Results 1  10
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13
with the same algebraic geometry
 Proceedings of the International Conference on Mathematical Logic, Algebra and Set Theory, dedicated to 100 anniversary of P.S.Novikov, Proceedings MIAN
, 2002
"... Abstract. Some basic notions of classical algebraic geometry can be defined in arbitrary varieties of algebras Θ. For every algebra H in Θ one can consider algebraic geometry in Θ over H. Correspondingly, algebras in Θ are considered with the emphasis on equations and geometry. We give examples of g ..."
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Abstract. Some basic notions of classical algebraic geometry can be defined in arbitrary varieties of algebras Θ. For every algebra H in Θ one can consider algebraic geometry in Θ over H. Correspondingly, algebras in Θ are considered with the emphasis on equations and geometry. We give examples of geometric properties of algebras in Θ and of geometric relations between them. The main problem considered in the paper is when different H1 and H2 have the same geometry.
Plotkin Automorphisms of categories of free modules and free Lie algebras
"... Abstract. Let Θ 0 be a category of finitly generated free algebras in the variety of algebras Θ. Solutions to problems in algebraic geometry over Θ are often determined by the structure of the group of automorphisms AutΘ 0 of category Θ 0. Here we consider two varieties Θ: noetherian modules and Lie ..."
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Cited by 6 (2 self)
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Abstract. Let Θ 0 be a category of finitly generated free algebras in the variety of algebras Θ. Solutions to problems in algebraic geometry over Θ are often determined by the structure of the group of automorphisms AutΘ 0 of category Θ 0. Here we consider two varieties Θ: noetherian modules and Lie algebras. We show that every automorphism in AutΘ 0, where Θ is the variety of modules over noetherian rings, is semiinner. A similar result for the variety of Lie algebras over a infinite field has been recently obtained in [9]. Here we present a different approach allowing us to shorten the proof in the Lie case.
AUTOMORPHISMS OF THE SEMIGROUP OF ENDOMORPHISMS OF FREE ASSOCIATIVE ALGEBRAS
, 2005
"... Abstract. Let A = A(x1,...,xn) be a free associative algebra in the variety of associative algebras A freely generated over K by a set X = {x1,...,xn}, End A be the semigroup of endomorphisms of A, and Aut EndA be the group of automorphisms of the semigroup EndA. We investigate the structure of the ..."
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Cited by 2 (0 self)
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Abstract. Let A = A(x1,...,xn) be a free associative algebra in the variety of associative algebras A freely generated over K by a set X = {x1,...,xn}, End A be the semigroup of endomorphisms of A, and Aut EndA be the group of automorphisms of the semigroup EndA. We investigate the structure of the groups Aut EndA and Aut A ◦ , where A ◦ is the category of finitely generated free algebras from A. We prove that the group Aut EndA is generated by semiinner and mirror automorphisms of EndF and the group Aut A ◦ is generated by semiinner and mirror automorphisms of the category A ◦. This result solves an open Problem formulated in [14]. 1.
Automorphisms of the semigroup of endomorphisms of free algebras of homogeneous varieties, Preprint. Atxiv: math. RA//0511654v1
, 2005
"... Abstract. We consider homogeneous varieties of linear algebras over an associativecommutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F = F(x1,...,xn) be a free algebra of some variety Θ of linear algebras over K freely generated by a set X = {x1,...,xn}, EndF be t ..."
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Cited by 2 (2 self)
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Abstract. We consider homogeneous varieties of linear algebras over an associativecommutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F = F(x1,...,xn) be a free algebra of some variety Θ of linear algebras over K freely generated by a set X = {x1,...,xn}, EndF be the semigroup of endomorphisms of F, and Aut EndF be the group of automorphisms of the semigroup EndF. We investigate structure of the group Aut End F and its relation to the algebraical and categorical equivalence of algebras from Θ. We define a wide class of R1MFdomains containing, in particular, Bezout domains, unique factorization domains, and some other domains. We show that every automorphism Φ of semigroup EndF, where F is a free finitely generated Lie algebra over an R1MFdomain, is semiinner. This solves the Problem 5.1 left open in [21]. As a corollary, semiinnerity of all automorphism of the category of free Lie algebras over R1MFdomains is obtained. Relations between categorical and geometrical equivalence of Lie algebras over R1MFdomains are clarified. The group Aut EndF for the variety of mnilpotent associative algebras over R1MFdomains is described. As a consequence, a complete description of the group of automorphisms of the full matrix semigroup of n × n matrices over R1MFdomains is obtained. We give an example of the variety Θ of linear algebras over a Dedekind domain such that not all automorphisms of Aut EndF are quasiinner. The results obtained generalize the previous studies of various special cases of varieties of linear algebras over infinite fields. 1.
On automorphisms of categories of universal algebras. Preprint. Arxiv: math.CT/0411408
, 2004
"... Abstract. Given a variety V of universal algebras. A new approach is suggested to characterize algebraically automorphisms of the category of free Valgebras. It gives in many cases an answer to the problem set by the first of authors, if automorphisms of such a category are inner or not. This quest ..."
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Abstract. Given a variety V of universal algebras. A new approach is suggested to characterize algebraically automorphisms of the category of free Valgebras. It gives in many cases an answer to the problem set by the first of authors, if automorphisms of such a category are inner or not. This question is important for universal algebraic geometry [5, 9]. Most of results will actually be proved to hold for arbitrary categories with a represented forgetful functor.
COMPUTING AUTOMORPHISMS OF SEMIGROUPS
"... ABSTRACT. In this paper an algorithm is presented that can be used to calculate the automorphism group of a finite transformation semigroup. The general algorithm employs a special method to compute the automorphism group of a finite simple semigroup. As applications of the algorithm all the automor ..."
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ABSTRACT. In this paper an algorithm is presented that can be used to calculate the automorphism group of a finite transformation semigroup. The general algorithm employs a special method to compute the automorphism group of a finite simple semigroup. As applications of the algorithm all the automorphism groups of semigroups of order at most 7 and of the multiplicative semigroups of some group rings are found. We also consider which groups occur as the automorphism groups of semigroups of several distinguished types. 1.
A PROBLEM OF B. PLOTKIN FOR SACTS: AUTOMORPHISMS OF CATEGORIES OF FREE
, 2006
"... Abstract. In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ 0) of automorphisms of the category Θ 0 of finitely generated free algebras of Θ is of great importance. In this paper, we prove that all automorphisms of categories of free Sacts are semiinner, which solves a ..."
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Abstract. In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ 0) of automorphisms of the category Θ 0 of finitely generated free algebras of Θ is of great importance. In this paper, we prove that all automorphisms of categories of free Sacts are semiinner, which solves a variation of Problem 12 in [12] for monoids. We also give a description of automorphisms of categories of finitely generated free algebras of varieties of unary algebras, and show that among varieties of unary algebras only the variety of monounary algebras is perfect [7]. 1.
Hebrew University
, 2002
"... We prove that every automorphism of the category of free Lie algebras is a semiinner automorphism. This solves the problem 3.9 from [19] for Lie algebras. ..."
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We prove that every automorphism of the category of free Lie algebras is a semiinner automorphism. This solves the problem 3.9 from [19] for Lie algebras.
AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE COMMUTATIVE ALGEBRA
, 903
"... Abstract. We describe the automorphism group of the endomorphism semigroup End(K[x1,..., xn]) of ring K[x1,..., xn] of polynomials over an arbitrary field K. A similar result is obtained for automorphism group of the category of finitely generated free commutativeassociative algebras of the variety ..."
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Abstract. We describe the automorphism group of the endomorphism semigroup End(K[x1,..., xn]) of ring K[x1,..., xn] of polynomials over an arbitrary field K. A similar result is obtained for automorphism group of the category of finitely generated free commutativeassociative algebras of the variety CA commutative algebras. This solves two problems posed by B. Plotkin ( [18], Problems 12 and 15). More precisely, we prove that if ϕ ∈ AutEnd(K[x1,..., xn]) then there exists a semilinear automorphism s: K[x1,..., xn] → K[x1,..., xn] such that ϕ(g) = s ◦g ◦s −1 for any g ∈ End(K[x1,..., xn]). This extends the result by A. Berzins obtained for an infinite field K. 1.