Tame and wild refinement monoids

TL;DR

This paper classifies refinement monoids into tame and wild types, showing tame monoids have desirable properties while wild ones can lack these despite sharing some features.

Contribution

It introduces the distinction between tame and wild refinement monoids and demonstrates the properties of tame monoids as well as constructing wild examples.

Findings

Tame refinement monoids have separative and multiplicative cancellation.

Wild refinement monoids can lack these properties despite sharing some features.

Examples of wild monoids are constructed that exhibit certain good properties but are not tame.

Abstract

The class of refinement monoids (abelian monoids satisfying the Riesz refinement property) is subdivided into those which are tame, defined as being an inductive limit of finitely generated refinement monoids, and those which are wild, i.e., not tame. It is shown that tame refinement monoids enjoy many positive properties, including separative cancellation ($2x=2y=x+y \implies x=y$) and multiplicative cancellation with respect to the algebraic ordering ($mx\le my \implies x\le y$). In contrast, examples are constructed to exhibit refinement monoids which enjoy all the mentioned good properties but are nonetheless wild.