Primitive elements of free nonassociative algebras

Improved algorithms to construct complements of primitive systems of elements of free nonassociative algebras with respect to free generating sets and algorithms to realize the rank of a system of elements are constructed and implemented. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 5, pp. 171–192, 2007.     

A class of simple nonassociative algebras

Translated from Matematicheskie Zametki, Vol. 45, No. 1, pp. 101–105, January, 1989.     

The arens algebras of a nonassociative algebra

The Arens construction of dii algebra in the biduai bpcice of an algebra has provided in functional analysis a useful method for constructing an algebra containing a rich supply of idempotcnts As idera-potents are an Important too] in the study of non-associative algebras, we begin in this paper the study of the algebraic biduai of a nonassociative algebra. In particular, we obtain a partial spectral decomposition for a transcendental element of a power associative algebra by analyzing the biduai of the associative subaigebra it generates.     

Deformations of nonassociative algebras and integrable differential equations

A new class of nonassociative algebras related to integrable PDE's and ODE's is introduced. These algebras can be regarded as a noncommutative generalization of Jordan algebras. Their deformations are investigated. Relationships between such algebras and graded Lie algebras are established.     

0-dialgebras with bar-unity and nonassociative Rota-Baxter algebras

We describe all homogeneous structures of Rota-Baxter algebras on a 0-dialgebra with associative bar-unity and give a corollary on the structure of a Rota-Baxter algebra on an arbitrary associative dialgebra with bar-unity as well as a unital associative conformal algebra. We prove that an arbitrary alternative dialgebra may be embedded into an alternative dialgebra with associative barunity. We suggest the definition of variety of dialgebras in the sense of Eilenberg which is equivalent to that introduced earlier by Kolesnikov.     

Semisimple classes and lower radicals of topological nonassociative algebras

In this article semisimple classes of topological algebras are characterized in a series of varieties of rings (in particular, all subvarieties of the varieties of alternative and Jordan algebras). Characterizations of semisimple classes of hereditary radicals are obtained, and the heredity of semisimple classes of radicals is demonstrated. It is proved that the construction of lower radicals stabilizes on the first infinite ordinal in the varieties of topological algebras considered.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 250–261, 1989.     

Free jordan algebras

Translated from Algebra i Logika, Vol. 27, No. 2, pp. 172–200, March–April, 1988.     

Free alternative algebras

The author's abstract of a dissertation submitted as a requirement for the advanced degree of Doctor of Physicomathematical Sciences. The dissertation was defended on Feb. 2, 1978, at a session of the Specialized Council D 002.23.01 of the Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR. The official examiners were: Corresponding Member of the Academy of Sciences of the USSR, Doctor of Physicomathematical Sciences A. I. Kostrikin; Doctor of Physicomathematical Sciences V. D. Mazurov; Doctor of Physicomathematical Sciences Yu. M. Ryabukhin.Translated from Matematicheskie Zametki, Vol. 25, No. 5, pp. 775–783, May, 1979.The author would llke to express his sincere gratitude to A. I. Shirshov for his unfailing interest and total support. The author is particularly indebted to his teacher, K. A. Zhevlakov, who greatly influenced his scientific interests.     

Free monadic Tarski algebras

The free monadic Tarski algebra FMT(n) with a finite set G of n free generators, was determined by A. Figallo in [7]. In this paper we indicate a formula, as a function of n, which is an easier way of calculating the number of elements of FMT(n). Received November 3, 1995; accepted in final form August 26, 1996.     

Free monadic Tarski algebras

A. Monteiro and L. Iturrioz [7], introduced the notion of monadic Tarski algebras as a generalization of the monadic Boolean algebras considered by P. Halmos [6]. They represent the algebraic version of a fragment of the functional monadic calculus. In this paper we determine the structure of the monadic Tarski algebras with a finite set of free generators.Presented by B. Jónsson.The author is grateful for many valuable comments and suggestions from the referee.     

Free affine Albert algebras

Dedicated to the 70th birthday of A. I. Shirshov.     

Free ω-complete algebras

In this paper we prove that for an arbitrary type Ω and an arbitrary strict ω-complete posetX the free ω-complete algebra of type Ω overX exists. Moreover, we prove, that for an arbitrary type (not necessary finitary!) this free algebra is, obtained by Adamek's construction in ω steps.