Weak heirs, coheirs and the Ellis semigroups

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 \(S_{ext,G}(N)\).