The Demazure product extended to biwords
William Q. Erickson

TL;DR
This paper extends the Demazure product to biwords and matrices, establishing an isomorphism with Monge matrices, which aids in analyzing their growth series and has implications for optimization theory.
Contribution
The paper introduces a diagrammatic extension of the Demazure product to biwords and matrices, linking it to Monge matrices in optimization.
Findings
Semigroup of Monge matrices is isomorphic to biwords with extended Demazure product
Generated functions for growth series of Monge matrices are derived
Diagrammatic representation via kelp beds generalizes previous seaweed models
Abstract
The symmetric group (and more generally, any Coxeter group) admits an associative operation known as the Demazure product. In this paper, we first extend the Demazure product to the (infinite) set of all biwords on , or equivalently, the set of all nonnegative integer matrices. We define this product diagrammatically, via braid-like graphs we call kelp beds, since they significantly generalize the seaweeds introduced by Tiskin (2015). Our motivation for this extended Demazure product arises from optimization theory, in particular the semigroup of all simple nonnegative integer Monge matrices equipped with the distance (i.e., min-plus) product. As our main result, we show that this semigroup of Monge matrices is isomorphic to the semigroup of biwords equipped with the extended Demazure product. We exploit this…
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Taxonomy
TopicsLanguage, Linguistics, Cultural Analysis · Speech and dialogue systems · Syntax, Semantics, Linguistic Variation
