Correspondence between factorability and normalisation in monoids
Alen Đurić
TL;DR
The paper unifies two approaches to monoid structure by establishing a precise correspondence between factorability structures and quadratic normalisations. It shows that every factorable monoid induces a quadratic normalisation mod $1$, typically of class $(5,4)$, and that with a strengthened local factorability condition the class can be reduced to $(4,3)$, which then guarantees termination of the associated rewriting system. Conversely, a quadratic normalisation of class $(4,3)$ (mod $1$) yields a local factorability structure, with graded monoids enabling the converse from factorability to class $(4,3)$. This bidirectional framework enables transfer of homological insights from factorability to quadratic normalisations and highlights termination criteria linked to the weaker and stronger domino-type conditions, enriching both theories and guiding future generalisations to higher classes.
Abstract
Abstract. This article determines relations between two notions concerning monoids: factorability structure, introduced to simplify the bar complex; and quadratic normalisation, introduced to generalise quadratic rewriting systems and normalisations arising from Garside families. Factorable monoids are characterised in the axiomatic setting of quadratic normalisations. Additionally, quadratic normalisations of class (4,3) are characterised in terms of factorability structures and a condition ensuring the termination of the associated rewriting system.