Automorphisms of categories of free algebras of some varieties

Plotkin, B.; Zhitomirski, Grigori

doi:10.1016/j.jalgebra.2006.07.028

Cited by 30 publications

(53 citation statements)

References 8 publications

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“…In this Subsection we will prove again one result of [10] . will be a variety of all the representations of groups over linear spaces over field k. We assume that k has a characteristic 0.…”

Section: Representations Of Groupsmentioning

confidence: 77%

“…The relation between strongly stable automorphisms and systems of verbal operations, which fulfill some specific conditions was discussed for the case of one-sorted algebras in [10] and [12]. Here we will explain this relation more formally and precisely.…”

Section: Strongly Stable Automorphisms and Systems Of Verbal Operationsmentioning

confidence: 98%

“…As in the proof of the second part of the Theorem 1 from [10], we check that σ is isomorphism. Definition 2.1 An automorphism ϒ of an arbitrary category K is inner, if it is isomorphic as a functor to the identity automorphism of the category K.…”

Section: Theorem 22mentioning

confidence: 99%

“…The following decomposition theorem is a generalization of the Theorem 2 from [10]. Hear we omit the considerations of the [10, Lemma 1, Lemma 2], which are valid for arbitrary categories, and give the short proof of this Theorem in which we consider only the category 0…”

Section: Theorem 22mentioning

confidence: 99%

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Automorphic Equivalence of Many-Sorted Algebras

Tsurkov

2015

Appl Categor Struct

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For every variety of universal algebras we can consider the category 0 of the finite generated free algebras of this variety. The quotient group A/Y, where A is a group of all the automorphisms of the category 0 and Y is a subgroup of all the inner automorphisms of this category measures difference between the geometric equivalence and automorphic equivalence of algebras from the variety . In Plotkin and Zhitomirski (J. Algebra 306(2), 2006) the simple and strong method of the verbal operations was elaborated on for the calculation of the group A/Y in the case when the is a variety of one-sorted algebras. In the first part of our paper (Sections 1, 2 and 3) we prove that this method can be used in the case of many-sorted algebras. In the second part of our paper (Section 4) we apply the results of the first part to the universal algebraic geometry of many-sorted algebras and prove again and refine the results of Plotkin (2003) and Tsurkov (Int. J. Algebra Comput. 17(5/6), 1263-1271, 2007) for these algebras. For example we prove in the Theorem 4.3 that the automorphic equivalence of algebras can be reduced to the geometric equivalence if we change the operations in one of these algebras. In the third part of this paper (Section 5) we consider some varieties of many-sorted algebras. We prove that automorphic equivalence coincides with geometric equivalence in the variety of all the actions of semigroups over sets and in the variety of all the automatons, because the group A/Y is trivial for these varieties. We also consider the variety of all the representations of groups and all the representations of Lie algebras. The group A/Y is not trivial for these varieties and for both these varieties we give an examples of the representations which are automorphically equivalent but not geometrically equivalent.A. Tsurkov

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Section: Representations Of Groupsmentioning

confidence: 77%

Section: Strongly Stable Automorphisms and Systems Of Verbal Operationsmentioning

confidence: 98%

Section: Theorem 22mentioning

confidence: 99%

Section: Theorem 22mentioning

confidence: 99%

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Automorphic Equivalence of Many-Sorted Algebras

Tsurkov

2015

Appl Categor Struct

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“…In our next paper [13], we apply this method and characterize automorphisms of categories Θ 0 (V) in the case V is the variety of all associative K−algebras, where K is a infinite field, and in the case V is the variety of all group representations in unital R− modules, where R is an associative commutative ring with unit.…”

mentioning

confidence: 99%

On Automorphisms of Categories of Universal Algebras

Plotkin

Zhitomirski

2007

Int. J. Algebra Comput.

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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 V-algebras. 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.

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On automorphisms of categories with applications to universal algebraic geometry

Zhitomirski

2023

Algebra Univers.

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Automorphisms of categories of free algebras of some varieties

Cited by 30 publications

References 8 publications

Automorphic Equivalence of Many-Sorted Algebras

Automorphic Equivalence of Many-Sorted Algebras

On Automorphisms of Categories of Universal Algebras

On automorphisms of categories with applications to universal algebraic geometry

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