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Classification theory for a 1-ary function. (English) Zbl 0696.03018

The paper contains a stability theoretic analysis of complete first order theories over a language with a single 1-ary function. First it is shown that these theories are superstable, and non-algebraic 1-types over models are regular and have U-rank \(\leq \omega\). Then orthogonality of types and depth are examined. These results are used to study which theories are classifiable in Shelah’s sense. The main theorem states that every theory of a 1-ary function is presentable and satisfies the existence property, hence shallowness is sufficient to guarantee classifiability. The paper finishes with a complete characterization of which theories are non-multidimensional, unidimensional and categorical. A second paper “1-ary function and the f.c.p.” will contain a full classification of theories satisfying the finite cover property. An alternative approach to this matter can be found in A. N. Ryaskin’s paper in Tr. Inst. Mat. 8, 162-182 (1988; Zbl 0675.03022).
Reviewer: C.Toffalori

MSC:

03C45 Classification theory, stability, and related concepts in model theory

Citations:

Zbl 0675.03022
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