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This article is cited in 2 scientific papers (total in 2 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.
Received: 02.05.2021 Revised: 02.10.2021
Citation:
A. Umar, “On certain semigroups of contraction mappings of a finite chain”, Algebra Discrete Math., 32:2 (2021), 299–320
Linking options:
https://www.mathnet.ru/eng/adm823 https://www.mathnet.ru/eng/adm/v32/i2/p299
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