Conjugacy in Patience Sorting monoids

TL;DR

This paper investigates the structure of cyclic shift graphs in patience sorting monoids, focusing on the diameter of connected components and exploring conjugacy relations within these algebraic objects using computational tools.

Contribution

It provides new insights into the diameter of cyclic shift graph components of patience sorting monoids and examines conjugacy relations, extending understanding of their algebraic structure.

Findings

Connected components have uniform symbol counts.

Diameter bounds for cyclic shift graph components are established.

Conjugacy relations are characterized for these monoids.

Abstract

The cyclic shift graph of a monoid is the graph whose vertices are the elements of the monoid and whose edges connect elements that are cyclic shift related. The Patience Sorting algorithm admits two generalizations to words, from which two kinds of monoids arise, the $\mathrm{rps}$ monoid and the $\mathrm{lps}$ (also known as Bell) monoid. Like other monoids arising from combinatorial objects such as the plactic and the sylvester, the connected components of the cyclic shift graph of the $\mathrm{rps}$ monoid consists of elements that have the same number of each of its composing symbols. In this paper, with the aid of the computational tool SageMath, we study the diameter of the connected components from the cyclic shift graph of the $\mathrm{rps}$ monoid. Within the theory of monoids, the cyclic shift relation, among other relations, generalizes the relation of conjugacy for groups. We examine several of these relations for both the $\mathrm{rps}$ and the $\mathrm{lps}$ monoids.