A generalization of the first Malcev theorem on nilpotent semigroups and nilpotency of the wreath product of semigroups

A. V. Tishchenko

Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 2, pp. 201-221.

Résumé

We describe all [0-]simple semigroups that are nilpotent in the sense of Malcev. This generalizes the first Malcev theorem on nilpotent (in the sense of Malcev) semigroups. It is proved that if the extended standard wreath product of semigroups is nilpotent in the sense of Malcev and the passive semigroup is not nilpotent, then the active semigroup of the wreath product is a finite nilpotent group. In addition to that, the passive semigroup is uniform periodic. There are found necessary and sufficient conditions under which the extended standard wreath product of semigroups is nilpotent in the sense of Malcev in the case where each of the semigroups of the wreath product generates a variety of finite step.

Bibliographie

[1] Golubchik I. Z., Mikhalëv A. V., “O mnogoobraziyakh algebr s polugruppovym tozhdestvom”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1982, no. 2, 8–11 | MR | Zbl

[2] Zimin A. I., “O polugruppakh, nilpotentnykh v smysle Maltseva”, Izv. vyssh. uchebn. zaved. Matematika, 1980, no. 6, 23–29 | MR | Zbl

[3] Zimin A. I., “Blokiruyuschie mnozhestva termov”, Mat. sb., 119(161):3 (1982), 363–375 | MR | Zbl

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

[5] Klifford A., Preston G., Algebraicheskaya teoriya polugrupp, v. 1, Mir, M., 1972 | Zbl

[6] Klifford A., Preston G., Algebraicheskaya teoriya polugrupp, v. 2, Mir, M., 1972

[7] Koshelev Yu. G., “Ob odnoi assotsiativnoi operatsii na mnozhestve vsekh mnogoobrazii monoidov”, Sovrem. algebra, 4, 1976, 107–117 | Zbl

[8] Koshelev Yu. G., “Regulyarnost i idempotentnost spleteniya polugrupp”, Sovrem. algebra, 4, 1976, 97–106 | Zbl

[9] Koshelev Yu. G., “Assotsiativnost umnozheniya mnogoobrazii polugrupp”, Mezhdunar. konf. po algebre, posv. pamyati A. I. Maltseva, Tezisy dokl. po teorii modelei i algebr. sistem, Novosibirsk, 1989, 63

[10] Krasilnikov A. N., “O polugruppovoi i lievskoi nilpotentnosti assotsiativnykh algebr”, Mat. zametki, 62:4 (1997), 510–519 | DOI | MR | Zbl

[11] Maltsev A. I., “Nilpotentnye polugruppy”, Izbrannye trudy, v. 1, Klassicheskaya algebra, Nauka, M., 1976, 335–339

[12] Maltsev A. I., “O vklyuchenii assotsiativnykh sistem v gruppy”, Izbrannye trudy, v. 1, Klassicheskaya algebra, Nauka, M., 1976, 39–45

[13] Sapir M. V., “Suschestvenno beskonechno baziruemye konechnye polugruppy”, Mat. sb., 133(175):2 (1987), 154–166 | MR | Zbl

[14] Sapir M. V., Sukhanov E. V., “O mnogoobraziyakh periodicheskikh polugrupp”, Izv. vyssh. uchebn. zaved. Matematika, 1981, no. 4, 48–55 | MR | Zbl

[15] Tischenko A. V., “O nilpotentnosti v smysle A. I. Maltseva spleteniya polugrupp”, Uspekhi mat. nauk, 43:5 (1988), 221–222 | MR | Zbl

[16] Tischenko A. V., “Zamechanie o polugruppovykh mnogoobraziyakh konechnogo indeksa”, Izv. vyssh. uchebn. zaved. Matematika, 1990, no. 7, 79–83 | MR | Zbl

[17] Tischenko A. V., “O razlichnykh opredeleniyakh spleteniya polugruppovykh mnogoobrazii”, Fundament. i prikl. mat., 2:1 (1996), 233–249 | MR | Zbl

[18] Tischenko A. V., “Cpleteniya mnogoobrazii i poluarkhimedovy mnogoobraziya polugrupp”, Tr. MMO, 57, 1996, 318–338

[19] Tischenko A. V., “Uporyadochennyi monoid polugruppovykh mnogoobrazii otnositelno spleteniya”, Fundament. i prikl. mat., 5:1 (1999), 283–305 | MR | Zbl

[20] Tischenko A. V., “Spletenie polugrupp i mnogoobraziya konechnogo indeksa”, Fundament. i prikl. mat., 6:3 (2000), 889–902 | MR | Zbl

[21] Shevrin L. N., Vernikov B. M., Volkov M. V., “Reshëtki mnogoobrazii polugrupp”, Izv. vyssh. uchebn. zaved. Matematika, 2009, no. 3, 3–36 | MR | Zbl

[22] Shevrin L. N., Volkov M. V., “Tozhdestva polugrupp”, Izv. vyssh. uchebn. zaved. Matematika, 1985, no. 11, 3–47 | MR | Zbl

[23] Shevrin L. N., Sukhanov E. V., “Strukturnye aspekty teorii mnogoobrazii polugrupp”, Izv. vyssh. uchebn. zaved. Matematika, 1989, no. 6, 3–39 | MR | Zbl

[24] Baumslag G., “Wreath products and $p$-groups”, Proc. Cambridge Philos. Soc., 55 (1959), 224–231 | DOI | MR | Zbl

[25] Golubchik I. Z., Mikhalev A. V., “A note on varieties of semiprime rings with semigroup identities”, J. Algebra, 54 (1978), 42–45 | DOI | MR | Zbl

[26] Hunter R. P., “Some results on wreath product of semigroups”, Bull. Soc. Math. Belg., 18 (1966), 3–16 | MR | Zbl

[27] Jones P. R., “Analogues of the bicyclic semigroup in simple semigroups without idempotents”, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 11–24 | DOI | MR | Zbl

[28] Neumann B. H., Taylor T., “Subsemigroups of nilpotent groups”, Proc. Roy. Soc. London Ser. A, 274:1 (1963), 1–4 | DOI | MR | Zbl

[29] Skornyakov L. A., “Regularity of the wreath product of monoids”, Semigroup Forum, 18:1 (1979), 83–86 | DOI | MR

[30] Tilson B., “Categories as algebra: an essential ingredient in the theory of monoids”, J. Pure Appl. Algebra, 48:1–2 (1987), 83–198 | DOI | MR | Zbl

[31] Tishchenko A. V., “On the Brown–McCoy radical in semigroups”, Math. Nachr., 63 (1974), 401–411 | DOI | MR | Zbl

[32] Tishchenko A. V., “Simplicity of wreath product of semigroups with fixed passive semigroup”, Semigroup Forum, 49 (1994), 275–287 | DOI | MR | Zbl