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TWISTED CONJUGACY IN FREE GROUPS AND MAKANIN’S QUESTION
, 2008
"... Abstract. We discuss the following question of G. Makanin from “Kourovka notebook”: does there exist an algorithm to determine is for an arbitrary pair of words U and V of a free group Fn and an arbitrary automorphism ϕ ∈ Aut(Fn) the equation ϕ(X)U = V X solvable in Fn? We give the affirmative answe ..."
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Abstract. We discuss the following question of G. Makanin from “Kourovka notebook”: does there exist an algorithm to determine is for an arbitrary pair of words U and V of a free group Fn and an arbitrary automorphism ϕ ∈ Aut(Fn) the equation ϕ(X)U = V X solvable in Fn? We give the affirmative answer in the case when an automorphism is virtually inner, i.e. some its nonzero power is an inner automorphism of Fn. 1. Conjugacy and twisted conjugacy Suppose G is a group given by a presentation in generators and defining relations. Three following decision problems formulated by M. Dehn [5] in 1912 (see also [8, Ch. 1, § 2; Ch. 2, § 1]) are fundamental in the group theory. Word problem: Does there exist an algorithm to determine if an arbitrary group word W given in the generators of G defines the identity element of G? Conjugacy problem: Does there exist an algorithm to determine is an arbitrary pair of group words U, V in the generators of G define conjugate elements of G? Isomorphism problem: Does there exist an algorithm to determine for any two arbitrary finite presentations whether the groups they present are isomorphic or not? All three of these problems have negative answers in general (see, for example [1], [3, Ch. 6.7]). These results together with solutions of Dehn’s problems in restricted cases have been of central importance in the combinatorial group theory. For this reason combinatorial group theory has always searched for and studied classes of groups in which
GRÖBNERSHIRSHOV BASIS FOR THE BRAID SEMIGROUP
, 806
"... Abstract. We found GröbnerShirshov basis for the braid semigroup B + n+1. It gives a new algorithm for the solution of the word problem for the braid semigroup and so for the braid group. 1. ..."
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Abstract. We found GröbnerShirshov basis for the braid semigroup B + n+1. It gives a new algorithm for the solution of the word problem for the braid semigroup and so for the braid group. 1.
Solid Pseudovarieties
"... Departing from the well known notion of a solid variety, we introduce the notion of a solid pseudovariety and give equational characterizations of solid pseudovarieties using both identities and pseudoidentities. We also establish a connection between lattices of solid varieties and pseudovariet ..."
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Departing from the well known notion of a solid variety, we introduce the notion of a solid pseudovariety and give equational characterizations of solid pseudovarieties using both identities and pseudoidentities. We also establish a connection between lattices of solid varieties and pseudovarieties. Introduction Recall that a pseudovariety is a class of finite algebras that is closed under the formation of homomorphic images, subalgebras, and finitary direct products. The notion of a pseudovariety is rather natural per se but it has got a strong additional motivation coming from recent developments in finite automata and formal languages [11, 20, 2]. It is clear from the very definition that the notion of a pseudovariety is somehow related to the classical notion of a variety. Therefore it appears to be reasonable to extend to the pseudovariety setting the approaches that have been proved to be successful for studying varieties. One of the most interesting concepts recently aris...
Algorithmic Problems for Finite Groups and Finite 0Simple Semigroups
, 1996
"... It is shown that the embeddability of a finite 4nilpotent semigroup into a 0simple finite semigroup with maximal groups from a pseudovariety V is decidable if and only if the universal theory of the class V is decidable. We show that it is impossible to replace 4 by 3 in this statement. We also sho ..."
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It is shown that the embeddability of a finite 4nilpotent semigroup into a 0simple finite semigroup with maximal groups from a pseudovariety V is decidable if and only if the universal theory of the class V is decidable. We show that it is impossible to replace 4 by 3 in this statement. We also show that if the membership in V is decidable then the membership in the pseudovariety generated by the class of all finite 0simple semigroups with subgroups from V is decidable while the membership in the quasivariety generated by this class of 0simple semigroups may be undecidable. 1 Introduction One of the most important classes of semigroups is the class of 0simple finite semigroups. Recall that a semigroup is called 0simple if it does not have ideals except itself and possibly f0g. Every finite semigroup may be obtained from 0simple semigroups by a sequence of ideal extensions. The classic theorem of Sushkevich [3] (which was arguably the first theorem in the algebraic theory of sem...
Hyperidentities of Type (2)
, 1997
"... ace se budeme drzet teto denice. Pritom, i kdybychom povolili hyperidentity vce typu pri zachovan vyhodnocovan v jedinem pevnem typu, vyjadrovac sla teto logiky by se v podstate nezmenila. Druhy prstup povoluje brat systemy hyperidentit libovolnych typu a dosazovat do nich termy vsech typu. Ukazuje ..."
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ace se budeme drzet teto denice. Pritom, i kdybychom povolili hyperidentity vce typu pri zachovan vyhodnocovan v jedinem pevnem typu, vyjadrovac sla teto logiky by se v podstate nezmenila. Druhy prstup povoluje brat systemy hyperidentit libovolnych typu a dosazovat do nich termy vsech typu. Ukazuje se, ze tmto zpusobem lze popsat vce objektu nez vyse uvedenym, ovsem za tyto objekty jiz nemuzeme brat variety, ale variety variet neboli hypervariety, poprve denovane v [Neu78]. Ani v tomto prpade ale neztracme uplnost logiky, nebot' k dokazovan zde lze pouzt aparat teorie mnohodruhov ych algeber, presneji dokazovan ve variete klonu. Teorie hyperidentit nen zdaleka jedinou moznost, jak zobecnit ekvacionaln logiku do jazyka druheho radu. Dals z prirozenych variant je naprklad uzit Mal'cevovsk ych podmnek. Rkame, ze trda variet K je denovana silnou Mal'cevovskou
Algorithms for Computing Finite Semigroups
, 1997
"... The aim of this paper is to present algorithms to compute nite semigroups. ..."
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The aim of this paper is to present algorithms to compute nite semigroups.
The Ekaterinburg seminar “Algebraic Systems”: 40 years of activities
"... The aim of the present article is to give a characterization of distinctive features of a scientific seminar founded and led by the author as well as to show the main sides of its activities during four decades. 1. Origin of the seminar The seminar indicated in the title of the article started its w ..."
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The aim of the present article is to give a characterization of distinctive features of a scientific seminar founded and led by the author as well as to show the main sides of its activities during four decades. 1. Origin of the seminar The seminar indicated in the title of the article started its work in 1966. By that time several younger researchers were grouped around the present writer at the Ural State University. Naturally, I discussed with each of them different problems belonged to the area of his/her investigations. However, besides these individual meetings, a usual need in such situations had arisen – to gather regularly for discussing results obtained and, in general, for diverse discussions concerning our investigations in algebra. It should be noted that Ekaterinburg (Sverdlovsk from 1924 to 1991) is a city with considerable scientific algebraic traditions. In a great degree their beginning is due to activities of Professor P. G. Kontorovich (1905–1968) who worked for several decades at the Ural State University and was one of the leading Soviet algebraists. The scientific school created by P. G.