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On automorphisms of categories of free algebras of some varieties
 Journal of Algebra
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The group of automorphisms of the category of free associative algebras
"... In this paper, the problem formulated in [8] is solved. We prove, that the group of automorphisms of the category of free associative algebras is generated by semiinner and mirror automorphisms. 1 ..."
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In this paper, the problem formulated in [8] is solved. We prove, that the group of automorphisms of the category of free associative algebras is generated by semiinner and mirror automorphisms. 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.
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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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
, 2005
"... 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 semig ..."
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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.
A generalization and a new proof of Plotkin’s Reduction theorem
"... Abstract. It is known that Plotkin’s reduction theorem is very important for his theory of universal algebraic geometry [1, 2]. It turns out that this theorem can be generalized to arbitrary categories containing two special objects and in this case its proof becomes considerable more simple. This n ..."
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Abstract. It is known that Plotkin’s reduction theorem is very important for his theory of universal algebraic geometry [1, 2]. It turns out that this theorem can be generalized to arbitrary categories containing two special objects and in this case its proof becomes considerable more simple. This new proof and applications are the subject of the present paper.
ACTION TYPE GEOMETRICAL EQUIVALENCE OF REPRESENTATIONS OF GROUPS.
, 2008
"... In the paper we prove (Theorem 8.1) that there exists a continuum of non isomorphic simple modules over KF2, where F2 is a free group with 2 generators (compare with [Ca] where a continuum of non isomorphic simple 2generated groups is constructed). Using this fact we give an example of a non action ..."
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In the paper we prove (Theorem 8.1) that there exists a continuum of non isomorphic simple modules over KF2, where F2 is a free group with 2 generators (compare with [Ca] where a continuum of non isomorphic simple 2generated groups is constructed). Using this fact we give an example of a non action type logically Noetherian representation (Section 9). In general, the topic of this paper is the action type algebraic geometry of representations of groups. For every variety of algebras Θ 1 and every algebra H ∈ Θ we can consider an algebraic geometry in Θ over H. Algebras in Θ may be many sorted (not necessarily one sorted) algebras. A set of sorts Γ is fixed for each Θ. This theory can be applied to the variety of representations of groups over fixed commutative ring K with unit. We consider a representation as two sorted algebra (V,G), where V is a Kmodule, and G is a group acting on
AUTOMORPHISMS OF THE SEMIGROUP OF ALL ENDOMORPHISMS OF FREE ALGEBRAS
, 2005
"... Abstract. Last years a number of papers were devoted to describing automorphisms of semigroups of endomorphisms of free finitely generated universal algebras of some varieties: groups, semigroups, associative commutative algebras, inverse semigroups, modules, Lie algebras and some others (see refere ..."
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Abstract. Last years a number of papers were devoted to describing automorphisms of semigroups of endomorphisms of free finitely generated universal algebras of some varieties: groups, semigroups, associative commutative algebras, inverse semigroups, modules, Lie algebras and some others (see references). All these researches were inspired by the questions prof. B. Plotkin set in connection with so called universal algebraic geometry [11], [12]. The aim of this paper is to suggest a method of describing Aut End(F) for a free algebra F of an arbitrary variety of universal algebras. This method allows to obtain easily all known results as well as new ones. 1. introduction The aim of this paper is to suggest a method of describing Aut End(F) where F is a free algebra of an arbitrary variety of universal algebras. The interest to this problem was inspired in most cases by questions which were set by B. Plotkin in connection with universal algebraic geometry developed in his papers (see for example [11], [12]).