On independently based varieties of monoids

V. Yu. Popov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 3, pp. 829-876.

Résumé

There exists finitely based variety of monoids $\mathfrak Z$ such that there is no algorithm to decide, given a recursive system $\Sigma$ of semigroup identities, whether the variety of monoids given by the system $\Sigma$ in $\mathfrak Z$ is an independently based variety. There is no algorithm to decide, given an infinite recursive system $\Sigma$ of semigroup identities, whether the variety of monoids defined by the system $\Sigma$ is a finitely based variety.

Bibliographie

[1] Bakhturin Yu. A., Tozhdestva v algebrakh Li, Nauka, M., 1985 | MR | Zbl

[2] Kleiman Yu. G., “O nekotorykh voprosakh teorii mnogoobrazii grupp”, Izv. AN SSSR. Ser. matem., 47:1 (1983), 37–74 | MR | Zbl

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

[4] Aizenshtat A. Ya., “O pokrytiyakh v reshetke mnogoobrazii polugrupp”, Sovremennyi analiz i geometriya, Leningrad, 1972, 3–11 | MR

[5] Martynova T. A., “Gruppoid $0$-privedennykh mnogoobrazii polugrupp”, Issledovaniya po sovremennoi algebre, Sverdlovsk, 1979, 96–115 | MR | Zbl

[6] Olshanskii A. Yu., “O nekotorykh beskonechnykh sistemakh tozhdestv”, Tr. seminara im. I. G. Petrovskogo, 13, 1978, 139–145

[7] Trakhtman A. N., “Mnogoobrazie polugrupp bez neprivodimogo bazisa tozhdestv”, Mat. zametki, 21:6 (1977), 865–872 | MR | Zbl

[8] Pollak G., “Some lattices of varieties containing elements without cover”, Noncommutative Structures in Algebra and Geometric Combinatorics, Rome, 1981, 91–96 | MR | Zbl

[9] Sapir M. V., “Malye, krossovy i predelnye mnogoobraziya polugrupp”, 16 Vsesoyuzn. algebraichesk. konf., Tezisy dokladov. Ch. 2, Leningrad, 1981, 142–143

[10] Trakhtman A. N., “Shestielementnaya polugruppa, porozhdayuschaya mnogoobrazie s kontinuumom podmnogoobrazii”, Algebraicheskie sistemy i ikh mnogoobraziya, Sverdlovsk, 1988, 138–143 | MR | Zbl

[11] Kleiman E. I., “Ob uslovii pokrytiya v reshetke mnogoobrazii inversnykh polugrupp”, Issledovaniya algebraicheskikh sistem po svoistvam ikh podsistem, Sverdlovsk, 1980, 76–91 | MR | Zbl

[12] Gorbunov V. A., “Pokrytiya v reshetkakh kvazimnogoobrazii i nezavisimaya aksiomatiziruemost”, Algebra i logika, 16:5 (1977), 340–369 | MR | Zbl

[13] Gorbunov V. A., “Kvazitozhdestva dvukhelementnykh algebr”, Algebra i logika, 22:2 (1983), 121–127 | MR | Zbl

[14] Thue A., “Über unendliche Zeichenreihen”, Norske Vid. Selsk. Skr., I. Mat. Nat. Kl., Christiana, 7 (1906), 1–22

[15] Arshon S. E., “Dokazatelstvo suschestvovaniya $n$-znachnykh beskonechnykh asimmetrichnykh posledovatelnostei”, Mat. sbornik, 2(44):4 (1937), 769–779 | Zbl

[16] Adyan S. I., Problema Bernsaida i tozhdestva v gruppakh, Nauka, M., 1975 | MR | Zbl

[17] Murskii V. L., “Neskolko primerov mnogoobrazii polugrupp”, Mat. zametki, 3:6 (1968), 663–670 | MR | Zbl

[18] Maltsev A. I., Algoritmy i rekursivnye funktsii, Nauka, M., 1965 | MR

[19] Gurevich Yu. Sh., “Problema ravenstva slov dlya nekotorykh klassov polugrupp”, Algebra i logika, 5:5 (1966), 25–35 | MR | Zbl

[20] Kharlampovich O. G., Sapir M. V., “Algorithmic problems in varieties”, Internat. J. Algebra Comput., 5:4–5 (1995), 379–602 | DOI | MR | Zbl

[21] Minsky M. L., “Recursive unsolvability of Post's problem of “TAG” and topics in theory of Turing machines”, Ann. Math., 74 (1961), 437–455 | DOI | MR | Zbl

[22] Perkins P., “Bases for equational theories of semigroups”, J. Algebra, 11:2 (1969), 298–314 | DOI | MR | Zbl