Narrow Results

In the general context of presentations of monoids, we study normalization process that are determined by their restriction to length-two words. Garside’s greedy normal forms and quadratic convergent rewriting systems, in particular those associated with the plactic monoids, are typical examples. Having introduced a parameter, called the class and measuring the complexity of the normalization of length-three words, we analyze the normalization of longer words and describe a number of possible behaviors. We fully axiomatize normalizations of class $(4,3)$, show the convergence of the associated rewriting systems, and characterize those deriving from a Garside family.

This paper determines relations between two notions concerning monoids: factorability structure, introduced to simplify the bar complex; and quadratic normalization, introduced to generalize quadratic rewriting systems and normalizations arising from Garside families. Factorable monoids are characterized in the axiomatic setting of quadratic normalizations. Additionally, quadratic normalizations of class $\left(4,3\right)$ are characterized in terms of factorability structures and a condition ensuring the termination of the associated rewriting system.