Relative ranks of some partial transformation semigroups

Let Pn , Tn , In , and Sn be the partial transformation semigroup, the (full) transformation semigroup, thesymmetric inverse semigroup, and the symmetric group on Xn = {1, . . . , n}, respectively. For 1 ≤ r ≤ n−1, let PKn,rbe the subsemigroup consisting α ∈ Pn such that |imα| ≤ r and let SPKn,r = PKn,r Tn . In this paper, we firstexamine the subsemigroup In,r = In ∪ PKn,r and we find the necessary and sufficient conditions for any nonemptysubset of PKn,r to be a (minimal) relative generating set of the subsemigroup In,r modulo In . Then we examine thesubsemigroups PIn,r = SIn ∪ PKn,r and SIn,r = SIn ∪ SPKn,r for 1 ≤ r ≤ n − 1 where SIn = In Sn and computetheir relative rank.

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[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.
[2] Ayık G, Ayık H, Howie JM. On factorisations and generators in transformation semigroup. Semigroup Forum 2005; 70: 225-237.
[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.
[4] Ayık H, Bugay L. Generating sets of finite transformation semigroups PK(n, r) and K(n, r) . Communications in Algebra 2015; 43: 412-422.
[5] Bugay L, Yağcı M, Ayık H. The ranks of certain semigroups of partial isometries. Semigroup Forum 2018; 97: 214-222.
[6] East J. Infinite partition monoids. International Journal of Algebra and Computation 2014; 24: 429-460.
[7] Evseen AE, Podran NE. Semigroups of transformations generated by idempotents of given defect. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika 1972; 117: 44-50.
[8] Ganyushkin O, Mazorchuk V. Classical Finite Transformation Semigroups. Berlin, Germany: Springer-Verlag, 2009.
[9] Garba GU. Idempotents in partial transformation semigroups. Proceedings of the Royal Society of Edinburgh 1990; 116A: 359-366.
[10] Garba GU. On the nilpotent ranks of certain semigroups of transformations. Glasgow Mathematical Journal 1994; 36: 1-9.
[11] Gomes GMS, Howie JM. Nilpotents in finite symmetric inverse semigroups. Proceedings of the Royal Society of Edinburgh 1987; 30: 383-395.
[12] Hardy GH, Wright EM. An Introduction to the Theory of Numbers. New York, USA: Oxford University Press, 1979.
[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.
[15] Howie JM, McFadden RB. Idempotent rank in finite full transformation semigroups. Proceedings of the Royal Society of Edinburgh 1990; 114A: 161-167.
[16] Kearnes KA, Szendrei Á, Wood J. Generating singular transformations. Semigroup Forum 2001; 63: 441-448.
[17] Levi I, McFadden RB. Sn -Normal semigroups. Proceedings of the Royal Society of Edinburgh 1994; 37: 471-476.
[18] Lipscomb S. Symmetric Inverse Semigroups, Mathematical Surveys and Monographs. Providence, RI, USA: American Mathematical Society, 1996.
[19] Yiğit E, Ayık G, Ayık H. Minimal relative generating sets of some partial transformation semigroups. Communications in Algebra 2017; 45: 1239-1245.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

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