Product of Invariant Types Modulo Domination-Equivalence
Rosario Mennuni
TL;DR
The paper investigates how the product of global invariant types interacts with domination-equivalence, focusing on when the tensor product $\otimes$ descends to quotients and under what hypotheses this descent yields a well-defined, commutative ordered semigroup. It provides counterexamples showing that domination-equivalence need not be a congruence for $\otimes$ in general, and that commutativity can fail (notably in the Random Graph and in a specially constructed ternary theory). It identifies properties preserved by domination—finite satisfiability, definability, generic stability, and weak orthogonality—to build stable, well-behaved substructures (e.g., $\operatorname{\widetilde{Inv}}^{\mathrm{gs}}(\mathfrak U)$) and derives sufficient conditions (stationary domination/equidominance, algebraic domination) ensuring compatibility with $\otimes$. The analysis extends to dependence on the monster model, the role of IP, and stability-theoretic specializations, yielding a decomposition in thin/unidimensional stable theories and clarifying when $\operatorname{\widetilde{Inv}}(\mathfrak U)$ is monster-model independent. Overall, the work maps a nuanced landscape where classical stability phenomena largely govern descent and structure, while pathologies arise in broader theories, motivating refined invariants focused on generically stable types.
Abstract
We investigate the interaction between the product of invariant types and domination-equivalence. We present a theory where the latter is not a congruence with respect to the former, provide sufficient conditions for it to be, and study the resulting quotient when it is.