible sequence in (M,I). A complete L-theory T has the second (M,I)-Pseudo-finite Homogeneity Property iff whenever (N,J) is first-order equivalent to (M,I), A and B are pseudo-finite subsets of J in the model N, C and D are finite subsets of N, and h:(A∪C)→(B∪D) is an elementary map in (N,J) with ω-saturated (N,J,A,B,h), for any a∈N there is b∈N such that h∪{(a,b)} is an elementary map in (N,J).

Theorem 2a

Let M be first-order equivalent to U and an infinite set I be an indiscernible sequence in (M,I). Suppose the first-order theory of a universe U has the second (M,I)-Pseudo-finite Homogeneity Property. Let an extended query ψ be locally generic over finite states over U. Then ψ is equivalent over finite states over U to a restricted query.

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