It is shown that if is an o-minimal structure such that is a dense total order and ≾ is a parameter-definable partial order on M, then ≾ has an extension to a definable total order.

Macpherson and Steinhorn 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 165–209), and allows us to find many more examples. (shrink)

We prove that has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank ω. Additionally, our methods yield other superstable expansions such as equipped with the set of factorial elements.