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 OJDM  Vol.11 No.3 , July 2021
On the Number of Idempotent Partial Contraction Mappings of a Finite Chain
Abstract: Let be the partial symmetric semigroup on and let and be its subsemigroups of order-preserving contractions and order-preserving, order-decreasing contractions mappings of , respectively. In this paper we investigate the cardinalities of and , the set idempotents of and , respectively. We also investigate the cardinalities of certain equivalences on and .
Cite this paper: Ojo, O. , Al-Kharousi, F. and Umar, A. (2021) On the Number of Idempotent Partial Contraction Mappings of a Finite Chain. Open Journal of Discrete Mathematics, 11, 94-101. doi: 10.4236/ojdm.2021.113007.
References

[1]   Zhao, P. and Yang, M. (2012) Regularity and Green’s Relations on Semigroups of Transformations Preserving Order and Compression. Bulletin of the Korean Mathematical Society, 49, 1015-1025.
https://doi.org/10.4134/BKMS.2012.49.5.1015

[2]   Ali, B., Umar, A. and Zubairu, M.M. (2018) Regularity and Green’s Relations on the Semigroup of Partial Contractions of a Finite Chain.

[3]   Adeshola, A.D. and Umar, A. (2018) Combinatorial Results for Certain Semigroups of Order-Preserving Full Contraction Mappings of a Finite Chain. JCMCC, 106, 37-49.

[4]   Ali, B., Umar, A. and Zubairu, M.M. (2018) Regularity and Green’s Relations on the Semigroup of Partial and Full Contractions of a Finite Chain.

[5]   Umar, A. and Zubairu, M.M. (2018) On Certain Semigroups of Partial Contractions of a Finite Chain.

[6]   Borwein, D., Rankin, S. and Renner, L. (1989) Enumeration of Injective Partial Transformations. Discrete Mathematics, 73, 291-296.
https://doi.org/10.1016/0012-365X(89)90272-0

[7]   Clifford, A.H. and Preston, G.B. (1961) The Algebraic Theory of Semigroups, Vol. 1. American Mathematical Society, Providence.

[8]   Ganyushkin, O. and Mazorchuk, V. (2009) Classical Finite Transformation Semigroups: An Introduction. Springer, London.
https://doi.org/10.1007/978-1-84800-281-4

[9]   Howie, J.M. (1995) Fundamentals of Semigroup Theory. Clarendon Press, Oxford.

[10]   Howie, J.M. (1971) Products of Idempotents in Certain Semigroups of Transformations. Proceedings of the Edinburgh Mathematical Society, 17, 223-236.
https://doi.org/10.1017/S0013091500026936

[11]   Laradji, A. and Umar, A. (2004) Combinatorial Results for Semigroups of Order-Preserving Partial Transformations. Journal of Algebra, 278, 342-359.

[12]   Laradji, A. and Umar, A. (2004) Combinatorial Results for Semigroups of Order-Decreasing Partial Transformations. Journal of Integer Sequences, 7, Article 04.3.8.

[13]   Laradji, A. and Umar, A. (2006) Combinatorial Results for Semigroups of Order-Preserving Full Transformations. Semigroup Forum, 72, 51-62.
https://doi.org/10.1007/s00233-005-0553-6

[14]   Tainiter, T. (1968) A Characterization of Idempotents in Semigroups. The Journal of Combinatorial Theory, 5, 370-373.
https://doi.org/10.1016/S0021-9800(68)80012-2

[15]   Umar, A. (1992) On the Semigroups of Order-Decreasing Finite Full Transformations. Proceedings of the Royal Society of Edinburgh Section A, 120, 129-142.
https://doi.org/10.1017/S0308210500015031

[16]   Umar, A. (1998) Enumeration of Certain Finite Semigroups of Transformations. Discrete Mathematics, 89, 291-297.

[17]   Umar, A. (2014) Some Combinatorial Problems in the Theory of Partial Transformation Semigroups. Algebra Discrete Mathematics, 17, 110-134.

[18]   Garba, G.U. (1990) Idempotents in Partial Transformation Semigroups. Proceedings of the Royal Society of Edinburgh Section A, 116, 359-366.

[19]   Sloane, N.J.A. (2011) The On-Line Encyclopedia of Integer Sequences.
https://oeis.org

[20]   Laradji, A. (2019) On Order-Preserving Partial Transformations with Prescribed Number of Fixed Points. Technical Report No. 298, (June) Department of Mathematical Sciences, KFUPM, Dhahran.

[21]   Garba, G.U. (1994) Nilpotents in Semigroups of Partial Order-Preserving Transformations. Proceedings of the Edinburgh Mathematical Society, 37, 361-377.
https://doi.org/10.1017/S001309150001885X

 
 
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