Right-topological semigroup operations on inclusion hyperspaces

TL;DR

This paper extends semigroup operations to inclusion hyperspaces on discrete semigroups, analyzing their algebraic and topological properties, including minimal ideals, centers, and specific structures for finite cyclic groups.

Contribution

It introduces a method to extend semigroup operations to inclusion hyperspaces and investigates their algebraic and topological structures, including minimal ideals and centers.

Findings

Extension of semigroup operation to inclusion hyperspaces on any discrete semigroup.

Identification of important subsemigroups and analysis of minimal ideals and centers.

Structural description of G(IZ_n) for small n.

Abstract

We show that for any discrete semigroup $X$ the semigroup operation can be extended to a right-topological semigroup operation on the space $G(X)$ of inclusion hyperspaces on $X$. We detect some important subsemigroups of $G(X)$, study the minimal ideal, the (topological) center, left cancelable elements of $G(X)$, and describe the structure of the semigroups $G(\IZ_n)$ for small numbers $n$.