Malcolmson semigroups

TL;DR

This paper introduces the matrix and finitely presented module Malcolmson semigroups for unital rings, exploring their properties and relations to the Cuntz semigroup in C*-algebras, revealing new structural insights.

Contribution

It defines new semigroups inspired by the Cuntz semigroup and establishes their algebraic properties and connections to existing invariants in operator algebras.

Findings

The semigroups have isomorphic Grothendieck groups generally.

They are isomorphic for von Neumann regular rings.

A surjective homomorphism exists from the matrix Malcolmson semigroup to the Cuntz semigroup.

Abstract

Inspired by the construction of the Cuntz semigroup for a C*-algebra, we introduce the matrix Malcolmson semigroup and the finitely presented module Malcolmson semigroup for a unital ring. These two semigroups are shown to have isomorphic Grothendieck group in general and be isomorphic for von Neumann regular rings. For unital C*-algebras, it is shown that the matrix Malcolmson semigroup has a natural surjective order-preserving homomorphism to the Cuntz semigroup, every dimension function is a Sylvester matrix rank function, and there exist Sylvester matrix rank functions which are not dimension functions.