Abelian C-minimal groups Original Research Article

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.