A categorical study on the generalized type semigroup

Abstract

In this short note, we show that the generalized type semigroup $\CW(X, Γ)$ introduced by the author in \cite{M3} belongs to the category \textnormal{W}. In particular, we demonstrate that $\CW(X, Γ)$ satisfies axioms (W1)-(W4) and (W6). When $X$ is zero-dimensional, we also establish (W5) for the semigroup. This supports the analogy between the generalized type semigroup and pre-completed Cuntz semigroup $W(\cdot )$ for $C^{\ast }$-algebras.

Major revision. To appear in Proceedings of AMS