Related papers: On automorphisms of categories

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,…

Given a variety of universal algebras. A method is suggested for describing automorphisms of a category of free algebras of this variety. Applying this general method all automorphisms of such categories are found in two cases: 1) for the…

A new approach is suggested to characterize algebraically automorphisms of the category of free algebras of a given variety. It gives in many cases an answer to the problem set by the first of authors, if automorphisms of such a category…

In the first part of our paper (Sections 1, 2 and 3) we reprove results of B. Plotkin, G. Zhitomirski. On automorphisms of categories of free algebras of some varieties, Journal of Algebra, 306:2, (2006), 344 -- 367 for the case of…

In this paper, we prove that all automorphisms of categories of free S-acts are semi-inner, which solves a variation of a well known B. Plotkin's problem in the case of monoids. We also give a description of automorphisms of categories of…

In universal algebraic geometry the category of the finite generated free algebras of some fixed variety of algebras and the quotient group A/Y are very important. Here A is a group of all automorphisms of this category and Y is a group of…

In algebraic geometry over a variety of universal algebras $\Theta $, the group $Aut(\Theta ^{0})$ of automorphisms of the category $\Theta ^{0}$ of finitely generated free algebras of $\Theta $ is of great importance. In this paper,…

Universal algebraic geometry allows considering of geometric properties of every universal algebra. When two algebras have same algebraic geometry? We must consider the categories of algebraic closed sets of these algebras to answer this…

Prof. Boris I. Plotkin [arXiv:math.GM/0210194, arXiv:math.GM/0204245] drew attention to the question when an equivalence between two categories is isomorphic as a functor to an isomorphism between them. It turns out that it is quite…

In 1976 S. Shelah posed the following problem: for which variety V of algebras the automorphism group of any free algebra F from V of "large" infinite rank interprets by means of first-order logic set theory (according to his results, for…

We prove that the group of automorphisms of the category of free modules over Noterian rings is semi-inner and present also a short proof of a similar result for the variery of Lie algebras.

This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In…

Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra $\mathfrak{g}_{\infty}$ defined over $\mathbb{C}_{\infty}$-the reduction of $\mathbb{C}$ mod infinitely large prime, and show that…

A monomial algebra B is defined as a quotient of a polynomial ring by a monomial ideal, which is an ideal generated by a finite set of monomials. In this paper, we determine the automorphism group of a monomial algebra B, under the…

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 semi-inner and mirror automorphisms

Let (S, B) be the log pair associated with a projective completion of a smooth quasi-projective surface V . Under the assumption that the boundary B is irreducible, we obtain an algorithm to factorize any automorphism of V into a sequence…

We show that pseudovarieties of finitely generated algebras, i.e., classes $C$ of finitely generated algebras closed under finite products, homomorphic images, and subalgebras, can be described via a uniform structure $U$ on the free…

We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F be a free algebra of some variety A of linear algebras over K freely generated…

We describe the automorphism group of the endomorphism semigroup $\End(K[x_1,...,x_n])$ of ring $K[x_1,...,x_n]$ of polynomials over an {\it arbitrary} field $K$. A similar result is obtained for automorphism group of the category of…

This is a survey of recent progress in several areas of combinatorial algebra. We consider combinatorial problems about free groups, polynomial algebras, free associative and Lie algebras. Our main idea is to study automorphisms and, more…