Structure of quasivariety lattices. II. Undecidable problems
Algebra i logika, Tome 58 (2019) no. 2, pp. 179-199.

Voir la notice de l'article provenant de la source Math-Net.Ru

Sufficient conditions are specified under which a quasivariety contains continuum many subquasivarieties having an independent quasi-equational basis but for which the quasi-equational theory and the finite membership problem are undecidable. A number of applications are presented.
Keywords: quasi-identity, quasivariety, membership problem, undecidable theory, $Q$-universality, independent basis.
@article{AL_2019_58_2_a2,
     author = {A. V. Kravchenko and A. M. Nurakunov and M. V. Schwidefsky},
     title = {Structure of quasivariety lattices. {II.} {Undecidable} problems},
     journal = {Algebra i logika},
     pages = {179--199},
     publisher = {mathdoc},
     volume = {58},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2019_58_2_a2/}
}
TY  - JOUR
AU  - A. V. Kravchenko
AU  - A. M. Nurakunov
AU  - M. V. Schwidefsky
TI  - Structure of quasivariety lattices. II. Undecidable problems
JO  - Algebra i logika
PY  - 2019
SP  - 179
EP  - 199
VL  - 58
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2019_58_2_a2/
LA  - ru
ID  - AL_2019_58_2_a2
ER  - 
%0 Journal Article
%A A. V. Kravchenko
%A A. M. Nurakunov
%A M. V. Schwidefsky
%T Structure of quasivariety lattices. II. Undecidable problems
%J Algebra i logika
%D 2019
%P 179-199
%V 58
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2019_58_2_a2/
%G ru
%F AL_2019_58_2_a2
A. V. Kravchenko; A. M. Nurakunov; M. V. Schwidefsky. Structure of quasivariety lattices. II. Undecidable problems. Algebra i logika, Tome 58 (2019) no. 2, pp. 179-199. http://geodesic.mathdoc.fr/item/AL_2019_58_2_a2/

[1] A. V. Kravchenko, A. M. Nurakunov, M. V. Shvidefski, “O stroenii reshetok kvazimnogoobrazii. I. Nezavisimaya aksiomatiziruemost”, Algebra i logika, 57:6 (2018), 684–710 | MR

[2] A. Basheyeva, A. Nurakunov, M. Schwidefsky, A. Zamojska-Dzienio, “Lattices of subklasses. III”, Sib. elektron. matem. izv., 14 (2017), 252–263 http://semr.math.nsc.ru/v14/p252-263.pdf | MR | Zbl

[3] A. I. Maltsev, Algebraicheskie sistemy, Nauka, M., 1970 | MR

[4] W. Dziobiak, Selected topics in quasivarieties of algebraic systems, Manuscript of the lecture notes delivered at the Workshop on Comput. Algebra and Appl. to Semigroup Theory (the Center of Algebra of Lisbon Univ., Portugal, 17–21 November, 1997), 61 pp.

[5] V. A. Gorbunov, Algebraicheskaya teoriya kvazimnogoobrazii, Sibirskaya shkola algebry i logiki, Nauch. kniga, Novosibirsk, 1999

[6] S. S. Goncharov, Yu. L. Ershov, Konstruktivnye modeli, Sibirskaya shkola algebry i logiki, Nauch. kniga, Novosibirsk, 1999

[7] V. K. Kartashov, “Kvazimnogoobraziya unarov s konechnym chislom tsiklov”, Algebra i logika, 19:2 (1980), 173–193 | MR

[8] S. V. Sizyi, “Kvazimnogoobraziya grafov”, Sib. matem. zh., 35:4 (1994), 879–892 | MR

[9] V. A. Gorbunov, “Pokrytiya v reshetkakh kvazimnogoobrazii i nezavisimaya aksiomatiziruemost”, Algebra i logika, 16:5 (1977), 507–548 | MR | Zbl

[10] V. Yu. Popov, “O nezavisimo razbivaemykh sistemakh kvazitozhdestv”, Izv. UrGU, ser. Matem., mekh. Kompyuter. n., 2005, no. 36, 139–144 | Zbl

[11] A. V. Kravchenko, “O reshetochnoi slozhnosti kvazimnogoobrazii grafov i endografov”, Algebra i logika, 36:3 (1997), 273–281 | MR | Zbl

[12] M. E. Adams, W. Dziobiak, “${\mathcal Q}$-universal quasivarieties of algebras”, Proc. Am. Math. Soc., 120:4 (1994), 1053–1059 | MR | Zbl

[13] M. S. Sheremet, “Quasivarieties of Cantor algebras”, Algebra Univers., 46:1/2 (2001), 193–201 | DOI | MR | Zbl

[14] A. M. Nurakunov, “Reshetki kvazimnogoobrazii tochechnykh abelevykh grupp”, Algebra i logika, 53:3 (2014), 372–400 | MR | Zbl

[15] R. Freese, “Projective geometries as projective modular lattices”, Trans. Am. Math. Soc., 251:2 (1979), 329–342 | DOI | MR | Zbl

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

[17] M. E. Adams, W. Dziobiak, “Finite-to-finite universal quasivarieties are ${\mathcal Q}$-universal”, Algebra Univers, 46:1/2 (2001), 253–283 | DOI | MR | Zbl

[18] P. Goralčík, V. Koubek, J. Sichler, “Universal varieties of $(0,1)$-lattices”, Can. J. Math., 42:3 (1990), 470–490 | DOI | MR | Zbl

[19] M. E. Adams, W. Dziobiak, “The lattice of quasivarieties of undirected graphs”, Algebra Univers, 47:1 (2002), 7–11 | DOI | MR | Zbl

[20] A. V. Kravchenko, “${\mathcal Q}$-universalnye kvazimnogoobraziya grafov”, Algebra i logika, 41:3 (2002), 311–325 | MR | Zbl

[21] V. Dzebyak, “Kvazimnogoobraziya polureshetok Sugikhara s involyutsiei”, Algebra i logika, 39:1 (2000), 47–65 | MR

[22] M. E. Adams, V. Koubek, J. Sichler, “Homomorphisms and endomorphisms of distributive lattices”, Houston J. Math., 11 (1985), 129–145 | MR | Zbl

[23] A. V. Kravchenko, A. V. Yakovlev, “Kvazimnogoobraziya grafov i nezavisimaya baziruemost”, Matem. tr., 20:2 (2017), 80–89 | Zbl

[24] A. V. Kravchenko, A. M. Nurakunov, M. V. Schwidefsky, “On quasi-equational bases for differential groupoids and unary algebras”, Sib. elektron. matem. izv., 14 (2017), 1330–1337 http://semr.math.nsc.ru/v14/p1330-1337.pdf | MR | Zbl

[25] A. V. Kravchenko, “O slozhnosti reshetok kvazimnogoobrazii dlya mnogoobrazii unarnykh algebr. II”, Sib. elektron. matem. izv., 13 (2016), 388–394 http://semr.math.nsc.ru/v13/p388-394.pdf | Zbl

[26] A. Romanowska, J. D. H. Smith, Modes, World Scientific, Singapore, 2002 | MR | Zbl

[27] A. V. Kravchenko, “Slozhnost reshetok kvazimnogoobrazii dlya mnogoobrazii unarnykh algebr”, Matem. tr., 4:2 (2001), 113–127 | MR | Zbl

[28] L. V. Shabunin, “Teorema vlozheniya dlya mnogoobrazii Kantora”, Algebra i logika, 40:3 (2001), 352–369 | MR | Zbl

[29] A. V. Kravchenko, “Slozhnost reshetok kvazimnogoobrazii dlya mnogoobrazii differentsialnykh gruppoidov”, Matem. tr., 12:1 (2009), 26–39 | MR | Zbl

[30] M. V. Shvidefski, “O slozhnosti reshetok kvazimnogoobrazii”, Algebra i logika, 54:3 (2015), 381–398 | MR

[31] A. M. Nurakunov, “Unreasonable lattices of quasivarieties”, Internat. J. Algebra Comput., 22:3 (2012), 1250006, 17 pp. | DOI | MR | Zbl