Weak heirs, coheirs and the Ellis semigroups

TL;DR

This paper explores algebraic relationships between Ellis semigroups of certain flows associated with groups and their extensions, with applications to model theory and the concepts of weak heirs and coheirs.

Contribution

It establishes new algebraic connections between Ellis semigroups of flows for groups and their extensions using weak heirs and coheirs, with implications in model theory.

Findings

Ellis groups of $S_{ext,G}(M)$ are isomorphic to subgroups of those of $S_{ext,G}(N)$ under certain conditions.

Algebraic connections between Ellis semigroups of $G$-flows and $H$-flows are identified.

Results apply to model-theoretic contexts involving definable groups and their extensions.

Abstract

Assume $G\prec H$ are groups and ${\cal A}\subseteq{\cal P}(G),\ {\cal B}\subseteq{\cal P}(H)$ are algebras of sets closed under left group translation. Under some additional assumptions we find algebraic connections between the Ellis [semi]groups of the $G$-flow $S({\cal A})$ and the $H$-flow $S({\cal B})$. We apply these results in the model theoretic context. Namely, assume $G$ is a group definable in a model $M$ and $M\prec^* N$. Using weak heirs and weak coheirs we point out some algebraic connections between the Ellis semigroups $S_{ext,G}(M)$ and $S_{ext,G}(N)$. Assuming every minimal left ideal in $S_{ext,G}(N)$ is a group we prove that the Ellis groups of $S_{ext,G}(M)$ are isomorphic to closed subgroups of the Ellis groups of $S_{ext,G}(N)$.