Infinite injective transformations whose centralizers have simple structure

Janusz Konieczny

Open Mathematics, Tome 9 (2011), p. 23-35 / Harvested from The Polish Digital Mathematics Library

Résumé

For an infinite set X, denote by Γ(X) the semigroup of all injective mappings from X to X under function composition. For α ∈ Γ(X), let C(α) = β ∈ g/g(X): αβ = βα be the centralizer of α in Γ(X). The aim of this paper is to determine those elements of Γ(X) whose centralizers have simple structure. We find α ∈ (X) such that various Green’s relations in C(α) coincide, characterize α ∈ Γ(X) such that the 𝒥 -classes of C(α) form a chain, and describe Green’s relations in C(α) for α with so-called finite ray-cycle decomposition. If α is a permutation, we also find the structure of C(α) in terms of direct and wreath products of familiar semigroups.

Détail

Publié le : 2011-01-01

Zbl 1214.20057

EUDML-ID : urn:eudml:doc:269142

Bibliographie

[1] Araújo J., Kinyon M., Konieczny J., Minimal paths in the commuting graphs of semigroups, European J. Combin., 2011, 32(1), 178–197 http://dx.doi.org/10.1016/j.ejc.2010.09.004 | Zbl 1227.05161

[2] Araújo J., Konieczny J., Automorphism groups of centralizers of idempotents, J. Algebra, 2003, 269(1), 227–239 http://dx.doi.org/10.1016/S0021-8693(03)00499-X | Zbl 1037.20062

[3] Araújo J., Konieczny J., Semigroups of transformations preserving an equivalence relation and a cross-section, Comm. Algebra, 2004, 32(5), 1917–1935 http://dx.doi.org/10.1081/AGB-120029913 | Zbl 1068.20061

[4] Araújo J., Konieczny J., General theorems on automorphisms of semigroups and their applications, J. Aust. Math. Soc., 2009, 87(1), 1–17 http://dx.doi.org/10.1017/S1446788709000019 | Zbl 1194.20059

[5] Araújo J., Konieczny J., The largest subsemilattices of the semigroup of transformations on a finite set, preprint available at http://arxiv.org/abs/1007.4845

[6] Bates C., Bundy D., Perkins S., Rowley P., Commuting involution graphs for symmetric groups, J. Algebra, 2003, 266(1), 133–153 http://dx.doi.org/10.1016/S0021-8693(03)00302-8 | Zbl 1034.20003

[7] Dixon J.D., Mortimer B., Permutation Groups, Grad. Texts in Math., 163, Springer, New York, 1996

[8] Dummit D.S., Foote R.M., Abstract Algebra, 3rd ed., John Wiley & Sons, Hoboken, 2004

[9] Higgins P.M., Digraphs and the semigroup of all functions on a finite set, Glasg. Math. J., 1988, 30(1), 41–57 http://dx.doi.org/10.1017/S0017089500007011 | Zbl 0634.20034

[10] Howie J.M., Fundamentals of Semigroup Theory, London Math. Soc. Monogr. (N.S.), Oxford University Press, New York, 1995 | Zbl 0835.20077

[11] Hyman J., Automorphisms of 1-unary algebras, Algebra Universalis, 1974, 4(1), 61–77 http://dx.doi.org/10.1007/BF02485708 | Zbl 0322.08002

[12] Hyman J., Nation J.B., Automorphisms of 1-unary algebras. II, Algebra Universalis, 1974, 4(1), 127–131 http://dx.doi.org/10.1007/BF02485714 | Zbl 0322.08003

[13] Iranmanesh A., Jafarzadeh A., On the commuting graph associated with the symmetric and alternating groups, J. Algebra Appl., 2008, 7(1), 129–146 http://dx.doi.org/10.1142/S0219498808002710 | Zbl 1151.20013

[14] Jacobson N., Basic Algebra. I, 2nd ed., W.H. Freeman and Co., New York, 1985

[15] Jakubíková D., Systems of unary algebras with common endomorphisms. I, II, Czechoslovak Math. J., 1979, 29(104)(3), 406–420, 421–429 | Zbl 0401.08010

[16] Konieczny J., Green's relations and regularity in centralizers of permutations, Glasg. Math. J., 1999, 41(1), 45–57 http://dx.doi.org/10.1017/S0017089599970301 | Zbl 0924.20049

[17] Konieczny J., Semigroups of transformations commuting with idempotents, Algebra Colloq., 2002, 9(2), 121–134 | Zbl 1005.20046

[18] Konieczny J., Semigroups of transformations commuting with injective nilpotents, Comm. Algebra, 2004, 32(4), 1551–1569 http://dx.doi.org/10.1081/AGB-120028798 | Zbl 1068.20065

[19] Konieczny J., Centralizers in the semigroup of injective transformations on an infinite set, Bull. Aust. Math. Soc., 2010, 82(2), 305–321 http://dx.doi.org/10.1017/S0004972710000304 | Zbl 1205.20063

[20] Konieczny J., Lipscomb S., Centralizers in the semigroup of partial transformations, Math. Japon., 1998, 48(3), 367–376 | Zbl 0917.20051

[21] Levi I., Normal semigroups of one-to-one transformations, Proc. Edinburgh Math. Soc., 1991, 34(1), 65–76 http://dx.doi.org/10.1017/S0013091500005009 | Zbl 0701.20040

[22] Lipscomb S.L., The structure of the centralizer of a permutation, Semigroup Forum, 1988, 37(3), 301–312 http://dx.doi.org/10.1007/BF02573142 | Zbl 0641.20041

[23] Lipscomb S., Konieczny J., Centralizers of permutations in the partial transformation semigroup, Pure Math. Appl., 1995, 6(4), 349–354 | Zbl 0856.20044

[24] Liskovec V.A., Feĭnberg V.Z., On the permutability of mappings, Dokl. Akad. Nauk BSSR, 1963, 7, 366–369 (in Russian)

[25] Novotný M., Sur an problème de la théorie des applications, Publ. Fac. Sci. Univ. Massaryk, 1953, 344, 53–64 (in Czech)

[26] Novotný M., Über Abbildungen von Mengen, Pacific J. Math., 1963, 13(4), 1359–1369 | Zbl 0137.25304

[27] Pei H., Regularity and Green's relations for semigroups of transformations that preserve an equivalence, Comm. Algebra, 2005, 33(1), 109–118 http://dx.doi.org/10.1081/AGB-200040921 | Zbl 1072.20082

[28] Pei H., Deng W., A note on Green's relations in the semigroups T(X, ρ), Semigroup Forum, 2009, 79(1), 210–213 http://dx.doi.org/10.1007/s00233-009-9151-3 | Zbl 1172.20046

[29] Suzuki M., Group Theory I, Grundlehren Math. Wiss., 247, Springer, Berlin-New York, 1982

[30] Szechtman F., On the automorphism group of the centralizer of an idempotent in the full transformation monoid, Semigroup Forum, 2005, 70(2), 238–242 http://dx.doi.org/10.1007/s00233-004-0141-1 | Zbl 1080.20057

[31] Weaver M.W., On the commutativity of a correspondence and a permutation, Pacific J. Math., 1960, 10(2), 705–711 | Zbl 0094.03203