On certain semigroups of contraction mappings of a finite chain

Abdullahi Umar, Muhammad Mansur Zubairu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Let [n] = {1, 2, …, n} be a őnite chain and let Pn (resp., Tn ) be the semigroup of partial transformations on [n] (resp., full transformations on [n]). Let CPn = {α ∈ Pn: (for all x, y ∈Domα) |xα − yα| ⩽ |x − y|} (resp., CTn = {α ∈ Tn: (for all x, y ∈ [n]) |xα−yα| ⩽ |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 CPn 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 CPn and CTn, and some of their subsemigroups are left abundant semigroups for all n but not right abundant for n ⩾ 4. We further show that the set of regular elements of the semigroup CTn and its subsemigroup of order preserving or order reversing full contractions on [n], each forms a regular subsemigroup and an orthodox semigroup, respectively.

Original languageEnglish
Pages (from-to)299-320
Number of pages22
JournalAlgebra and Discrete Mathematics
Volume32
Issue number2
DOIs
Publication statusPublished - 2021
Externally publishedYes

Keywords

  • orthodox semigroups
  • quasiadequate semigroups
  • regularity
  • starred Green’s relations

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics

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