Grammic monoids with three generators

TL;DR

This paper investigates a specific infinite monoid generated by actions on rows of Young tableaux over a three-letter alphabet, revealing it as a quotient of the free monoid with added relations.

Contribution

It characterizes the structure of the monoid acting on rows for a three-letter alphabet, extending previous work on columns and identifying its defining relations.

Findings

The monoid is a quotient of the free monoid with classical Knuth relations plus one extra rule.

It provides a presentation of the monoid in terms of generators and relations.

The monoid is infinite but finitely presented with specific relations.

Abstract

Young tableaux are combinatorial objects whose construction can be achieved from words over a finite alphabet by row or column insertion as shown by Schensted sixty years ago. Recently Abram and Reutenauer studied the action the free monoid on the set of columns by slightly adapting the insertion algorithm. Since the number of columns is finite, this action yields a finite transformation monoid. Here we consider the action on the set of rows. We investigate this infinite monoid in the case of a 3 letter alphabet. In particular we show that it is the quotient of the free monoid relative to a congruence generated by the classical Knuth rules plus a unique extra rule.