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Krull dimension of types in a class of first-order theories

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We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula—plus a few other technical requirements. The theory of vector spaces and the theory fields are examples. | Turk J Math 35 (2011) , 323 – 331. ¨ ITAK ˙ c TUB doi:10.3906/mat-0905-26 Krull dimension of types in a class of first-order theories Domenico Zambella Abstract We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula—plus a few other technical requirements. The theory of vector spaces and the theory fields are examples. We prove the amalgamation property and the existence of a model-companion. We show that the model-companion is strongly minimal. We also prove that the length of any increasing sequence of prime types is bounded, so every formula has finite Krull dimension. 1. Introduction Krull-minimal theories are defined in Definition 1 below. The main requirement is that every complete quantifier-free type with one free variable either has a trivial positive part or it is isolated by a positive quantifierfree formula. This means that the formula x = x has Krull-dimension ≤ 1 , as defined in Section 3 below. We show that Krull-minimal theories have the amalgamation property in Theorem 11 and that they are model-companionable in Theorem 12. In Corollary 13 we show that the model-companion of a Krull-minimal theory is a strongly minimal theory. This reproduces in general the usual aguments used to prove elimination of quantifiers for vector spaces, torsion-free divisible groups (see e.g. Section 3.1 in [1]), and fields (see e.g. Section 3.1 in [1] and/or Section 1 in [2]). In Section 3 we consider two notions of dimension. We prove that in a Krull-minimal theory the length of any increasing sequence of prime types is bounded by the maximal degree of transcendence of its solutions. So every formula has finite Krull dimension. In the past there has been some interest in first-order theories with formulas that satisfy descending chain conditions, see [5], [4], and references cited therein—as we found out when the final draft of this paper .