On the monoid of order-preserving transformations of a finite chain whose ranges are intervals

TL;DR

This paper provides a presentation for the monoid of order-preserving transformations with interval ranges on a finite chain, and analyzes a submonoid of order-decreasing transformations including its size, rank, and presentation.

Contribution

It introduces a presentation for the monoid of order-preserving transformations with interval ranges and studies the properties of its submonoid of order-decreasing transformations.

Findings

Presented a presentation for the monoid $IO_n$.

Determined the cardinality and rank of the submonoid $IO_n^-$.

Provided a presentation for the submonoid $IO_n^-$.

Abstract

In this note we give a presentation for the monoid $IO_n$ of all order-preserving transformations of a $n$-chain whose ranges are intervals. We also consider the submonoid $IO_n^-$ of $IO_n$ consisting of order-decreasing transformations, for which we determine the cardinality, the rank and a presentation.