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On the structure of groups endowed with a compatible C-relation
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Abstract
We use quasi-orders to describe the structure of C-groups. We do this by associating a quasi-order to each compatible C-relation of a group, and then give the structure of such quasi-ordered groups. We also reformulate in terms of quasi-orders some results concerning C-minimal groups given by Macpherson and Steinhorn.
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This memoir is devoted to a study of four types of relational structures, namely semilinear orderings, betweenness relations (and, more generally, B-sets), C-sets, and D-sets. Semilinear orderings and betweenness relations are tree-like structures; C-sets and D-sets are structures designed to capture the behaviour of 'leaves' or 'points at infinity' in semilinear orderings and B-sets. Classification theorems are proved and the automorphism groups of suitably homogeneous relational structures are analyzed. A need for the information recorded here emerged in our study of Jordan groups in the theory of permutation groups. The work is therefore focussed on connections with that subject.
We show that a quasi-ordered field (see below for definition) is either an ordered field or a Krull valued field.
Macpherson and Steinhorn (Macpherson and Steinhorn, Ann. Pure Appl. Logic 79 (1996) 165–209) introduce some variants of the notion of o-minimality. One of the most interesting is C-minimality, which provides a natural setting to study algebraically closed-valued fields and some valued groups. In this paper we go further in the study of the structure of C-minimal valued groups, giving a partial characterization in the abelian case. We obtain the following principle: for abelian valued groups G for which the valuation satisfies some kind of compatibility with the multiplication by any prime number p, being C-minimal is equivalent to the o-minimality of the expansion of the chain of valuations by the maps induced by the multiplication by each prime number in G, and by some unary predicates controlling the cardinality of the residual structures. This result is quite nice, because the class considered contains all the natural examples of abelian C-minimal valued groups of (Macpherson and Steinhorn, Ann. Pure Appl. Logic 79 (1996) 165–209), and allows us to find many more examples.