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Sibirsk. Mat. Zh., 2006, Volume 47, Number 3, Pages 670–694 (Mi smj886)  

This article is cited in 14 scientific papers (total in 14 papers)

Operads and varieties of algebras defined by polylinear identities

S. N. Tronin

Kazan State University, Faculty of Mechanics and Mathematics

Abstract: We show that varieties of algebras over abstract clones and over the corresponding operads are rationally equivalent. We introduce the class of operads (which we call commutative for definiteness) such that the varieties of algebras over these operads resemble in a sense categories of modules over commutative rings. In particular, the notions of a polylinear mapping and the tensor product of algebras. The categories of modules over commutative rings and the category of convexors are examples of varieties over commutative operads. By analogy with the theory of linear multioperator algebras, we develop a theory of $C$-linear multioperator algebras; in particular, of algebras, defined by $C$-polylinear identities (here $C$ is a commutative operad). We introduce and study symmetric $C$-linear operads. The main result of this article is as follows: A variety of $C$-linear multioperator algebras is defined by $C$-polylinear identities if and only if it is rationally equivalent to a variety of algebras over a symmetric $C$-linear operad.

Keywords: operad, algebra, variety, identity, rational equivalence

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English version:
Siberian Mathematical Journal, 2006, 47:3, 555–573

Bibliographic databases:

UDC: 512
Received: 11.03.2005

Citation: S. N. Tronin, “Operads and varieties of algebras defined by polylinear identities”, Sibirsk. Mat. Zh., 47:3 (2006), 670–694; Siberian Math. J., 47:3 (2006), 555–573

Citation in format AMSBIB
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\paper Operads and varieties of algebras defined by polylinear identities
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\issue 3
\pages 670--694
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\pages 555--573
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. N. Tronin, “Multicategories and varieties of many-sorted algebras”, Siberian Math. J., 49:5 (2008), 944–958  mathnet  crossref  mathscinet  isi
    2. S. N. Tronin, L. T. Abdulmyanova, “The operad of finite labeled tournaments”, Russian Math. (Iz. VUZ), 53:2 (2009), 59–67  mathnet  crossref  mathscinet  zmath
    3. S. N. Tronin, “Superalgebras and operads. I”, Siberian Math. J., 50:3 (2009), 503–514  mathnet  crossref  mathscinet  isi  elib  elib
    4. S. N. Tronin, “On superalgebras over operads”, Siberian Math. J., 50:6 (2009), 1106–1114  mathnet  crossref  mathscinet  isi  elib  elib
    5. S. N. Tronin, “Algebras over operad of spheres”, Russian Math. (Iz. VUZ), 54:3 (2010), 63–71  mathnet  crossref  mathscinet
    6. S. N. Tronin, “Natural multitransformations of multifunctors”, Russian Math. (Iz. VUZ), 55:11 (2011), 49–60  mathnet  crossref  mathscinet
    7. A. N. Abyzov, Yu. A. Alpin, N. A. Koreshkov, M. F. Nasrutdinov, S. N. Tronin, “Algebraicheskie issledovaniya v Kazanskom universitete ot V. V. Morozova do nashikh dnei”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 154, no. 2, Izd-vo Kazanskogo un-ta, Kazan, 2012, 44–59  mathnet
    8. S. N. Tronin, “Verbalnye kategorii i tozhdestva universalnykh algebr”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 154, no. 2, Izd-vo Kazanskogo un-ta, Kazan, 2012, 125–141  mathnet
    9. S. N. Tronin, R. N. Tuktamyshov, “Operads of hypergraphs”, Russian Math. (Iz. VUZ), 57:4 (2013), 52–62  mathnet  crossref
    10. S. N. Tronin, “On algebras over multicategories”, Russian Math. (Iz. VUZ), 60:2 (2016), 52–61  mathnet  crossref  isi
    11. Gaynullina A.R., Tronin S.N., “Some New Platforms For Algebraic Cryptography and One Method of Increasing the Security”, Lobachevskii J. Math., 37:6 (2016), 768–776  crossref  mathscinet  zmath  isi  scopus
    12. Gaynullina A.R., Tronin S.N., “Towards An Operad-Based Cryptography: Applications of Commutative Operads”, Lobachevskii J. Math., 37:3 (2016), 234–239  crossref  mathscinet  zmath  isi  scopus
    13. Gaynullina A., “on One Class of Commutative Operads”, Asian-Eur. J. Math., 10:1 (2017), 1750007  crossref  mathscinet  zmath  isi  scopus
    14. A. R. Gainullina, S. N. Tronin, “Algebry nad operadoi polykh kubov i novaya metrika v neotritsatelnom kvadrante evklidovoi ploskosti”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 159, no. 1, Izd-vo Kazanskogo un-ta, Kazan, 2017, 21–32  mathnet  elib
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