More on the Ehrenfeucht - Fräıssé game of length

This paper is a continuation of [8]. Let A and B be two first order structures of the same vocabulary L. We denote the domains of A and B by A and B respectively. All vocabularies are assumed to be relational. The Ehrenfeucht-Fräıssé-game of length γ of A and B denoted by EFGγ(A,B) is defined as follows: There are two players called ∀ and ∃. First ∀ plays x0 and then ∃ plays y0. After this ∀ plays x1, and ∃ plays y1, and so on. If 〈(xβ, yβ) : β < α〉 has been played and α < γ, then ∀ plays xα after which ∃ plays yα. Eventually a sequence 〈(xβ, yβ) : β < γ〉 has been played. The rules of the game say that both players have to play elements of A ∪ B. Moreover, if ∀ plays his xβ in A (B), then ∃ has to play his yβ in B (A). Thus the sequence 〈(xβ, yβ) : β < γ〉 determines a relation π ⊆ A×B. Player ∃ wins this round of the game if π is a partial isomorphism. Otherwise ∀ ∗Partially supported by the Academy of Finland grant #40734. †Research partially supported by the United States-Israel Binational Science Foundation. Publication number [776] ‡Partially supported by the Academy of Finland grant #40734.