Abstract:
Let [n]={1,2,…,n} be a finite 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.
\Bibitem{Uma21}
\by A.~Umar
\paper On certain semigroups of contraction mappings of a~finite chain
\jour Algebra Discrete Math.
\yr 2021
\vol 32
\issue 2
\pages 299--320
\mathnet{http://mi.mathnet.ru/adm823}
\crossref{https://doi.org/10.12958/adm1816}
Linking options:
https://www.mathnet.ru/eng/adm823
https://www.mathnet.ru/eng/adm/v32/i2/p299
This publication is cited in the following 2 articles:
Muhammad Mansur Zubairu, Abdullahi Umar, Jaafar Abdulkadir Aliyu, Trends in Mathematics, Recent Developments in Algebra and Analysis, 2024, 35
Bashir Ali, Abdullahi Umar, Muhammad Mansur Zubairu, “Regularity and Green's relations for the semigroups of partial and full contractions of a finite chain”, Scientific African, 21 (2023), e01890