$\mathscr{R}$-cross-sections of the monoid of order-preserving transformations on a finite chain

TL;DR

This paper classifies the $ $-cross-sections of the monoid of order-preserving transformations on a finite chain using binary trees, providing a structural understanding of these algebraic objects.

Contribution

It introduces a classification of $ $-cross-sections for the monoid of order-preserving transformations on finite chains via binary trees, a novel structural approach.

Findings

Complete classification of $ $-cross-sections

Representation of classifications through binary trees

Structural insights into the monoid of order-preserving transformations

Abstract

We classify the $\mathscr{R}$-cross-sections of the monoid of order-preserving transformations on the $n$-element chain in terms of certain binary trees.