On 
 the radical of  ${\mathcal 
F}{\mathcal P}^+(S_n)$

O.  Ganyushkin; V. Mazorchuk

We prove that the semigroups $\mathfrak B_n$ of all binary relations and the factor-power ${\mathcal F}{\mathcal P}^+(S_n)$ of the symmetric group can be asymptotically approximated by nilpotent semigroups. Further, we show that almost all elements of these semigroups satisfy the equation $x^2=0$, where $0$ denotes the full binary relation. All these facts are obtained from a careful study of the radical $\mathfrak R_n$ in ${\mathcal F}{\mathcal P}^+(S_n)$. Along this study we also derive some corollaries for doubly stochastic matrices.

1. D. F. Cowan, R. Reilly, Partial cross-sections of symmetric inverse semigroups, Internat. J. Algebra Comput. 5 (1995), no. 3, 259–287.

2. O. G. Ganyushkin, V. S. Mazorchuk, Factor powers of semigroups of transformations, Dopov. Akad. Nauk Ukrainy (1993), no. 12, 5–9. (in Ukrainian)

3. A. G. Ganyushkin, V. S. Mazorchuk, Factor powers of finite symmetric groups, Mat. Zametki 58 (1995), no. 2, 176–188; (in Russian) translation in Math. Notes 58 (1995), no. 1-2, 794–802.

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

5. D. J. Kleitman, B. R. Rothschild, J. H. Spencer, The number of semigroups of order $n$, Proc. Amer. Math. Soc. 55 (1976), no. 1, 227--232.

6. A. D. Korshunov, Basic properties of random graphs with big number of verticies and arrows, Uspekhi Mat. Nauk, 40 (1985), № 1, 107–173.

7. V. Mazorchuk, All automorphisms of $ F P^+(S_n)$ are inner, Semigroup Forum 60 (2000), no. 3, 486--490.

8. V. S. Mazorchuk, Green's relations on $ F P^+(S_n)$, Mat. Stud. 15 (2001), № 2, 151-155.

9. R. McKenzie, B. M. Schein, Every semigroup is isomorphic to a transitive semigroup of binary relations, Trans. Amer. Math. Soc. 349 (1997), no. 1, 271–285.

10. M. Putcha, Monoids with idempotent cross-sections, Internat. J. Algebra Comput. 11 (2001), no. 4, 457–466.

11. S. Satoh, K. Yama, K., M. Tokizawa, Semigroups of order $8$, Semigroup Forum 49 (1994), no. 1, 7--29.

12. B. M. Schein, Representation of semigroups by means of binary relations, Mat. Sb. (N.S.) 60 (102) (1963), 293–303. (in Russian)

	On the radical of ${\mathcal F}{\mathcal P}^+(S_n)$
Author	O. Ganyushkin, V. Mazorchuk Department of Mechanics ,and Mathematics, Kyiv Taras Shevchenko University, ,64, Volodymyrska st., ,01033, Kyiv, Ukraine , Department of Mathematics, Uppsala University, ,Box 480, SE 751 06, Uppsala, ,Sweden, mazor64math.uu.se
Abstract	We prove that the semigroups $\mathfrak B_n$ of all binary relations and the factor-power ${\mathcal F}{\mathcal P}^+(S_n)$ of the symmetric group can be asymptotically approximated by nilpotent semigroups. Further, we show that almost all elements of these semigroups satisfy the equation $x^2=0$, where $0$ denotes the full binary relation. All these facts are obtained from a careful study of the radical $\mathfrak R_n$ in ${\mathcal F}{\mathcal P}^+(S_n)$. Along this study we also derive some corollaries for doubly stochastic matrices.
Keywords	semigroups $\mathfrak B_n$, binary relations, factor-power ${\mathcal F}{\mathcal P}^+(S_n)$, radical $\mathfrak R_n$
DOI	doi:10.30970/ms.20.1.17-26
Reference	1. D. F. Cowan, R. Reilly, Partial cross-sections of symmetric inverse semigroups, Internat. J. Algebra Comput. 5 (1995), no. 3, 259–287. 2. O. G. Ganyushkin, V. S. Mazorchuk, Factor powers of semigroups of transformations, Dopov. Akad. Nauk Ukrainy (1993), no. 12, 5–9. (in Ukrainian) 3. A. G. Ganyushkin, V. S. Mazorchuk, Factor powers of finite symmetric groups, Mat. Zametki 58 (1995), no. 2, 176–188; (in Russian) translation in Math. Notes 58 (1995), no. 1-2, 794–802. 4. A. G. Ganyushkin, V. S. Mazorchuk, The structure of subsemigroups of factor powers of finite symmetric groups, Mat. Zametki 58 (1995), no. 3, 341–354. (in Russian); translation in Math. Notes 58 (1995), no. 3-4, 910–920. 5. D. J. Kleitman, B. R. Rothschild, J. H. Spencer, The number of semigroups of order $n$, Proc. Amer. Math. Soc. 55 (1976), no. 1, 227--232. 6. A. D. Korshunov, Basic properties of random graphs with big number of verticies and arrows, Uspekhi Mat. Nauk, 40 (1985), № 1, 107–173. 7. V. Mazorchuk, All automorphisms of $ F P^+(S_n)$ are inner, Semigroup Forum 60 (2000), no. 3, 486--490. 8. V. S. Mazorchuk, Green's relations on $ F P^+(S_n)$, Mat. Stud. 15 (2001), № 2, 151-155. 9. R. McKenzie, B. M. Schein, Every semigroup is isomorphic to a transitive semigroup of binary relations, Trans. Amer. Math. Soc. 349 (1997), no. 1, 271–285. 10. M. Putcha, Monoids with idempotent cross-sections, Internat. J. Algebra Comput. 11 (2001), no. 4, 457–466. 11. S. Satoh, K. Yama, K., M. Tokizawa, Semigroups of order $8$, Semigroup Forum 49 (1994), no. 1, 7--29. 12. B. M. Schein, Representation of semigroups by means of binary relations, Mat. Sb. (N.S.) 60 (102) (1963), 293–303. (in Russian)
Pages	17-26
Volume	20
Issue	1
Year	2003
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On the radical of ${\mathcal F}{\mathcal P}^+(S_n)$