This article is cited in 5 papers

RESEARCH ARTICLE

On certain semigroups of contraction mappings of a finite chain

A. Umar

Department of Mathematics, The Petroleum Institute, Sas Nakhl, Khalifa University of Science and Technology, P.O. Box 2533, Abu Dhabi, UAE

Abstract: Let $[n]=\{1,2,\dots,n\}$ be a finite chain and let $\mathcal{P}_{n}$ (resp., $\mathcal{T}_{n}$) be the semigroup of partial transformations on $[n]$ (resp., full transformations on $[n]$). Let $\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}\colon (\text{for all }x,y\in \operatorname{Dom}\alpha)\ |x\alpha-y\alpha|\leq|x-y|\}$ (resp., $\mathcal{CT}_{n}=\{\alpha\in \mathcal{T}_{n}\colon (\text{for all }x,y\in [n])\ |x\alpha-y\alpha|\leq|x-y|\}$) be the subsemigroup of partial contraction mappings on $[n]$ (resp., subsemigroup of full contraction mappings on $[n]$). We characterize all the starred Green's relations on $\mathcal{CP}_{n}$ and it subsemigroup of order preserving and/or order reversing and subsemigroup of order preserving partial contractions on $[n]$, respectively. We show that the semigroups $\mathcal{CP}_{n}$ and $\mathcal{CT}_{n}$, and some of their subsemigroups are left abundant semigroups for all $n$ but not right abundant for $n\geq 4$. We further show that the set of regular elements of the semigroup $\mathcal{CT}_{n}$ and its subsemigroup of order preserving or order reversing full contractions on $[n]$, each forms a regular subsemigroup and an orthodox semigroup, respectively.

Keywords: starred Green's relations, orthodox semigroups, quasi-adequate semigroups, regularity.

MSC: 20M20

Received: 02.05.2021
Revised: 02.10.2021

Language: English

DOI: 10.12958/adm1816