Квазимногообразия коммутативных колец
DOI:
https://doi.org/10.26577/jmmcs-2018-1-485Ключевые слова:
квазиэквациональная теория, неразрешимая теория, квазитождество, квазимногообразие, базис квазитождеств, независимый базис, !-незавиcимый базис, рекурсивный незавиcимый базиcАннотация
Работа посвящена вопросам неразрешимости квазиэквациональных теорий и проблеме
конечной аксиоматизируемости. В 1966 году Тарский озвучил следующую проблему:
Существует ли алгоритм, определяющий является ли эквациональная теория конечного
множества конечных алгебр конечно аксиоматизируемой? В 1986 году Мальцевым был задан
следующий вопрос: Существует ли конечно базируемые полугруппы, группы и кольца с
неразрешимой эквациональной теорией? Нуракунов А.М. (Nurakunov, 2012) доказал, что есть
континуум квазимногообразий унаров с неразрешимой квазиэквациональной теорией, для
которых проблема вхождения для конечных унаров неразрешима. В работе (Basheyeva, 2017)
получены результаты для графов, дифференциальных группоидов и точечных абелевых
групп. В данной работе мы доказываем аналогичные результаты для комммутативных
колец с единицей. Мы доказываем, что квазимногообразие коммутативных колец с
единицей содержит континуум подквазимногообразий с неразрешимой квазиэквациональной
теорией, для которых проблема вхождения для конечных колец также неразрешима.
Кроме того, мы доказываем здесь, что в многообразии коммутативных колец с единицей
существует континуум подквазимногообразий с !-независимым базиcом квазитождеcтв,
которые, однако, не имеют незавиcимого базиcа квазитождеcтв. Кроме того, переcечение
этих квазимногообразий имеет незавиcимый базиc квазитождеcтв.
Библиографические ссылки
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III."Siberian Electronic Mathematical Reports. 14 (2017): 252–263.
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106–120.
[12] Kartashova, Anna. "Antivarieties of unars."Algebra and Logic. 50 (2011): 357–364
[13] Kravchenko, Aleksandr. "Complexity of quasivariety lattices for varieties of unary algebras. II."Siberian Electronic Mathematical
Reports. 13 (2016): 388–394.
[14] Kravchenko, Aleksandr, and Anvar Nurakunov, and Marina Schwidefsky. "Complexity of quasivariety lattices. I. Covers
and independent axiomatizability."manuscript, 2017.
[15] Kravchenko, Aleksandr, and Anvar Nurakunov, and Marina Schwidefsky. "On quasi-equational bases for differential
groupoids and unary algebras."Siberian Electronic Mathematical Reports. 14 (2017): 1330–1337
[16] Kravchenko, Aleksandr, and Andrei Yakovlev. "Quasivarieties of graphs and independent axiomatizability."Siberian Electronic
Mathematical Reports. 20 (2017): 80–89
[17] Kruse, Robert. "Identities satisfied by a finite ring."Journal of Algebra. 26 (1973): 298—318.
[18] Lyndon, Rodger. "Two notes on nilpotent groups."Proc. Amer. LVTcztk. Sot. 3 (1952): 579-583.
[19] L’vov, I.V. "Varieties of associative rings, I, II"Algebra and logika 12 (1973): 267–298, 667–668, 735.
[20] Maltsev, Anatolij. "Universally axiomatizable subclasses of locally finite classes of models."Siberian Math. J. 8 (1967):
764–770.
[21] Medvedev N.Y. "Quasivarieties of l-groups and groups". Siberian Math. J. 26 (1985): 717–723.
[22] McKenzie, Ralf. "Tarski’s finite basis problem is undecidable."International Journal of Algebra and Computation. V. 6
(1996): 49-104.
[23] Murskii, Vadim. "Examples of varieties of semigroups". Algebra and logica. 3 (1968): 423–427.
[24] Neumann, Bernhard. Varieties of Groups. Berlin, Springer-Verlag, Ergeb. d. Math. B. 37, 1967.
[25] Nurakunov, Аnvar. "Unreasonable lattices of quasivarieties."International Journal of Algebra and Computation. V. 22
(2012): 1-17.
[26] Nurakunov, Аnvar. Quasi-indentities of relatively distributive and relatively cocontinuous quasivarieties of algebras. Darmstadt:
Arbeitstangung Allgemeine Algebra, 1995.
[27] Semenova, Marina, Anna Zamojska-Dzienio. "Lattices of subclasses"Siberian Math. J., 53 (2012): 889–905.
[28] Sizyi, Sergei. "Quasivarieties of graphs."Siberian Math. J. 35 (1994): 783–794.
[29] Specht, Wilhelm. "Gesetze in Ringen. I."Math. Z. 52 (1950): 557–589.
[30] Schwidefsky, Marina, Anna Zamojska-Dzienio. "Lattices of subclasses, II"Internat. J. Algebra Comput., 24 (2014): 1099-
1126.
[31] Tarski, Аlfred. "Equational logic and equational theories of algebras."Contrib. Math. Logic. 8 (1966): 275–288.
[32] Tarski, Аlfred. "Some methodological results concerning the calculus of relations."J. Symbolic Logic. 18 (1953): 188–189.
[33] Tarski, Аlfred., Givant S. A Formalization of set theory without variables AMS: Colloquium publications, Providence,
Rhode Island. 41 (1987).
[34] Tropin, Mihail. "Finite pseudo-Boolean and topological algebras not having an independent basis of quasiindentities."
Algebra and Logic. 27 (1988): 79–99.
[35] Tumanov, Vladimir. "Finite lattice with independent quasi-equational basises"Math. notes, 36 (1984): 811–815.
[36] Fedorov, Aleksandr . "Quasi-identities of a free 2-nilpotent group"Math. notes, 40 (1986): 837–841.
[37] Kleiman, Y. (1982) O tojdestvax v gruppax [On indentities in groups]. Tr. Mosk. Mat. obs. - Proseeding of Moskow
mathematical sosiety vol.44, pp. 62–108.
[38] Maltsev, A. (1939) O vkluchenyi assosiativnykh sistem v gruppy [On including of associative systems in groups]. Mat.
sbornik - Math. collection, vol. 6, no. 2, pp.187–189.
[39] Maltsev, A. (1966) O nekotorykh pogranichnikh voprosakh algebry i matematicheskoi logiki [Some boundary questions
of algebra and Mathematical logic]. Trudy kongressa matematikov - Proseeding of mathematic congress (Moskva, 1966),
М.: Мir, pp.217–231.
Symbolic Logic. 57 (1992): 179–192.
[2] Baker, Kirby. "Finite equational bases for finite algebras in congruence-distributive equational classes."Advances in Math.
24 (1977): 207–243.
[3] Basheyeva, Ainur, and Nurakunov Anvar, and Schwidefsky Marina, and Zamojska-Dzienio Anna. "Lattices of subclasses.
III."Siberian Electronic Mathematical Reports. 14 (2017): 252–263.
[4] Birkgoff, Garret. "On the structure of abstract algebras."Proc. Cambridge Philos. Soc. 31 (1935): 433 – 454.
[5] Gorbunov, Viktor. "Covers in lattices of quasivarieties and independent axiomatizability."Algebra and Logic. 16 (1977):
340–369.
[6] Gorbunov, Viktor. Algebraic Theory of Quasivarieties. New York: Plenum, 1998.
[7] Cohen, Daniel. "On the laws of a metabelian variety."J. Alg. 5 (1967): 267–273.
[8] Chin, Luogeng Hua, and Tarski Аlfred. "Distributive and modular lows in the arithmetic of relation algebras."University
of California Publications in Mathematics 9 (1951): 341-384.
[9] J´onsson, Bjarni. "Algebras whose congruence lattices are distributive."Mathematica Scandinavica 21 (1967): 110–121.
[10] J´onsson, Bjarni. "Equational classes of lattices."Mathematica Scandinavica 22 (1968): 187–196.
[11] Kartashov, Vladimir. "Quasivarieties of unary algebras with a finite number of cycles."Algebra and Logic. 19 (1980):
106–120.
[12] Kartashova, Anna. "Antivarieties of unars."Algebra and Logic. 50 (2011): 357–364
[13] Kravchenko, Aleksandr. "Complexity of quasivariety lattices for varieties of unary algebras. II."Siberian Electronic Mathematical
Reports. 13 (2016): 388–394.
[14] Kravchenko, Aleksandr, and Anvar Nurakunov, and Marina Schwidefsky. "Complexity of quasivariety lattices. I. Covers
and independent axiomatizability."manuscript, 2017.
[15] Kravchenko, Aleksandr, and Anvar Nurakunov, and Marina Schwidefsky. "On quasi-equational bases for differential
groupoids and unary algebras."Siberian Electronic Mathematical Reports. 14 (2017): 1330–1337
[16] Kravchenko, Aleksandr, and Andrei Yakovlev. "Quasivarieties of graphs and independent axiomatizability."Siberian Electronic
Mathematical Reports. 20 (2017): 80–89
[17] Kruse, Robert. "Identities satisfied by a finite ring."Journal of Algebra. 26 (1973): 298—318.
[18] Lyndon, Rodger. "Two notes on nilpotent groups."Proc. Amer. LVTcztk. Sot. 3 (1952): 579-583.
[19] L’vov, I.V. "Varieties of associative rings, I, II"Algebra and logika 12 (1973): 267–298, 667–668, 735.
[20] Maltsev, Anatolij. "Universally axiomatizable subclasses of locally finite classes of models."Siberian Math. J. 8 (1967):
764–770.
[21] Medvedev N.Y. "Quasivarieties of l-groups and groups". Siberian Math. J. 26 (1985): 717–723.
[22] McKenzie, Ralf. "Tarski’s finite basis problem is undecidable."International Journal of Algebra and Computation. V. 6
(1996): 49-104.
[23] Murskii, Vadim. "Examples of varieties of semigroups". Algebra and logica. 3 (1968): 423–427.
[24] Neumann, Bernhard. Varieties of Groups. Berlin, Springer-Verlag, Ergeb. d. Math. B. 37, 1967.
[25] Nurakunov, Аnvar. "Unreasonable lattices of quasivarieties."International Journal of Algebra and Computation. V. 22
(2012): 1-17.
[26] Nurakunov, Аnvar. Quasi-indentities of relatively distributive and relatively cocontinuous quasivarieties of algebras. Darmstadt:
Arbeitstangung Allgemeine Algebra, 1995.
[27] Semenova, Marina, Anna Zamojska-Dzienio. "Lattices of subclasses"Siberian Math. J., 53 (2012): 889–905.
[28] Sizyi, Sergei. "Quasivarieties of graphs."Siberian Math. J. 35 (1994): 783–794.
[29] Specht, Wilhelm. "Gesetze in Ringen. I."Math. Z. 52 (1950): 557–589.
[30] Schwidefsky, Marina, Anna Zamojska-Dzienio. "Lattices of subclasses, II"Internat. J. Algebra Comput., 24 (2014): 1099-
1126.
[31] Tarski, Аlfred. "Equational logic and equational theories of algebras."Contrib. Math. Logic. 8 (1966): 275–288.
[32] Tarski, Аlfred. "Some methodological results concerning the calculus of relations."J. Symbolic Logic. 18 (1953): 188–189.
[33] Tarski, Аlfred., Givant S. A Formalization of set theory without variables AMS: Colloquium publications, Providence,
Rhode Island. 41 (1987).
[34] Tropin, Mihail. "Finite pseudo-Boolean and topological algebras not having an independent basis of quasiindentities."
Algebra and Logic. 27 (1988): 79–99.
[35] Tumanov, Vladimir. "Finite lattice with independent quasi-equational basises"Math. notes, 36 (1984): 811–815.
[36] Fedorov, Aleksandr . "Quasi-identities of a free 2-nilpotent group"Math. notes, 40 (1986): 837–841.
[37] Kleiman, Y. (1982) O tojdestvax v gruppax [On indentities in groups]. Tr. Mosk. Mat. obs. - Proseeding of Moskow
mathematical sosiety vol.44, pp. 62–108.
[38] Maltsev, A. (1939) O vkluchenyi assosiativnykh sistem v gruppy [On including of associative systems in groups]. Mat.
sbornik - Math. collection, vol. 6, no. 2, pp.187–189.
[39] Maltsev, A. (1966) O nekotorykh pogranichnikh voprosakh algebry i matematicheskoi logiki [Some boundary questions
of algebra and Mathematical logic]. Trudy kongressa matematikov - Proseeding of mathematic congress (Moskva, 1966),
М.: Мir, pp.217–231.
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Как цитировать
Basheyeva, A. O. (2018). Квазимногообразия коммутативных колец. Вестник КазНУ. Серия математика, механика, информатика, 97(1), 54–66. https://doi.org/10.26577/jmmcs-2018-1-485
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