## Abstract

Let [n] = {1, 2, …, n} be a őnite chain and let P_{n} (resp., T_{n} ) be the semigroup of partial transformations on [n] (resp., full transformations on [n]). Let CP_{n} = {α ∈ P_{n}: (for all x, y ∈Domα) |xα − yα| ⩽ |x − y|} (resp., CT_{n} = {α ∈ T_{n}: (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 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 CP_{n} and CT_{n}, 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 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.

Original language | English |
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Pages (from-to) | 299-320 |

Number of pages | 22 |

Journal | Algebra and Discrete Mathematics |

Volume | 32 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2021 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics