TY - JOUR

T1 - On certain semigroups of contraction mappings of a finite chain

AU - Umar, Abdullahi

AU - Zubairu, Muhammad Mansur

N1 - Funding Information:
The second author would like to thank Bayero University and TET Fund for őnancial support. He would also like to thank The Petroleum Institute, Khalifa University of Science and Technology for hospitality during his 3-months research visit (November 2017 to February 2018) to the institution.
Publisher Copyright:
© Algebra and Discrete Mathematics.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - orthodox semigroups

KW - quasiadequate semigroups

KW - regularity

KW - starred Green’s relations

UR - http://www.scopus.com/inward/record.url?scp=85128371260&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85128371260&partnerID=8YFLogxK

U2 - 10.12958/adm1816

DO - 10.12958/adm1816

M3 - Article

AN - SCOPUS:85128371260

VL - 32

SP - 299

EP - 320

JO - Algebra and Discrete Mathematics

JF - Algebra and Discrete Mathematics

SN - 1726-3255

IS - 2

ER -