Algebra in superextensions of twinic groups

TL;DR

This paper investigates the algebraic structure of the semigroup of maximal linked systems on twinic groups, revealing detailed descriptions of minimal ideals and their properties, and constructing faithful representations of these semigroups.

Contribution

It provides a new faithful representation of the semigroup of maximal linked systems and characterizes the minimal ideals for twinic groups, expanding understanding of their algebraic structure.

Findings

Describes the structure of minimal ideals in the semigroup of maximal linked systems.

Constructs a faithful representation of the semigroup in self-maps of the power-set.

Identifies classes of groups (amenable, with periodic commutators) where these structures are analyzed.

Abstract

Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $\lambda(X)$ consisting of maximal linked systems on $X$. This semigroup contains the semigroup $\beta(X)$ of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup $\lambda(X)$ in the semigroup of all self-maps of the power-set of $X$ and using this representation describe the structure of minimal ideal and minimal left ideals of $\lambda(X)$ for each twinic group $X$. The class of twinic groups includes all amenable groups and all groups with periodic commutators but does not include the free group with two generators.