Acts over semigroups

I. B. Kozhukhov ; A. V. Mikhalev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 3, pp. 141-199.

Résumé

We present a review of results obtained mainly in the last two decades in a number of areas of the theory of acts over semigroups. The authors limited themselves to the structural theory of acts. The acts over completely ($0$-)simple semigroups, with conditions on the congruence lattice, diagonal acts, biacts and multiacts, and also partial acts are considered. Our work is an expanded version of the report made by the authors in October 2017 at the conference of Institute of Mathematics of Technical University of Berlin, dedicated to the 75th anniversary of Professor Ulrich Knauer, supplemented by the results of later works.

Bibliographie

[1] M. Arbib (red.), Algebraicheskaya teoriya avtomatov, yazykov i polugrupp, Statistika, M., 1975

[2] Aleshin S. V., “Avtomaty v algebre”, Fundament. i prikl. matem., 15:3 (2002), 23–32 | MR

[3] Aleshin S. V., Algebraicheskie sistemy avtomatov, MAKS Press, M., 2016

[4] Apraksina T. V., “Diagonalnye poligony nad polugruppami preobrazovanii”, Chebyshevskii sb., 12:1 (2011), 10–16 | MR | Zbl

[5] Apraksina T. V., “Tsiklichnost i konechnoporozhdennost poligonov nad polugruppami preobrazovanii”, Matem. vestn. pedvuzov i univ. Volgo-Vyatskogo regiona, 2012, no. 14, 51–57

[6] Apraksina T. V., “O sistemakh obrazuyuschikh diagonalnykh poligonov nad polugruppami izotonnykh i nepreryvnykh preobrazovanii”, Prikl. diskretn. matem., 26:4 (2014), 5–12 | Zbl

[7] Apraksina T. V., Barkov I. V., Kozhukhov I. B., “Dva primera diagonalnykh bipoligonov”, Fundament. i prikl. matem., 18:3 (2013), 3–9

[8] Apraksina T. V., Kozhukhov I. B., “Zamechaniya o diagonalnykh poligonakh invariantnykh polugrupp”, Vestn. MGOU. Ser. Fizika—Matematika, 2015, no. 1, 22–31

[9] Apraksina T. V, Maksimovskii M. Yu., “Poligony i chastichnye poligony nad polureshetkami”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 12:1 (2012), 3–7 | MR | Zbl

[10] Barkov I. V., Kozhukhov I. B., “O porozhdayuschikh mnozhestvakh diagonalnykh poligonov”, Elektron. inform. sist., 2015, no. 3(6), 101–104

[11] Barkov I. V., Shakirov R. R., “Konechnye polugruppy minimalnogo ranga”, Matem. vestn. pedvuzov i univ. Volgo-Vyatskogo regiona, 2014, no. 16, 54–62

[12] Bogomolov A. M., Salii V. N., Algebraicheskie osnovy teorii diskretnykh sistem, Nauka, M., 1997 | MR

[13] Glushkov V. M., “Abstraktnaya teoriya avtomatov”, UMN, 16:5(101) (1961), 3–62 | MR

[14] Grettser G., Obschaya teoriya reshetok, Mir, M., 1982

[15] Ershov Yu. L., Palyutin E. A., Matematicheskaya logika, Nauka, M., 1987 | MR

[16] Kartashov V. K., “Nezavisimye sistemy porozhdayuschikh i svoistvo Khopfa dlya unarnykh algebr”, Diskretn. matem., 20:4 (2008), 79–84 | Zbl

[17] Kartashov V. K., “O nekotorykh rezultatakh i nereshennykh zadachakh teorii unarnykh algebr”, Chebyshevskii sb., 12:2 (2011), 18–26 | MR | Zbl

[18] Klifford A., Preston G., Algebraicheskaya teoriya polugrupp, Mir, M., 1972

[19] Kozhukhov I. B., “Polugruppy, nad kotorymi vse poligony rezidualno konechny”, Fundament. i prikl. matem., 4:4 (1998), 135–1344 | MR

[20] Kozhukhov I. B., “Usloviya konechnosti dlya podpryamo nerazlozhimykh poligonov i modulei”, Fundament. i prikl. matem., 4:2 (1998), 763–767 | MR | Zbl

[21] Kozhukhov I. B., “Kommutativnye polugruppy s ogranicheniyami na podpryamo nerazlozhimye poligony”, Informatika i kibernetika (to appear)

[22] Kozhukhov I. B., Kozhukhova Yu. I., “O prodolzheniyakh chastichnykh poligonov”, Mater. 16-i Mezhdunar. konf. «Problemy teoreticheskoi kibernetiki» (N. Novgorod, 2011), 207–209

[23] Kozhukhov I. B., Kolesnikova K. A., “O khopfovosti i kokhopfovosti poligonov nad gruppami”, Fundament. i prikl. matem., 23:3 (2020), 131–139 | MR

[24] Kozhukhov I. B., Mikhalev A. V., “Poligony nad polugruppami, unarnye algebry i avtomaty”, Informatika i kibernetika, 2019, no. 2(16), 96–100

[25] Kozhukhov I. B., Petrikov A. O., “In'ektivnye i proektivnye poligony nad vpolne 0-prostoi polugruppoi”, Chebyshevskii sb., 17:4 (2016), 65–78 | MR | Zbl

[26] Kozhukhov I. B., Petrikov A. O., “Podpryamo nerazlozhimye poligony nad pryamougolnymi gruppami”, Elektron. inform. sist., 2017, no. 2, 101–109

[27] Kozhukhov I. B., Petrikov A. O., “Proektivnye i in'ektivnye poligony nad vpolne prostymi polugruppami”, Fundament. i prikl. matem., 21:1 (2016), 123–133 | MR

[28] Kozhukhov I. B., Pryanichnikov A. M., Poligony s tozhdestvami v reshetke kongruentsii, V pechati

[29] Kozhukhov I. B., Pryanichnikov A. M., Simakova A. R., “Usloviya modulyarnosti reshetki kongruentsii poligona nad pryamougolnoi svyazkoi”, Izv. RAN. Ser. matem., 84:2 (2020), 90–125 | MR | Zbl

[30] Kozhukhov I. B., Revyakin A. M., “Primer poligona, ne yavlyayuschegosya bipoligonom”, Mater. 15-go Mizhvuz. nauk.-prakt. sem. «Kombinatorni konfiguratsii ta ikh zastosuvannya» (12—13 kvitnya 2013 r.), DLAU, Kirovograd, 2013, 55–58

[31] Kozhukhov I. B., Reshetnikov A. V., “Algebry, u kotorykh vse otnosheniya ekvivalentnosti yavlyayutsya kongruentsiyami”, Fundament. i prikl. matem., 16:3 (2010), 161–192 | MR

[32] Kozhukhov I. B., Sotov A. S., Ob usloviyakh kantorovosti poligonov nad polureshetkoi, V pechati

[33] Kozhukhov I. B., Khaliullina A. R., “In'ektivnost i proektivnost poligonov nad singulyarnymi polugruppami”, Elektron. inform. sist., 2014, no. 2(2), 45–56

[34] Kozhukhov I. B., Khaliullina A. R., “Polugruppy s finitno approksimiruemymi poligonami”, Matem. zametki SVFU, 21:3 (83) (2014), 60–67 | Zbl

[35] Kozhukhov I. B., Khaliullina A. R., “Kharakterizatsiya podpryamo nerazlozhimykh poligonov”, Prikl. diskret. matem., 2015, no. 1(27), 5–16 | Zbl

[36] Kozhukhov I. B., Tsarev A. V., “Abelevy gruppy s finitno approksimiruemymi poligonami”, Fundament. i prikl. matem., 22:5 (2019), 81–89

[37] Kozhukhov I. B., Yaroshevich V. A., “Polugruppy otobrazhenii, sokhranyayuschikh binarnoe otnoshenie”, Fundament. i prikl. matem., 14:7 (2008), 129–135

[38] Kon P., Universalnaya algebra, Mir, M., 1968 | MR

[39] Korobov M. S., Petrikov A. O., “Prodolzhenie chastichnykh operatsii v universalnykh algebrakh”, Mater. 5-i Mezhdunar. nauchno-tekhn. konf. «Sovrem. inf. tekhnologii v obrazovanii i nauchn. issled.» (Donetsk, 2018), 79–83

[40] Kurosh A. G., Teoriya grupp, Nauka, M., 1967 | MR

[41] Kertis Ch., Rainer I., Teoriya predstavlenii konechnykh grupp i assotsiativnykh algebr, Nauka, M., 1969 | MR

[42] Lalleman Zh., Polugruppy i kombinatornye prilozheniya, Mir, M., 1985

[43] Lyapin E. S., Evseev A. E., Chastichnye algebraicheskie deistviya, Obrazovanie, SPb., 1991 | MR

[44] Maksimovskii M. Yu., “O poligonakh nad polureshetkami”, Fundament. i prikl. matem., 14:7 (2008), 151–156

[45] Maksimovskii M. Yu., “O bipoligonakh i multipoligonakh nad polugruppami”, Matem. zametki, 87:6 (2010), 855–866 | MR

[46] Naimark M. A., Teoriya predstavlenii grupp, Nauka, M., 1976 | MR

[47] Olshanskii A. Yu., Geometriya opredelyayuschikh sootnoshenii v gruppakh, Nauka, M., 1989 | MR

[48] Plotkin B. I., Gruppy avtomorfizmov algebraicheskikh sistem, Nauka, M., 1966 | MR

[49] Plotkin B. I., Gringlaz L. Ya., Gvaramiya A. A., Elementy algebraicheskoi teorii avtomatov, Vyssh. shkola, M., 1994

[50] Ptakhov D. O., Stepanova A. A., “Reshetki kongruentsii poligonov”, Dalnevost. matem. zhurn., 13:1 (2013), 107–115 | MR | Zbl

[51] Roiz E. N., “O podpryamo nerazlozhimykh monarakh”, Uporyadochennye mnozhestva i reshetki, 2, Saratov, 1974, 80–84 | MR | Zbl

[52] Skornyakov L. A., “Gomologicheskaya klassifikatsiya kolets”, Matem. vesnik (Serb. AN), 19:4 (1967), 415–434 | MR | Zbl

[53] Skornyakov L. A., “O gomologicheskoi klassifikatsii monoidov”, Sib. matem. zhurn., 10:5 (1969), 1139–1143 | MR | Zbl

[54] Sotov A. S., “Teorema Kantora–Bernshteina dlya poligonov nad gruppami”, Mater. VI Mezhdunar. nauchno-tekhn. konf. «Sovrem. inf. tekhnologii v obrazovanii i nauchn. issled.» (Donetsk, 2019), 120–123

[55] Khaliullina A. R., “Kongruentsii poligonov nad gruppami”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:4-2 (2013), 133–137 | Zbl

[56] Khaliullina A. R., “Kongruentsii poligonov nad polugruppami pravykh nulei”, Chebyshevskii sb., 13:3 (2013), 142–146

[57] Khaliullina A. R., “Usloviya modulyarnosti reshetki kongruentsii poligona nad polugruppoi pravykh ili levykh nulei”, Dalnevost. matem. zhurn., 15:1 (2015), 102–120 | MR | Zbl

[58] Alahmadi A., Alsulami H., Jain S. K., Zelmanov E., Matrix wreath products of algebras and embedding theorems, 2017, arXiv: 1703.08734v2 [math.RA] | MR

[59] Apraksina T. V., Barkov I. V., Kozhukhov I. B., “Diagonal ranks of semigroups”, Semigroup Forum, 90:2 (2015), 386–400 | MR | Zbl

[60] Avdeyev A. Yu., Kozhukhov I. B., “Acts over completely $0$-simple semigroups”, Acta Cybernetica, 14:4 (2000), 523–531 | MR | Zbl

[61] Barkov I. V., “Bidiagonal ranks of completely ($0$-)simple semigroups”, J. Sib. Fed. Univ. Mathematics. Physics, 9:2 (2016), 144–148 | MR | Zbl

[62] Berthiaume P., “The injective envelope of S-sets”, Canad. Math. Bull., 10 (1967), 261–273 | MR | Zbl

[63] Bulman-Fleming S., Gilmour A., “Flatness properties of diagonal acts over monoids”, Semigroup Forum, 79:2 (2009), 298–314 | MR | Zbl

[64] Burris S., Sankappanavar H. P., A Course in Universal Algebra, Grad. Texts Math., 78, Springer, Berlin, 1981 | MR | Zbl

[65] Eilenberg S., Automata, Languages, and Machines, Academic Press, New York, 1974 ; 1976 | MR | Zbl

[66] Ershad M., Sedaghatjoo M., “On a conjecture of Bulman-Fleming and Gilmour”, Semigroup Forum, 82:3 (2011), 542–546 | MR | Zbl

[67] Ésik Z., Imreh B., “Subdirectly irreducible commutative automata”, Acta Cybernetica, 5:3 (1981), 251–260 | MR | Zbl

[68] Gallagher P., “On the finite and non-finite generation of diagonal acts”, Commun. Algebra, 34 (2006), 3123–3137 | MR | Zbl

[69] Gallagher P., Ruškuc N., “Finite generation of diagonal acts of some infinite semigroups of transformations and relations”, Bull. Austral. Math. Soc., 72 (2005), 139–146 | MR | Zbl

[70] Goldstein J. A., Semigroups of Linear Operators and Applications, Dover, 2017 | MR | Zbl

[71] Haddadi M., Sheykholislami S. M. N., On radical and torsion theory in the category of S-acts, 2018, arXiv: 1806.07075v1 | MR

[72] Howie J. M., Fundamentals of Semigroup Theory, Clarendon Press, 1995 | MR | Zbl

[73] Isbell J. R., “Perfect monoids”, Semigroup Forum, 2 (1971), 95–118 | MR | Zbl

[74] Jakubíková-Studenovská D., Pócs J., “Monounary algebras”, Math. Slovac., 61:1 (2009), 107–125 | MR

[75] Jipsen P., Rose H., Varieties of Lattices, Lect. Notes Math., 1553, Springer, Berlin, 1992 | MR

[76] Kearnes K. A., Kiss E. W., The shape of congruence lattices, Mem. Amer. Math. Soc., 222, no. 1046, 2013 | MR

[77] Kilp M., Knauer U., Mikhalev A. V., Monoids, Acts and Categories, Walter de Gruyter, Berlin, 2000 | MR | Zbl

[78] Kozhukhov I. B., “One characteristical property of semilattices”, Commun. Algebra, 25:8 (1997), 2569–2577 | MR | Zbl

[79] Kozhukhov I. B., Olshanski A. Yu., “Diagonal bi-acts over semigroups with finiteness conditions”, Semigroup Forum, 92:2 (2015), 538–542 | MR

[80] Meldrum J. D. P., Wreath Products of Groups and Semigroups, Wiley, 1995 | MR

[81] Miller C., Ruškuc N., “Right Noetherian semigroups”, Int. J. Algebra Comput., 30:1 (2020), 13–48 | MR | Zbl

[82] Moghaddasi Gh., “On injective and subdirectly irreducible S-acts over left zero semigroups”, Turk. J. Math., 36 (2012), 359–365 | MR | Zbl

[83] Moghaddasi Gh., Mahmoudi M., “Subdirectly irreducible acts over some semigroups”, Bull. Iran. Math. Soc., 43:6 (2017), 1913–1924 | MR | Zbl

[84] Nation J. B., Varieties of algebras whose congruence lattices satisfy lattice identities, Thesis, California Inst. Techn., Pasadena, 1973 | MR

[85] “Problems and solutions”, Amer. Math. Mon., 96:2 (1989), 155

[86] “Problems and solutions”, Amer. Math. Mon., 97:7 (1990), 617–618 | MR

[87] Pudlák P., Tůma J., “Every finite lattice can be embedded in a finite partition lattice”, Algebra Universalis, 10:1 (1980), 74–95 | MR | Zbl

[88] Rankin S. A., Reis C. M., Thierrin G., “Right subdirectly irreducible semigroups”, Pacific J. Math., 85:2 (1979), 403–412 | MR | Zbl

[89] Robertson E. F., Ruškuc N., Thomson M. R., “On diagonal acts on monoids”, Bull. Austral. Math. Soc., 63:1 (2001), 167–175 | MR | Zbl

[90] Roueentan M., Sedaghatjoo M., “On uniform acts over semigroups”, Semigroup Forum, 97:2 (2018), 229–243 | MR | Zbl

[91] Roueentan M., Sedaghatjoo M., “The structure of subdirectly irreducible and uniform acts over rectangular bands”, Semigroup Forum, 101:1 (2020), 192–201 | MR | Zbl

[92] Sedaghatjoo M., Khaksari A., Monoids over which products of indecomposable acts are indecomposable, 2016, arXiv: 1607.00806v1 | MR | Zbl

[93] Steinberg B., Representation Theory of Finite Monoids, Springer, 2016 | MR | Zbl

[94] Wenzel G. H., “Subdirect irreducibility and equational compactness in unary algebras $\langle A; f \rangle$”, Arch. Math., 21 (1970), 256–264 | MR | Zbl

[95] Wiegandt R., “Radical and torsion theory for acts”, Semigroup Forum, 72 (2006), 312–328 | MR | Zbl

[96] Wilson R. A., The Finite Simple Groups, Springer, London, 2009 | MR | Zbl

[97] Zhang R. Z., Shum K. P., “Hereditary torsion classes of S-systems”, Semigroup Forum, 52 (1996), 253–270 | MR | Zbl