TY - JOUR
T1 - On certain semigroups of contraction mappings of a finite chain
AU - Umar, Abdullahi
AU - Zubairu, Muhammad Mansur
N1 - 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
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U2 - 10.12958/adm1816
DO - 10.12958/adm1816
M3 - Article
AN - SCOPUS:85128371260
SN - 1726-3255
VL - 32
SP - 299
EP - 320
JO - Algebra and Discrete Mathematics
JF - Algebra and Discrete Mathematics
IS - 2
ER -