The Stylic Monoid
Antoine Abram, Christophe Reutenauer

TL;DR
The paper introduces the stylic monoid, a finite monoid derived from the free monoid on a totally ordered alphabet, exploring its structure, combinatorial properties, and connections to tableaux and the longest decreasing subword function.
Contribution
It defines the stylic monoid as a quotient of the plactic monoid, provides its presentation, and establishes its combinatorial and algebraic properties, including a bijection with N-tableaux and its role as a syntactic monoid.
Findings
Cardinality equals the number of set partitions on |A|+1 elements.
Presented as generated by A with plactic and idempotent relations.
Identified as the syntactic monoid for the longest decreasing subword function.
Abstract
The free monoid on a finite totally ordered alphabet acts at the left on columns, by Schensted left insertion. This defines a finite monoid, denoted and called the stylic monoid. It is canonically a quotient of the plactic monoid. Main results are: the cardinality of is equal to the number of partitions of a set on elements. We give a bijection with so-called -tableaux, similar to Schensted's algorithm, explaining this fact. Presentation of : it is generated by subject to the plactic (Knuth) relations and the idempotent relations , . The canonical involutive anti-automorphism on , which reverses the order on , induces an involution of , which similarly to the corresponding involution of the plactic monoid, may be computed by an evacuation-like operation (Sch\"utzenberger involution on tableaux) on…
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Taxonomy
Topicssemigroups and automata theory · Authorship Attribution and Profiling · Advanced Algebra and Logic
