Abstract
Let $[n]=\{1,2,\ldots,n\}$ be a finite chain. Let $\mathcal{P}_{n}$ and $\mathcal{T}_{n}$ be Semigroups of partial and full transformations on $[n]$ respectively. Let $\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}: |x\alpha-y\alpha|\leq|x-y| \ \ \forall x, y\in \dom~\alpha\}$ and $\mathcal{CT}_{n}=\{\alpha\in \mathcal{T}_{n}: |x\alpha-y\alpha|\leq|x-y| \ \ \forall x, y\in [n]\}$, then $\mathcal{CP}_{n}$ and $\mathcal{CT}_{n}$ are subsemigroups of $\mathcal{P}_{n}$ and $\mathcal{T}_{n}$ respectively. In this paper, we characterize the idempotent elements and computed the number of idempotents of height, $n-1$ and $n-2$ for the semigroups $\mathcal{CP}_{n}$ and $\mathcal{CT}_{n}$ respectively.
References
- Ali, B., Umar, A. and Zubairu, M. M. Regularity and Green’s relations on the semigroup of partial contractions of a finite chain. arXiv:1803.02146v1.
- Adeshola, A. D. and Umar, A. Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain. J. Combin. Math. Combin. Comput. 106, (2018) 37-49.
- Clifford, A. H. and Preston, G.B. The algebraic theory of semigroups, vol.1. Providence, R. I.: American Mathematical Society, 1961.
- Garba, G. U. Idempotents in partial transformation semifroups. Proc. Roy. Soc. Edinburghn 116 A. (1990), 359-366.
- Gracinda, M. S. Gomes and Howie, J. M. On the ranks of certain semigroups of order preserving transformations. Semigroup Forum 45 (1992), 272-282.
- Ganyushkin, O. and Mazorchuk, V. Classical Finite Transformation Semigroups. Springer−Verlag: London Limited (2009).
- Howie, J. M. Product of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc. 17 (1971) 223-236.
- Howie, J. M . Fundamental of semigroup theory. London Mathematical Society, New series 12. The Clarendon Press, Oxford University Press, 1995.
- Laradji, A. and Umar A. Combinatorial results for semifroups of order preserving partial transformations. Journal of Algebra 278 (2004), 342-358.
- Tainter, T. A characterization of idempotents in semigroups. J. Combinatorial Theory 5 (1968) 370-373.
- Umar, A. Some combinatorial problems in the theory of partial transformation semigroups. Journal of Algebra and Discrete Mathematics 17 (2014) 1 110-134.
- Umar, A. and Zubairu, M. M. On certain semigroups of partial contractions of a finite chain. arXiv:1803.02604.
- Umar, A. and Zubairu, M. M. On certain semigroups of full contractions of a finite chain. arXiv:1804.10057.
Abstract
Let $[n]=\{1,2,\ldots,n\}$ be a finite chain. Let $\mathcal{P}_{n}$ and $\mathcal{T}_{n}$ be Semigroups of partial and full transformations on $[n]$ respectively. Let $\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}: |x\alpha-y\alpha|\leq|x-y| \ \ \forall x, y\in \dom~\alpha\}$ and $\mathcal{CT}_{n}=\{\alpha\in \mathcal{T}_{n}: |x\alpha-y\alpha|\leq|x-y| \ \ \forall x, y\in [n]\}$, then $\mathcal{CP}_{n}$ and $\mathcal{CT}_{n}$ are subsemigroups of $\mathcal{P}_{n}$ and $\mathcal{T}_{n}$ respectively. In this paper, we characterize the idempotent elements and computed the number of idempotents of height, $n-1$ and $n-2$ for the semigroups $\mathcal{CP}_{n}$ and $\mathcal{CT}_{n}$ respectively.
References
- Ali, B., Umar, A. and Zubairu, M. M. Regularity and Green’s relations on the semigroup of partial contractions of a finite chain. arXiv:1803.02146v1.
- Adeshola, A. D. and Umar, A. Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain. J. Combin. Math. Combin. Comput. 106, (2018) 37-49.
- Clifford, A. H. and Preston, G.B. The algebraic theory of semigroups, vol.1. Providence, R. I.: American Mathematical Society, 1961.
- Garba, G. U. Idempotents in partial transformation semifroups. Proc. Roy. Soc. Edinburghn 116 A. (1990), 359-366.
- Gracinda, M. S. Gomes and Howie, J. M. On the ranks of certain semigroups of order preserving transformations. Semigroup Forum 45 (1992), 272-282.
- Ganyushkin, O. and Mazorchuk, V. Classical Finite Transformation Semigroups. Springer−Verlag: London Limited (2009).
- Howie, J. M. Product of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc. 17 (1971) 223-236.
- Howie, J. M . Fundamental of semigroup theory. London Mathematical Society, New series 12. The Clarendon Press, Oxford University Press, 1995.
- Laradji, A. and Umar A. Combinatorial results for semifroups of order preserving partial transformations. Journal of Algebra 278 (2004), 342-358.
- Tainter, T. A characterization of idempotents in semigroups. J. Combinatorial Theory 5 (1968) 370-373.
- Umar, A. Some combinatorial problems in the theory of partial transformation semigroups. Journal of Algebra and Discrete Mathematics 17 (2014) 1 110-134.
- Umar, A. and Zubairu, M. M. On certain semigroups of partial contractions of a finite chain. arXiv:1803.02604.
- Umar, A. and Zubairu, M. M. On certain semigroups of full contractions of a finite chain. arXiv:1804.10057.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | RESEARCH ARTICLES |
| Authors | |
| Publication Date | December 15, 2021 |
| Submission Date | September 24, 2020 |
| Acceptance Date | March 14, 2021 |
| Published in Issue | Year 2021 Volume: 4 Issue: 3 |
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