On Rudin–keisler Preorders in Small Theories
نویسنده
چکیده
We consider complete first-order theories T with infinite models. Additionally we assume that T are small, i. e., they have countably many types (|S(T )| = ω). So for any type q ∈ S(T ) and its realization ā, there exists a model M(ā), being prime over ā. Since all prime models over realizations of q are isomorphic, we often denote such by Mq. Let p and q be types in S(T ). We say that the type p is dominated by a type q, or p does not exceed q under the Rudin–Keisler preorder (written p ≤RK q), if Mq |= p, that is, Mp is an elementary submodel of Mq (written Mp 1Mq). Besides, we say that a model Mp is dominated by a model Mq, or Mp does not exceed Mq under the Rudin–Keisler preorder , and write Mp ≤RK Mq. Syntactically, the condition p ≤RK q (and hence also Mp ≤RK Mq) is expressed thus: there exists a formula φ(x̄, ȳ) such that the set q(ȳ) ∪ {φ(x̄, ȳ)} is consistent and q(ȳ) ∪ {φ(x̄, ȳ)} ` p(x̄). Since we deal with a small theory, φ(x̄, ȳ) can be chosen so that, for any formula ψ(x̄, ȳ), the
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