# A generalized type semigroup and dynamical comparison

## TL;DR

This paper introduces a higher-dimensional generalization of the type semigroup for group actions on compact spaces, providing a new characterization of dynamical comparison and proposing a broader comparison framework.

## Contribution

It constructs a generalized type semigroup for non-amenable, non-minimal actions, answering a question of Kerr and Schafhauser.

## Key findings

- New semigroup characterization of dynamical comparison
- Extension of comparison concepts to non-amenable, non-minimal systems
- Addresses open question in the field

## Abstract

In this paper, we construct a semigroup associated to an action of countable discrete group on a compact Hausdorff space, that can be regarded as a higher dimensional generalization of the type semigroup. Using this generalized type semigroup we obtain a new characterization of dynamical comparison. This answers a question of Kerr and Schafhauser. In addition, this paper suggests a definition of comparison for dynamical systems in which neither necessarily the acting group is amenable nor the action is minimal.