On the way from logic to algebra

On generic structures preserving elementary equivalence and elementary embeddability

“…Thus, for generic structures in Theorem 2.1 it suffices to consider generic finite partial isomorphisms in PG(M 1 , M 2 ), with their restrictions, and a modification of that theorem holds allowing syntactically, in terms of generative classes, characterize the elementary equivalence for generic structures. Below we consider that generic modification, whose proof can be easily obtained from the proof of [11,Theorem 5…”

“…The following well-known Tarski-Vaught test [11,13] is used for the checking that a substructure is an elementary one.…”

“…They are based on classical characterizations for the general case. The criterion for elementary equivalence uses the well known Fraïssé-Taimanov-Ehrenfeucht overturning method [7][8][9][10][11][12]. The criterion for elementary embeddability is based on the Tarski-Vaught test [11,13].…”