The domination monoid in o-minimal theories
Rosario Mennuni
TL;DR
This work develops a domination theory for o-minimal theories by reducing the domination monoid to products of invariant $1$-types via an Idempotency Lemma. It proves that if every global invariant type factors as a product of $1$-types, the domination monoid is a well-defined free commutative idempotent monoid generated by a maximal weakly orthogonal family of invariant $1$-types. The paper then computes the monoid in key o-minimal theories: in DOAG the monoid is the finite-subset lattice of invariant convex subgroups; in RCF it is generated by invariant convex subrings, and in RCVF it decomposes as a direct sum of residue-field and value-group contributions. Together with prior work, these results yield a decomposition of domination-like invariants in the real-closed valued-field setting, tying the structure to valuation-theoretic data. The findings advance understanding of how o-minimality constrains domination and point to natural open questions about general o-minimal theories and specific expansions such as $\mathbb R_{\mathrm{exp}}$.
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 1-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 arxiv:1702.06504, this allows us to compute the domination monoid in the weakly o-minimal theory of Real Closed Valued Fields.