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 .
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