Power Quotients of Plactic-like Monoids

TL;DR

This paper explores quotients of various plactic-like monoids by specific congruences, providing normal forms, idempotents, and size formulas, extending recent work on stylic monoids.

Contribution

It introduces a unified approach to describe quotients of plactic-like monoids with new relations, including normal forms and size formulas, expanding understanding of their algebraic structure.

Findings

Normal forms for each monoid type

Explicit formulas for the size of quotients

Characterization of idempotent elements

Abstract

In this paper we describe the quotients of several plactic-like monoids by the least congruences containing the relations $a^{\sigma(a)} = a$ with $\sigma(a)\ge 2$ for every generator $a$. The starting point for this description is the recent paper of Abram and Reutenauer about the so-called stylic monoid which happens to be the quotient of the plactic monoid by the relations $a^2 = a$ for every letter $a$. The plactic-like monoids considered are the plactic monoid itself, the Chinese monoid, and the sylvester monoid. In each case we describe: a set of normal forms, and the idempotents; and obtain formulae for their size.