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On variants of \(o\)-minimality. (English) Zbl 0858.03039

A \(C\)-structure \((M,C)\) consists of a structure \(M\) along with a ternary relation \(C(x;y,z)\) on the domain of \(M\), where the domain of \(M\) has at least 2 elements and \(C\) satisfies certain additional axioms. Let \((T,\leq)\) be an infinite tree in which there are incomparable elements above any node of the tree. The set \(M\) of maximal chains of \(T\) with the ternary relation \(C(x;y,z)\) induced on \(M\) by requiring that \(y\) and \(z\) branch above where \(x\) and \(y\) branch turns out to be a very representative \(C\)-structure. A \(C\)-minimal structure is an expansion \({\mathcal M} = (M,C, \dots)\) of a \(C\)-structure \((M,C)\) such that for every \({\mathcal N} \equiv {\mathcal M}\), every definable subset of \(N\) is quantifier-free definable in \((N,C)\).
This paper is an investigation of \(C\)-minimal structures as an interesting variant of \(o\)-minimality. This choice is influenced by work of Adeleke and Neumann on Jordan groups. The authors develop some general model theory for \(C\)-minimal structures. They go on to study \(C\)-minimal groups and fields. The paper ends with some other variants of \(o\)-minimality.

MSC:

03C50 Models with special properties (saturated, rigid, etc.)
03C60 Model-theoretic algebra
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