Nilpotent subsemigroups on the semigroup of order-decreasing transformations

A.O.Stronska

We consider the semigroup $D_n$ of order-decreasing transformations, i.e. transformations $\alpha$ of the set $\{1,...,n\}$, such that $x\alpha\leqslant x$ for any $x$ in $\{1,...,n\}$. We describe all the subsemigroups of $D_n$ which are maximal among the nilpotent subsemigroups of $D_n$ with nilpotency class $k$, prove that all these semigroups are pairwise nonisomorphic and derive formulas for calculation of number of such semigroups for nilpotency class 3 and 4.

1. Ganyushkin O.G., Mazorchuk V.S. On the structure of $IO_n$, Semigroup Forum, 66 (2003), 455--483.

2. Ganyushkin A.G., Mazorchuk V.S. The structure of subsemigroups of factor powers of finite symmetric groups, Mat.Zametki 58 (3) (1995), 341-354 (in Russian). Translation in Math. Notes 58 (1995), no.3-4, 910--920.

3. Ganyushkin O.G., Kormysheva T.V. Isolated and nilpotent subsemigroups of a finite inverse symmetric semigroup, Dopov. Akad. Nauk Ukrainy (1993), no.9, 5--9 (in Ukrainian).

4. Ganyushkin O.G., Kormysheva T.V. On nilpotent subsemigroups of a finite inverse symmetric group, Mat Zametki 56 (3)(1994), 29--35 (in Russian). Translation in Math. Notes 56 (1994), no.3-4, 896--899.

5. Howie J. M. Combinatorial and probabilistic results in transformation semigroups. Ito, Masami (ed.) et al., Words, languages and combinatorics II. Proceedings of the 2nd international conference, Kyoto, Japan, August 25-28, 1992. Singapore: World Scientific, (1994) 200--206.

6. Pin J.E. Varieties of formal languages(Paris: Masson,1984); English translation, translated by A.Howie(Oxford: North Oxford Academic Publishers Ltd.,1986).

7. Shafranova H. Maximal nilpotent subsemigroups of the semigroup $PAut_q(V_n)$, Problems in Algebra 13, Gomel Univ. Publisher, (1998) 69--83.

8. Umar A. On the ranks of certain finite semigroups of order-decreasing transformations, Portugalie Mathematica, 53 (1996) Fasc.1, 23--34

9. Umar A. On the semigroups of order-decreasing finite full transformations, Proceeding of the Royal Society of Edinburgh, 120A (1992), 129--142

10. Vernitskii A. Semigroups of order-decreasing graph endomorphism, Semigroup Forum Vol. 58 (1999), 222--240

	Nilpotent subsemigroups on the semigroup of order-decreasing transformations
Author	A.O.Stronska astra@univ.kiev.ua Kyiv Taras Shevchenko National University, Chair of Algebra and Logic,Kyiv, Glushkov av., 6.
Abstract	We consider the semigroup $D_n$ of order-decreasing transformations, i.e. transformations $\alpha$ of the set $\{1,...,n\}$, such that $x\alpha\leqslant x$ for any $x$ in $\{1,...,n\}$. We describe all the subsemigroups of $D_n$ which are maximal among the nilpotent subsemigroups of $D_n$ with nilpotency class $k$, prove that all these semigroups are pairwise nonisomorphic and derive formulas for calculation of number of such semigroups for nilpotency class 3 and 4.
Keywords	nilpotent subsemigroup, order-descreasing transformstion, nilpotency class
DOI	doi:10.30970/ms.22.2.184-197
Reference	1. Ganyushkin O.G., Mazorchuk V.S. On the structure of $IO_n$, Semigroup Forum, 66 (2003), 455--483. 2. Ganyushkin A.G., Mazorchuk V.S. The structure of subsemigroups of factor powers of finite symmetric groups, Mat.Zametki 58 (3) (1995), 341-354 (in Russian). Translation in Math. Notes 58 (1995), no.3-4, 910--920. 3. Ganyushkin O.G., Kormysheva T.V. Isolated and nilpotent subsemigroups of a finite inverse symmetric semigroup, Dopov. Akad. Nauk Ukrainy (1993), no.9, 5--9 (in Ukrainian). 4. Ganyushkin O.G., Kormysheva T.V. On nilpotent subsemigroups of a finite inverse symmetric group, Mat Zametki 56 (3)(1994), 29--35 (in Russian). Translation in Math. Notes 56 (1994), no.3-4, 896--899. 5. Howie J. M. Combinatorial and probabilistic results in transformation semigroups. Ito, Masami (ed.) et al., Words, languages and combinatorics II. Proceedings of the 2nd international conference, Kyoto, Japan, August 25-28, 1992. Singapore: World Scientific, (1994) 200--206. 6. Pin J.E. Varieties of formal languages(Paris: Masson,1984); English translation, translated by A.Howie(Oxford: North Oxford Academic Publishers Ltd.,1986). 7. Shafranova H. Maximal nilpotent subsemigroups of the semigroup $PAut_q(V_n)$, Problems in Algebra 13, Gomel Univ. Publisher, (1998) 69--83. 8. Umar A. On the ranks of certain finite semigroups of order-decreasing transformations, Portugalie Mathematica, 53 (1996) Fasc.1, 23--34 9. Umar A. On the semigroups of order-decreasing finite full transformations, Proceeding of the Royal Society of Edinburgh, 120A (1992), 129--142 10. Vernitskii A. Semigroups of order-decreasing graph endomorphism, Semigroup Forum Vol. 58 (1999), 222--240
Pages	184-197
Volume	22
Issue	2
Year	2004
Journal	Matematychni Studii
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