The domination monoid in o-minimal theories

Abstract

We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by classes of [Formula: see text]-types. We show this to hold in Real Closed Fields, where generators of this monoid correspond to invariant convex subrings of the monster model. Combined with [C. Ealy, D. Haskell and J. Maríková, Residue field domination in real closed valued fields, Notre Dame J. Formal Logic 60(3) (2019) 333–351], this allows us to compute the domination monoid in the weakly o-minimal theory of Real Closed Valued Fields.

The domination monoid in henselian valued fields

“…By Fact 3.2, U/‫ޚ‬ is divisible, and it is easy to see that U/‫ޚ‬ inherits saturation and strong homogeneity from U. The conclusion follows by lifting the analogous result [17,Corollary 13.11] (see also [23,Proposition 4.8]) from U/‫.ޚ‬ □…”

“…We assume familiarity with invariant types, and recall some basic definitions and facts about domination. See [23,Section 1.2], [21, Section 2.1.2] and [22] for a more thorough treatment.…”

“…As in Corollary 6.19, but using Corollary 4.9 instead of Corollary 4.15. □ In special cases, results such as the previous corollaries may also be obtained by using domination by a family of sorts in the sense of [12, Definition 1.7] (see [23,Section 6]). This kind of domination was proven in the algebraically closed case in [17], in the real closed case in [12], and in the equicharacteristic zero case in [31].…”

An invitation to extension domination

“…In practice, sometimes one can understand the space of invariant types by studying a collection of nice invariant types together with the domination relation. For example, the space of invariant types in ominimal structures and henselian valued fields [9,6] has been studied in precisely this way. Therefore, the question of whether a notion of domination exists for Keisler measures naturally arises.…”