Recurrence in a dynamical system over adequate partial semigroups
arXiv:2607.26548
The paper studies quasi‑central sets in commutative adequate partial semigroups, providing algebraic and dynamical characterizations of the associated idempotent ultrafilters and investigating minimal dynamical systems for such semigroups.
Abstract
Using tools from topological dynamics, H.~Furstenberg introduced the notion of \emph{central sets} and established the celebrated Central Sets Theorem. The sets which satisfy the conclusion of the Central Sets Theorem are called -sets. Hindman, Maleki, and Strauss first brought the concept of an important type of -sets called the quasi-central sets which are not central sets. In 2017, A. Ghosh gave the combinatorial treatment of -sets in commutative adequate partial semigroups, where -sets are the sets which satisfy the conclusion of Central Sets Theorem for commutative adequate partial semigroups. In this work, we discuss the Quasi-central sets algebraically and dynamically for commutative adequate partial semigroups. We give dynamical characterization of members of idempotent ultrafilters for commutative adequate partial semigroups, also we study the minimal dynamical systems for an adequate partial semigroup.