On the Ranks of Semigroup of Order-preserving or Order-reversing Partial Contraction Mappings on a Finite Chain

TL;DR

This paper determines the minimal generating sets (ranks) of semigroups consisting of order-preserving or order-preserving/reversing partial contraction mappings on a finite chain, advancing understanding of their algebraic structure.

Contribution

It provides the first explicit calculation of the ranks of the semigroups of order-preserving and order-preserving/reversing partial contraction mappings on finite chains.

Findings

Calculated the rank of CP_n.

Calculated the rank of CRCP_n.

Established structural properties of these semigroups.

Abstract

Let $\mathcal{CP}_n$ be the semigroup of partial contraction mappings on $[n]=\{1,2,\ldots,n\}$ and let $\mathcal{OCP}_n$ and $\mathcal{ORCP}_n$ be its subsemigroups consisting of all order-preserving and of all order-preserving or order-reversing, partial contraction mappings, respectively. In this paper we obtain the rank of the two semigroups, $\mathcal{OCP}_n$ and $\mathcal{ORCP}_n$, respectively.