On the idempotents of semigroup of partial contractions of a finite chain

Öz

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.

Anahtar Kelimeler

• Transformations semigroup
• Contractions maps
• idempotents

References

1. 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.
2. 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.
3. Clifford, A. H. and Preston, G.B. The algebraic theory of semigroups, vol.1. Providence, R. I.: American Mathematical Society, 1961.
4. Garba, G. U. Idempotents in partial transformation semifroups. Proc. Roy. Soc. Edinburghn 116 A. (1990), 359-366.
5. Gracinda, M. S. Gomes and Howie, J. M. On the ranks of certain semigroups of order preserving transformations. Semigroup Forum 45 (1992), 272-282.
6. Ganyushkin, O. and Mazorchuk, V. Classical Finite Transformation Semigroups. Springer−Verlag: London Limited (2009).
7. Howie, J. M. Product of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc. 17 (1971) 223-236.
8. Howie, J. M . Fundamental of semigroup theory. London Mathematical Society, New series 12. The Clarendon Press, Oxford University Press, 1995.

9. Laradji, A. and Umar A. Combinatorial results for semifroups of order preserving partial transformations. Journal of Algebra 278 (2004), 342-358.
10. Tainter, T. A characterization of idempotents in semigroups. J. Combinatorial Theory 5 (1968) 370-373.
11. Umar, A. Some combinatorial problems in the theory of partial transformation semigroups. Journal of Algebra and Discrete Mathematics 17 (2014) 1 110-134.
12. Umar, A. and Zubairu, M. M. On certain semigroups of partial contractions of a finite chain. arXiv:1803.02604.
13. Umar, A. and Zubairu, M. M. On certain semigroups of full contractions of a finite chain. arXiv:1804.10057.