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36
On the extension problem for partial permutations
 PROC. AMER. MATH. SOC
, 2003
"... A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no nontrivial group identity and a metabelian pseudovariety Q. For each of these pseudovarietie ..."
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A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no nontrivial group identity and a metabelian pseudovariety Q. For each of these pseudovarieties V, the inverse monoid pseudovariety Sl∗V has undecidable membership problem. As a consequence, it is proved that the pseudovariety operators ∗, ∗∗, m○, ♦, ♦n, andP do not preserve decidability. In addition, several joins, including A ∨ U, are shown to be undecidable.
Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras
 J. Algebra
"... Abstract. In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra K[x1,..., xm] in several variables over a field K of characteristic 0. The classical theorem of Weitzenböck states that the algebra of constants is finitely generated. (This ..."
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Abstract. In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra K[x1,..., xm] in several variables over a field K of characteristic 0. The classical theorem of Weitzenböck states that the algebra of constants is finitely generated. (This algebra coincides with the algebra of invariants of a single unipotent transformation.) In this paper we study the problem of finite generation of the algebras of constants of triangular linear derivations of finitely generated (not necessarily commutative or associative) algebras over K assuming that the algebras are free in some sense (in most of the cases relatively free algebras in varieties of associative or Lie algebras). In this case the algebra of constants also coincides with the algebra of invariants of some unipotent transformation. The main results are the following: 1. We show that the subalgebra of constants of a factor algebra can be lifted to the subalgebra of constants. 2. For all varieties of associative algebras which are not nilpotent in Lie sense the subalgebras of constants of the relatively free algebras of rank ≥ 2 are not finitely generated. 3. We describe the generators of the subalgebra of constants for all factor algebras K〈x, y〉/I modulo a GL2(K)invariant ideal I. 4. Applying known results from commutative algebra, we construct classes of automorphisms of the algebra generated by two generic 2 × 2 matrices. We obtain also some partial results on relatively free Lie algebras. 1.
Profinite Methods in Finite Semigroup Theory
, 2001
"... This paper is a survey of the authors' recent results in the theory of finite semigroups using profinite techniques. This involves the study of free profinite semigroups, whose structure encodes algebraic and combinatorial properties of finite semigroups. ..."
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This paper is a survey of the authors' recent results in the theory of finite semigroups using profinite techniques. This involves the study of free profinite semigroups, whose structure encodes algebraic and combinatorial properties of finite semigroups.
The Uniform Word Problem for Groups and Finite Rees Quotients of EUnitary Inverse Semigroups
, 2001
"... If C is a class of groups closed under taking subgroups, we show that the decidability of the uniform word problem for C is implied by the decidability of the membership problem for the class of nite Rees quotients of Eunitary inverse semigroups with maximal group image in C. The converse is sh ..."
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If C is a class of groups closed under taking subgroups, we show that the decidability of the uniform word problem for C is implied by the decidability of the membership problem for the class of nite Rees quotients of Eunitary inverse semigroups with maximal group image in C. The converse is shown if C is a pseudovariety. When C is a pseudovariety, the above problems are shown to be equivalent to the problem of embedding a nite labeled graph in the Cayley graph of a group in C. This latter problem is shown to be equivalent to deciding whether a nite labeled graph is a Schutzenberger graph of an Eunitary inverse semigroup with maximal group image in C. 1.
Using decision problems in public key cryptography
, 2007
"... There are several public key establishment protocols as well as complete public key cryptosystems based on allegedly hard problems from combinatorial (semi)group theory known by now. Most of these problems are search problems, i.e., they are of the following nature: given a property P and the info ..."
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Cited by 3 (3 self)
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There are several public key establishment protocols as well as complete public key cryptosystems based on allegedly hard problems from combinatorial (semi)group theory known by now. Most of these problems are search problems, i.e., they are of the following nature: given a property P and the information that there are objects with the property P, find at least one particular object with the property P. So far, no cryptographic protocol based on a search problem in a noncommutative (semi)group has been recognized as secure enough to be a viable alternative to established protocols (such as RSA) based on commutative (semi)groups, although most of these protocols are more efficient than RSA is. In this paper, we suggest to use decision problems from combinatorial group theory as the core of a public key establishment protocol or a public key cryptosystem. Decision problems are problems of the following nature: given a property P and an object O, find out whether or not the object O has the property P. By using a popular decision problem, the word problem, we design a cryptosystem with the following features: (1) Bob transmits to Alice an encrypted binary sequence which Alice decrypts correctly with probability “very close ” to 1; (2) the adversary, Eve, who is granted arbitrarily high (but fixed) computational speed, cannot positively identify (at least, in theory), by using a “brute force attack”, the “1” or “0 ” bits in Bob’s binary sequence. In other words: no matter what computational speed we grant Eve at the outset, there is no guarantee that her “brute force attack ” program will give a conclusive answer (or an answer which is correct with overwhelming probability) about any bit in Bob’s sequence.
Finitely based, finite sets of words
 Internat. J. Algebra Comput
"... For W a finite set of words, we consider the Rees quotient of a free monoid with respect to the ideal consisting of all words that are not subwords of W. This monoid is denoted by S(W). It is shown that for every finite set of words W, there are sets of words U ⊃ W and V ⊃ W such that the identities ..."
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For W a finite set of words, we consider the Rees quotient of a free monoid with respect to the ideal consisting of all words that are not subwords of W. This monoid is denoted by S(W). It is shown that for every finite set of words W, there are sets of words U ⊃ W and V ⊃ W such that the identities satisfied by S(V) are finitely based and those of S(U) are not finitely based (regardless of the situation for S(W)). The first examples of finitely based (not finitely based) aperiodic finite semigroups whose direct product is not finitely based (finitely based) are presented and it is shown that every monoid of the form S(W) with fewer than 9 elements is finitely based and that there is precisely one not finitely based 9 element example. 1
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.
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
The conjugacy problem and Higman embeddings
 Mem. Amer. Math. Soc
"... For every finitely generated recursively presented group G we construct a finitely presented group H containing G such that G is (Frattini) embedded into H and the group H has solvable conjugacy problem if and only if G has solvable conjugacy problem. Moreover G and H have the same r.e. Turing degre ..."
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For every finitely generated recursively presented group G we construct a finitely presented group H containing G such that G is (Frattini) embedded into H and the group H has solvable conjugacy problem if and only if G has solvable conjugacy problem. Moreover G and H have the same r.e. Turing degrees of the conjugacy