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28
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.
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.
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.
Using decision problems in public key cryptography, preprint. http://www.sci.ccny.cuny.edu/˜shpil/wppkc.pdf Department of Mathematics, The City College of
"... Abstract. 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 ..."
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Abstract. 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. 1.
On the join of two pseudovarieties
 Semigroups, Automata and Languages
, 1996
"... The aim of this lecture is to survey some recent developments in the theory of finite semigroups. More precisely, we shall consider the following problem about pseudovarieties of semigroups: given two pseudovarieties V and W, find a description of their join V ∨ W (that is, of the pseudovariety they ..."
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The aim of this lecture is to survey some recent developments in the theory of finite semigroups. More precisely, we shall consider the following problem about pseudovarieties of semigroups: given two pseudovarieties V and W, find a description of their join V ∨ W (that is, of the pseudovariety they generate). This question is motivated by the theory of rational languages: it appears in a natural way when considering parallel operation of automata. The lattice of semigroup varieties (in Birkhoff’s sense) has been studied for a long time. In particular, it was proved that the join of two finitely based varieties might not be finitely based (see for instance Taylor [21]). Other important contributions in this area were given by Biryukov [11], Fennemore [12, 13] and Gerhard [14] who described the lattice of idempotent semigroup varieties, and by Polák [18] who described the lattice of varieties of completely regular semigroups. The problems appearing in the study of the lattice of pseudovarieties are analogous. Reiterman’s theorem [19] is the starting point of an equational theory for pseudovarieties: just as varieties are defined by identities, pseudovarieties are defined by pseudoidentities. Numerous algorithmic problems on pseudovarieties were proposed, for instance by Rhodes [20],
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
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.
IDENTITIES IN THE ALGEBRA OF PARTIAL MAPS
"... Abstract. We consider the identities of a variety of semigrouprelated algebras modeling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting sys ..."
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Abstract. We consider the identities of a variety of semigrouprelated algebras modeling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable. 1. Introduction. The variety of agreeable semigroups was introduced by the authors in [12] as an abstraction of some properties of the semigroup of partial maps on a set. As well as the usual multiplication modelling composition there is a second binary operation ∗ giving information on where two elements ‘agree’. For a set X we let PX denote the semigroup of all partial maps on X (acting on the left)
Inherently Nonfinitely Based Lattices
 Ann. Pure Appl. Logic
, 1997
"... We give a general method for constructing lattices L whose equational theories are inherently nonfinitely based. This means that the equational class (that is, the variety) generated by L is locally finite and that L belongs to no locally finite finitely axiomatizable equational class. We also provi ..."
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We give a general method for constructing lattices L whose equational theories are inherently nonfinitely based. This means that the equational class (that is, the variety) generated by L is locally finite and that L belongs to no locally finite finitely axiomatizable equational class. We also provide an example of a lattice which fails to be inherently nonfinitely based but whose equational theory is not finitely axiomatizable. Key words: lattice, finite axiomatizability, inherently nonfinitely based 1 Introduction A variety is a class of algebras which can be axiomatized by a set of equations (that is, by a set of universal sentences whose quantifierfree parts are equations between terms). According to a classical result of Garrett Birkho# 1 This research was partially supported by NSF grants no. DMS9500752 and DMS9400511. The second author thanks the University of Hawaii for its hospitality. Preprint submitted to Elsevier Preprint 12 November 1997 [5] the varieties are exac...
The Conjugacy Problem and Higman Embeddings
"... 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 problem. This solves a problem by D. Collins.