  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.

Combinatorics;Group Theory