Orthogonality and domination in o-minimal expansions of ordered groups
Pablo Andújar Guerrero, Pantelis E. Eleftheriou, Rosario Mennuni
TL;DR
The paper analyzes domination among invariant types in o-minimal expansions of ordered groups, proving that the domination poset splits as a direct product of the domination poset from a real-closed-field expansion and a factor from a linear o-minimal structure. It leverages the short closure scl pregeometry to separate long versus short information, establishing that scl-independence yields weak orthogonality to short types and enabling a decomposition of invariant types into tensor products of 1-types. A central achievement is showing that, in o-minimal expansions of ordered groups, every definable type is domination-equivalent to a Morley product of 1-types, with the number of domination-equivalence classes determined by whether a global field is definable. The results provide a transfer principle: proving the Main Conjecture for RCFF expansions suffices to obtain it for DOAG expansions, and yield a full computation of definable types' domination monoids, with two or four possible classes depending on field definability. Overall, the work gives a structural, dimension-theoretic decomposition of invariant/definable types in semi-bounded o-minimal theories, connecting linear, semi-bounded, and field-expansion cases through orthogonality and scl-based analysis.
Abstract
We analyse domination between invariant types in o-minimal expansions of ordered groups, showing that the domination poset decomposes as the direct product of two posets: the domination poset of an o-minimal expansion of a real closed field, and one derived from a linear o-minimal structure. We prove that if the Morley product is well-defined on the former poset, then the same holds for the poset computed in the whole structure. We establish our results by employing the `short closure' pregeometry ($\mathrm{scl}$) in semi-bounded o-minimal structures, showing that types of $\mathrm{scl}$-independent tuples are weakly orthogonal to types of short tuples. As an application we prove that, in an o-minimal expansion of an ordered group, every definable type is domination-equivalent to a product of 1-types. Furthermore, there are precisely two or four classes of definable types up to domination-equivalence, depending on whether a global field is definable or not.