Some Remarks on Theories Without Finite Cover Property

B.S. Baizhanov

Let M be a model of a language L with complete theory T without finite cover property, A ⊂ M, M^{∗} = (M, L ∪ {P^{1}}) be a model of language L^{∗} = L ∪ {P^{1}}, such that P^{1}(M^{∗}) = A. We prove that ∀l < ω, ∀ᾱ, β̄ ∈ (M \ A)^{l},

[tp(ᾱ|M) = tp(β̄|M) ⟺ tp_{∗}(ᾱ|M) = tp_{∗}(β̄|M)] (∗∗)

iff there are not A-definable formulas E(x, y, z̄), D(z̄) such that

(1)∀β̄ ∈ D(M^{l(z̄)}), E(x, y, β̄) is relation of equivalence.

(2) ∃β̄_{1}, β̄_{2} ∈ (M \ A)^{l(z̄)} ∩ D(M^{l(z̄)}) such that we have

(a) Every E(x, y, β̄_{1})-class meets with A and there is E(x, y, β̄_{2})-class does not meet with A.

(b) ∀a, a′ ∈ A, [E(a, a′, β̄_{1}) ⟺ E(a, a′, β̄_{2}].

E.Bouscaren proved (**), when A is an elementary submodel of M and T is superstable theory ([1], Proposition 2.1).

[1] E. Bouscaren, ”Dimensional order property and pairs of models”, Annals of Pure and Applied Logic 41 (1989), pp.205-231.

Address: Institut of Informatics and Control Problems, Academy of Sciences of Kazakhstan, 125, Pushkin, Almaty, 480100, Kazakhstan

E-mail: baijan@ipic.academ.alma-ata.su