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

TL;DR

This work studies the monoid $\mathcal{PIO}_{n}$ of interval-domain, order-preserving partial transformations on a finite chain and its order-decreasing submonoid $\mathcal{PIO}_{n}^{-}$. It develops finite presentations for $\mathcal{PIO}_{n}^{-}$ and $\mathcal{PIO}_{n}$, leveraging connections to $\mathcal{IO}_{n}$ and canonical form techniques, and employs a Guess-and-Prove framework to validate the presentations. The authors provide exact enumerative data, describe regular elements, and count idempotents and nilpotents, establishing ranks $3n-3$ for $\mathcal{PIO}_{n}^{-}$ and $n+1$ for $\mathcal{PIO}_{n}$ (for $n\ge 2$), along with regularity criteria (regular iff $n\le 3$). Their results extend the theory of endomorphisms on paths to the broader $\mathcal{PIO}_{n}$ setting and offer concrete, GAP-assisted methods for computational semigroup analysis. The explicit presentations and canonical form descriptions enhance both theoretical understanding and practical computation of these monoids.

Abstract

In this paper, we consider the monoid $\mathcal{PIO}_{n}$, of all partial order-preserving transformations on a chain with $n$ elements whose domains and ranges are intervals, along with its submonoid $\mathcal{PIO}_{n}^-$ of order-decreasing transformations. Our main aim is to give presentations for $\mathcal{PIO}_{n}^-$ and $\mathcal{PIO}_{n}$. Moreover, for both monoids, we describe regular elements and determine their ranks, cardinalities and the numbers of idempotents and nilpotents.

Theorems & Definitions (52)

• Proposition 1
• Proposition 2: Guess and Prove method
• Theorem 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Proposition 6
• proof