Relative ranks of some partial transformation semigroups. (English) Zbl 1429.20047
Summary: Let \(P_n\), \(T_n\), \(I_n\), and \(S_n\) be the partial transformation semigroup, the (full) transformation semigroup, the symmetric inverse semigroup, and the symmetric group on \(X_n=\{1,\ldots,n \}\), respectively. For \(1\leq r\leq n-1\), let \(PK_{n,r}\) be the subsemigroup consisting \(\alpha\in P_n\) such that \(|operatorname{im}\alpha|\leq r\) and let \(SPK_{n,r}=PK_{n,r}\setminus T_n\). In this paper, we first examine the subsemigroup \(I_{n,r}=I_n\cup PK_{n,r}\) and we find the necessary and sufficient conditions for any nonempty subset of \(PK_{n,r}\) to be a (minimal) relative generating set of the subsemigroup \(I_{n,r}\) modulo \(I_n\). Then we examine the subsemigroups \(PI_{n,r}= SI_n\cup PK_{n,r}\) and \(SI_{n,r}=SI_n\cup SPK_{n,r}\) for \(1\leq r\leq n-1\) where \(SI_n=I_n\setminus S_n\) and compute their relative rank.
MSC:
| 20M20 | Semigroups of transformations, relations, partitions, etc. |
| 20M10 | General structure theory for semigroups |
| 20M18 | Inverse semigroups |
Keywords:
(partial) transformation semigroup; symmetric inverse semigroup; symmetric group; (minimal) generating set; relative rankReferences:
| [1] | Al-Kharousi F, Kehinde R, Umar A. Combinatorial results for certain semigroups of partial isometries of a finite chain. Australasian Journal of Combinatorics 2014; 58(3): 365-375. · Zbl 1296.05203 |
| [2] | Ayık G, Ayık H, Howie JM. On factorisations and generators in transformation semigroup. Semigroup Forum 2005; 70: 225-237. · Zbl 1072.20078 |
| [3] | Ayık G, Ayık H, Howie JM, Ünlü Y. Rank properties of the semigroup of singular transformations on a finite set. Communications in Algebra 2008; 36: 2581-2587. · Zbl 1143.20043 |
| [4] | Ayık H, Bugay L. Generating sets of finite transformation semigroupsP K(n, r)andK(n, r). Communications in Algebra 2015; 43: 412-422. · Zbl 1312.20056 |
| [5] | Bugay L, Yağcı M, Ayık H. The ranks of certain semigroups of partial isometries. Semigroup Forum 2018; 97: 214-222. · Zbl 1467.20046 |
| [6] | East J. Infinite partition monoids. International Journal of Algebra and Computation 2014; 24: 429-460. · Zbl 1305.20070 |
| [7] | Evseen AE, Podran NE. Semigroups of transformations generated by idempotents of given defect. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika 1972; 117: 44-50. · Zbl 0242.20071 |
| [8] | Ganyushkin O, Mazorchuk V. Classical Finite Transformation Semigroups. Berlin, Germany: Springer-Verlag, 2009. · Zbl 1166.20056 |
| [9] | Garba GU. Idempotents in partial transformation semigroups. Proceedings of the Royal Society of Edinburgh 1990; 116A: 359-366. · Zbl 0719.20033 |
| [10] | Garba GU. On the nilpotent ranks of certain semigroups of transformations. Glasgow Mathematical Journal 1994; 36: 1-9. · Zbl 0801.20050 |
| [11] | Gomes GMS, Howie JM. Nilpotents in finite symmetric inverse semigroups. Proceedings of the Royal Society of Edinburgh 1987; 30: 383-395. · Zbl 0629.20037 |
| [12] | Hardy GH, Wright EM. An Introduction to the Theory of Numbers. New York, USA: Oxford University Press, 1979. · Zbl 0423.10001 |
| [13] | Higgins PM. The product of the idempotents and an H-class of the finite full transformation semigroup. Semigroup Forum 2012; 84: 216-228. |
| [14] | Howie JM. Fundamentals of Semigroup Theory. New York, NY, USA: Oxford University Press, 1995. · Zbl 0835.20077 |
| [15] | Howie JM, McFadden RB. Idempotent rank in finite full transformation semigroups. Proceedings of the Royal Society of Edinburgh 1990; 114A: 161-167. · Zbl 0704.20050 |
| [16] | Kearnes KA, Szendrei Á, Wood J. Generating singular transformations. Semigroup Forum 2001; 63: 441-448. · Zbl 1014.20037 |
| [17] | Levi I, McFadden RB.Sn-Normal semigroups. Proceedings of the Royal Society of Edinburgh 1994; 37: 471-476. · Zbl 0814.20046 |
| [18] | Lipscomb S. Symmetric Inverse Semigroups, Mathematical Surveys and Monographs. Providence, RI, USA: American Mathematical Society, 1996. · Zbl 0857.20047 |
| [19] | Yiğit E, Ayık G, Ayık H. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
