A Lemma on a Free Group

Morimoto, Akihiko

doi:10.1017/s0027763000018146

Cited by 4 publications

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“…More precisely -w 1 = 1 (1) , w 0 = 0 (2) . By [4] we have two possibilities: w · x (1) (1) ∈ F x (1) :…”

Section: Theorem 53 A/y ∼ =Autk For the Variety Of All The Representmentioning

confidence: 99%

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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“…More precisely -w 1 = 1 (1) , w 0 = 0 (2) . By [4] we have two possibilities: w · x (1) (1) ∈ F x (1) :…”

Section: Theorem 53 A/y ∼ =Autk For the Variety Of All The Representmentioning

confidence: 99%

Automorphic Equivalence of Many-Sorted Algebras

Tsurkov

2015

Appl Categor Struct

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“…(1) = S X (1) , (F (X)) (2) = S X (1) • X (2) ∪ X (2) , where S X (1) • X (2) = s • x (2) | s ∈ S X (1) , x (2) ∈ X (2) , S X (1) is a free semigroup generated by the set of free generators X (1) .…”

Section: Actions Of Semigroups Over Setsmentioning

confidence: 99%

“…and the relation Q -the minimal equivalence in (F (X)) (2) which contain R. Quotient set (F (X)) (2) /Q has X (2) elements.…”

Section: Actions Of Semigroups Over Setsmentioning

confidence: 99%

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Automorphic Equivalence of Many-Sorted Algebras

Tsurkov

2013

Preprint

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“…with gi ~-g' for each i, and equipped with the bracket [gi,gj] = gi+j' with gi+j = ~ if i + j > k [2], [14]. We will also use the Taylor representation…”

Section: II Definitionmentioning

confidence: 99%

Completely integrable systems of evolution equations on KAC moody lie algebras

Dhooghe

Geometric Aspects of the Einstein Equations and Integrable Systems

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A Lemma on a Free Group

Cited by 4 publications

References 0 publications

Automorphic Equivalence of Many-Sorted Algebras

Automorphic Equivalence of Many-Sorted Algebras

Automorphic Equivalence of Many-Sorted Algebras

Completely integrable systems of evolution equations on KAC moody lie algebras

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