Correspondences of the Semigroup of Endomorphisms of an Equivalence Relation

Yu. V. Zhuchok ; E. A. Toichkina

Matematičeskie zametki, Tome 97 (2015) no. 2, pp. 217-230.

Résumé

In terms of different constructions of the wreath product of a transformation semigroup with a small category and of the direct product of wreath products of groups, faithful representations of three correspondences of the semigroup of all endomorphisms of an arbitrary equivalence relation are described, namely, of the semigroup of all endotopisms, of the monoid of all strong endotopisms, and, correspondingly, of the group of all autotopisms of a given equivalence relation.

Détail

Keywords: wreath product, correspondence
Mots-clés : endomorphism of an equivalence relation, endotopism, autotopism.

Bibliographie

[1] A. G. Kurosh, Obschaya algebra, Lektsii 1969–70 uchebnogo goda, Nauka, M., 1974 | MR | Zbl

[2] H. J. Zassenhaus, The Theory of Groups, Chelsea Publ., New York, 1958 | MR | Zbl

[3] O. G. Ganyushkin, T. V. Turka, “Poryadok napivgrupi vidpovidnostei skinchennoï grupi”, Visn. Kiïvsk. un-tu. Ser. fiz.-mat. nauki, 2009, no. 3, 9–13 | Zbl

[4] T. V. Turka, “Vidnoshennya Grina na napivgrupi vidpovidnostei skinchennoï grupi”, Visn. Kiïvsk. un-tu. Ser. fiz.-mat. nauki, 2010, no. 4, 38–42 | Zbl

[5] D. A. Bredikhin, “Svyazki sootvetstvii polugrupp”, Sovremennaya algebra, Postoyanno deistvuyuschii mezhvuz. (resp.) temat. sb. Vyp. 4, Izd-vo Leningr. gos. ped. in-ta, L., 1976, 31–47 | Zbl

[6] D. A. Bredikhin, “Ob opredelimosti nilpolugrupp svoimi svyazkami sootvetstvii”, Issledovaniya po sovremennoi algebre, Matem. zapiski, 11, no. 1, Izd-vo Uralskogo gos. un-ta, Sverdlovsk, 1978, 3–9 | MR | Zbl

[7] S. M. Goberstein, “Inverse semigroups determined by their bundles of correspondences”, J. Algebra, 125:2 (1989), 474–488 | DOI | MR | Zbl

[8] S. M. Goberstein, “On orthodox semigroups determined by their bundles of correspondences”, Pacific J. Math., 153:1 (1992), 71–84 | DOI | MR | Zbl

[9] M. S. Burgin, “Kategorii s involyutsiei i sootvetstviya v $\gamma$-kategoriyakh”, Tr. MMO, 22, Izd-vo Mosk. un-ta, M., 1970, 161–228 | MR | Zbl

[10] M. Sh. Tsalenko, “Klassifikatsiya kategorii sootvetstvii i tipy regulyarnosti kategorii”, Tr. MMO, 41, Izd-vo Mosk. un-ta, M., 1980, 241–285 | MR | Zbl

[11] I. M. Gelfand, V. A. Ponomarev, “Nerazlozhimye predstavleniya gruppy Lorentsa”, UMN, 23:2 (1968), 3–60 | MR | Zbl

[12] S. Mac Lane, “An algebra of additive relations”, Proc. Nat. Acad. Sci. USA, 47 (1961), 1043–1051 | DOI | MR | Zbl

[13] D. Puppe, “Korrespondenzen in Abelschen Kategorien”, Math. Ann., 148 (1962), 1–30 | DOI | MR | Zbl

[14] A. A. Iskander, “Struktura sootvetstvii universalnoi algebry”, Izv. AN SSSR. Ser. matem., 29:6 (1965), 1357–1372 | MR | Zbl

[15] A. A. Iskander, “Chastichnye universalnye algebry s zadannymi strukturami podalgebr i sootvetstvii”, Matem. sb., 70:3 (1966), 438–456 | MR | Zbl

[16] G. I. Zhitomirskii, “Stabilnye binarnye otnosheniya na universalnykh algebrakh”, Matem. sb., 82:2 (1970), 163–174 | MR | Zbl

[17] M. Petrich, “The semigroup of endomorphisms of a linear manifold”, Duke Math. J., 36 (1969), 145–152 | DOI | MR | Zbl

[18] A. Laradji, A. Umar, “Combinatorial results for semigroups of order-preserving full transformations”, Semigroup Forum, 72:1 (2006), 51–62 | DOI | MR | Zbl

[19] B. Steinberg, “On the endomorphism monoid of a profinite semigroup”, Port. Math., 68:2 (2011), 177–183 | DOI | MR | Zbl

[20] E. A. Bondar, Yu. V. Zhuchok, “Polugruppy silnykh endomorfizmov beskonechnykh grafov i gipergrafov”, Ukr. matem. zhurn., 65:6 (2013), 743–754 | MR | Zbl

[21] B. V. Popov, “Polugruppy endotopizmov $\mu$-arnykh otnoshenii”, Uchen. zap. LNPU im. A. Gertsena, 274 (1965), 184–201 | MR | Zbl

[22] Yu. V. Zhuchok, E. A. Toichkina, “Polugruppy endotopizmov otnosheniya ekvivalentnosti”, Matem. sb., 205:5 (2014), 37–54 | DOI | Zbl

[23] A. A. Albert, “Quasigroups. I”, Trans. Amer. Math. Soc., 54 (1943), 507–519 | DOI | MR | Zbl

[24] V. D. Belousov, “Uravnoveshennye tozhdestva v kvazigruppakh”, Matem. sb., 70:1 (1966), 55–97 | MR | Zbl

[25] A. Kh. Tabarov, “Avtotopii i antiavtotopii lineinykh kvazigrupp”, Dokl. AN Respubliki Tatarstan, 52 (2009), 10–16

[26] A. Kh. Tabarov, “On endotopisms of linear and alinear quasigroups”, The XIV Conference On Applied and Industrial Mathematics (Chisinau, August 17–19, 2006), 2006, 319–320

[27] Yu. V. Zhuchok, “Endomorfizmi vidnoshen ekvivalentnosti”, Visn. Kiïvsk. un-tu. Ser. fiz.-mat. nauki, 2007, no. 3, 22–26 | Zbl

[28] V. Fleisher, “O spletenii monoidov s kategoriyami”, Tr. AN Estonskoi SSR, 35:3 (1986), 237–243 | MR

[29] V. Fleischer, U. Knauer, “Endomorphism monoids of acts are wreath products of monoids with small categories”, Semigroups, Theory and Applications, Lecture Notes in Math., 1320, Springer-Verlag, Berlin, 1988, 84–96 | DOI | MR | Zbl

[30] U. Knauer, M. Nieporte, “Endomorphisms of graphs. I. The monoid of strong endomorphisms”, Arch. Math. (Basel), 52:6 (1989), 607–614 | DOI | MR | Zbl

[31] Yu. V. Zhuchok, “Polugruppy endomorfizmov nekotorykh svobodnykh proizvedenii”, Fundament. i prikl. matem., 17:3 (2012), 51–60 | MR | Zbl

[32] L. M. Gluskin, “Polugruppy izotonnykh preobrazovanii”, UMN, 16:5 (1961), 157–162 | MR | Zbl

[33] K. Čulík, “Zur Theorie der Graphen”, Časopis Pěst. Mat., 83 (1958), 133–155 | MR | Zbl

[34] M. Böttcher, U. Knauer, “Endomorphism spectra of graphs”, Discrete Math., 109:1-3 (1992), 45–57 | DOI | MR | Zbl

[35] M. I. Kargapolov, Yu. I. Merzlyakov, Osnovy teorii grupp, Nauka, M., 1972 | MR | Zbl