Asymptotic dimension, decomposition complexity, and Haver's property C

TL;DR

This paper introduces a new concept called straight decomposition complexity, compares it with existing notions like asymptotic property C, and explores game-theoretic analogs of Haver's property C in dimension theory.

Contribution

It defines and analyzes straight decomposition complexity and game-theoretic versions of Haver's property C, expanding the understanding of decomposition and dimension properties.

Findings

Straight decomposition complexity is comparable to asymptotic property C.

Game-theoretic analogs of Haver's property C are introduced and analyzed.

The relationships between these new notions and classical dimension properties are established.

Abstract

The notion of the decomposition complexity was introduced in \cite{GTY} using a game theoretical approach. We introduce a notion of straight decomposition complexity and compare it with the original as well with the asymptotic property C. Then we define a game theoretical analog of Haver's property C in the classical dimension theory and compare it with the original.