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 Uspekhi Mat. Nauk, 1975, Volume 30, Issue 4(184), Pages 61–106 (Mi umn4233)

Linear $\Omega$-algebras

T. M. Baranovich, M. S. Burgin

Abstract: In this paper we give a brief account of the basic results in the theory of linear $\Omega$-algebras. Particular attention is paid to research of recent years, and the connections of the theory of linear $\Omega$-algebras with other parts of algebra are shown. For some special cases of linear $\Omega$-algebras (ternary algebras, $\Gamma$-rings) only a survey of the literature is given.
With the help of linear $\Omega$-algebras new and simplified proofs of some known results in universal algebra are obtained. Various applications of linear $\Omega$-algebras to functional analysis and differential geometry are described.
A large number of open problems have been included, whose solution would apparently be of interest in the development of the theory of linear $\Omega$-algebras.

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English version:
Russian Mathematical Surveys, 1975, 30:4, 65–113

Bibliographic databases:

UDC: 512+519.4
MSC: 17A40, 17B35, 17A50, 16D40, 46A63

Citation: T. M. Baranovich, M. S. Burgin, “Linear $\Omega$-algebras”, Uspekhi Mat. Nauk, 30:4(184) (1975), 61–106; Russian Math. Surveys, 30:4 (1975), 65–113

Citation in format AMSBIB
\Bibitem{BarBur75} \by T.~M.~Baranovich, M.~S.~Burgin \paper Linear $\Omega$-algebras \jour Uspekhi Mat. Nauk \yr 1975 \vol 30 \issue 4(184) \pages 61--106 \mathnet{http://mi.mathnet.ru/umn4233} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=417027} \zmath{https://zbmath.org/?q=an:0366.08005} \transl \jour Russian Math. Surveys \yr 1975 \vol 30 \issue 4 \pages 65--113 \crossref{https://doi.org/10.1070/RM1975v030n04ABEH001512} 

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This publication is cited in the following articles:
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8. S. N. Tronin, “Operads and varieties of algebras defined by polylinear identities”, Siberian Math. J., 47:3 (2006), 555–573
9. S. N. Tronin, “Superalgebras and operads. I”, Siberian Math. J., 50:3 (2009), 503–514
10. J A de Azcárraga, J M Izquierdo, “Topics on n-ary algebras”, J. Phys.: Conf. Ser, 284 (2011), 012019
11. V. A. Artamonov, A. V. Klimakov, A. A. Mikhalev, A. V. Mikhalev, “Primitive and almost primitive elements of Schreier varieties”, J. Math. Sci., 237:2 (2019), 157–179
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