Showing papers in "International Journal of Algebra and Computation in 2000"
TL;DR: 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 finitelybased 9 element example.
Abstract: 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 resulting 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.
49 citations
43 citations
TL;DR: It is proved that every finite subgroup of a hyperbolic group G can be conjugated to a 2δ+1 neighborhood of the identity element, where δ is thehyperbolicity constant for G with respect to a given generating set.
Abstract: We prove that every finite subgroup of a hyperbolic group G can be conjugated to a 2δ+1 neighborhood of the identity element, where δ is the hyperbolicity constant for G with respect to a given generating set. This gives an upper bound for the size of such finite subgroups in terms of δ and the number of generators for G.
40 citations
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TL;DR: Allfinitely based words in a two-letter alphabet are described and some necessary and some sufficient conditions for a set of words to be finitely based are found.
Abstract: Let W be a finite language and let Wc be the closure of W under taking subwords. Let S(W) denote the Rees quotient of a free monoid over the ideal consisting of all words that are not in Wc. We call W finitely based if the monoid S(W) is finitely based. Although these semigroups have easy structure they behave "generically" with respect to the finite basis property [6]. In this paper, we describe all finitely based words in a two-letter alphabet. We also find some necessary and some sufficient conditions for a set of words to be finitely based.
37 citations
TL;DR: A simple algorithm for computing Kashiwara's global crystal basis of a finite-dimensional irreducible representation of Uq(sln) is described.
Abstract: We describe a simple algorithm for computing Kashiwara's global crystal basis of a finite-dimensional irreducible representation of Uq(sln). Resume: Nous decrivons un algorithme simple pour calculer la base cristalline globale de Kashiwara d'une representation irreductible de dimension finie de Uq(sln).
35 citations
33 citations
TL;DR: This paper proves that for any special subgroup AT of A, A(2)∩AT=(AT)(2) is a right-angled Artin group and shows that locally reducible Artin groups have a CAT(0) geometry.
Abstract: Given an Artin system (A,S), a conjecture of Tits states that the subgroup A(2) of A generated by the squares of the generators in S is subject only to the obvious commutator relations between generators. In particular, A(2) is a right-angled Artin group. We prove this conjecture for a class of infinite type Artin groups, called locally reducible Artin groups, for which the associated Deligne complex has a CAT(0) geometry. We also prove that for any special subgroup AT of A, A(2)∩AT=(AT)(2).
32 citations
TL;DR: Finite automata are extended by adding an element of a given group to each of their configurations, and a new characterization of the context-free languages as soon as the considered group is the binary free group.
Abstract: Finite automata are extended by adding an element of a given group to each of their configurations. An input string is accepted if and only if the neutral element of the group is associated to a final configuration reached by the automaton. We get a new characterization of the context-free languages as soon as the considered group is the binary free group. The result cannot be carried out in the deterministic case. Some remarks about finite automata over other groups are also presented.
29 citations
TL;DR: In this paper, it was shown that the lattice of permutations on a n -element set is bounded, and a characterization of the class of bounded lattices in terms of arrows relations defined on the join-irreducible elements of a lattice was given.
Abstract: The purpose of this paper is to show that the lattice of permutations on a n -element set is bounded. This result strengthens the semi-distributive nature of the lattice . To prove this property, we use a characterization of the class of bounded lattices in terms of arrows relations defined on the join-irreducible elements of a lattice or, more precisely, in terms of the A-table of a lattice.
29 citations
TL;DR: A share package of functions for computing with finite, permutation crossed modules, cat1-groups and their morphisms, written using the $\mathsf {GAP}$ group theory programming language is described.
Abstract: In this paper we describe a share package $\mathsf {XMOD}$ of functions for computing with finite, permutation crossed modules, cat1-groups and their morphisms, written using the $\mathsf {GAP}$ group theory programming language. The category XMod of crossed modules is equivalent to the category Cat1 of cat1-groups and we include functions emulating the functors between these categories. The monoid of derivations of a crossed module ${\mathcal X}$ , and the corresponding monoid of sections of a cat1-group ${\mathcal C}$ , are constructed using the Whitehead multiplication. The Whitehead group of invertible derivations, together with the group of automorphisms of ${\mathcal X}$ , are used to construct the actor crossed module of ${\mathcal X}$ which is the automorphism object in XMod. We include a table of the 350 isomorphism classes of cat1-structures on groups of order at most 30.
23 citations
TL;DR: This paper studies the pseudovariety of monoids corresponding to the variety of languages generated by the polynomial closure of the varieties of H-languages and the Pseudo-Pseudo-Monoid equivalent corresponding to ordered monoids, and obtains a basis of ordered pseudoidentities for such positive varieties.
Abstract: Suppose H is a pseudovariety of groups This paper studies the pseudovariety of monoids corresponding to the variety of languages generated by the polynomial closure of the variety of H-languages (which is shown to be J*H via a short syntactic argument) and the pseudovariety of ordered monoids corresponding to the polynomial closure of the variety of H-languages We also explore alternative descriptions of these pseudovarieties under additional hypotheses Decidability results are given In addition, we study the positive varieties of pro-V open and closed recognizable sets for a pseudovariety V of monoids In particular, we obtain a basis of ordered pseudoidentities for such positive varieties For pseudovarieties of groups H, we relate these positive varieties to the polynomial closure of the H-languages Again, decidability results are obtained
TL;DR: It is proved that the generalized Post Correspondence Problem (GPCP) is decidable for marked morphisms and gives as a corollary a shorter proof for the decidability of the binary PCP.
Abstract: We prove that the generalized Post Correspondence Problem (GPCP) is decidable for marked morphisms. This result gives as a corollary a shorter proof for the decidability of the binary PCP, proved in 1982 by Ehrenfeucht, Karhumaki and Rozenberg.
TL;DR: A generalized version of small cancellation theory is developed which is applicable to specific types of high-dimensional simplicial complexes, and arbitrary dimensional versions of the Poincare construction and the Cayley complex are described.
Abstract: In this article, a generalized version of small cancellation theory is developed which is applicable to specific types of high-dimensional simplicial complexes. The usual results on small cancellation groups are then shown to hold in this new setting with only slight modifications. For example, arbitrary dimensional versions of the Poincare construction and the Cayley complex are described.
TL;DR: It is shown that the order of a finite simple of Lie type is bounded by a small constant power of its exponent, which confirms, in a strengthened form, a conjecture of Vaughan-Lee and Zel'manov on the order and exponent of almost simple groups.
Abstract: We show that the order of a finite simple of Lie type is bounded by a small constant power of its exponent. This confirms, in a strengthened form, a conjecture of Vaughan-Lee and Zel'manov on the order and exponent of almost simple groups. We also obtain various structural restrictions on groups of polynomial index growth. Combining the above results we construct finitely generated residually finite groups of polynomial index growth which are neither linear nor boundedly generated. This answers questions of Segal and Platonov–Rapinchuk respectively. A further question of Platonov–Rapinchuk concerning a weakened polynomial index growth assumption is also answered.
TL;DR: Hyperdecidability is studied for pseudovarieties of completely simple semigroups as a means of illustrating the kind of techniques that may be used in this area.
Abstract: Recent results related to hyperdecidability and their applications to decidability of semidirect products are reviewed. Hyperdecidability is studied for pseudovarieties of completely simple semigroups as a means of illustrating the kind of techniques that may be used in this area.
TL;DR: This paper provides a complete analysis of these identities by giving a concrete description of the free theories in the variety axiomatized by the Conway identities and any given subcollection of the group-identities.
Abstract: It has been shown that the axioms of iteration theories capture the equational properties of iteration in several different models related to computer science. Iteration theories are axiomatizable by the Conway identities and the group-identities corresponding to the finite (simple) groups. In this paper we provide a complete analysis of these identities by giving a concrete description of the free theories in the variety axiomatized by the Conway identities and any given subcollection of the group-identities. It follows that when the group-identities are effectively given, the equational theory of the variety is decidable.
TL;DR: A characterization of the variety of languages recognized by semigroups in V is given and some join decompositions of pseudovarieties are derived.
Abstract: This paper is concerned with the structure of semigroups of implicit operations on various subpseudovarieties V of ∩ , where and are the pseudovarieties of all semigroups S in which each regular -class is, respectively, a rectangular group and a group, and where is the pseudovariety of semigroups locally in As an application, we give a characterization of the variety of languages recognized by semigroups in V and derive some join decompositions of pseudovarieties.
TL;DR: This work considers the G -spectra of the varieties of algebras generated by a single two-element algebra and shows that such spectra are either polynomial, exponential, or at least doubly exponential as a function of k.
Abstract: The G -spectrum of a class of algebras is the function that counts, up to isomorphism, the number of at most k -generated algebras in the class, for k =1,2,… . We consider the G -spectra of the varieties of algebras generated by a single two-element algebra. We show that such spectra are either polynomial, exponential, or at least doubly exponential as a function of k . For the varieties that have polynomial or exponential G -spectra we provide an exact formula for the function.
TL;DR: A criterion for a group to be representable as a vertex stabilizer of a transitive action on a locally finite graph is proved and it is shown that actions with finite vertex stabilizers are open in a natural topology.
Abstract: We prove a criterion for a group to be representable as a vertex stabilizer of a transitive action on a locally finite graph. We show that actions with finite vertex stabilizers are open in a natural topology.
TL;DR: In this paper, the Dehn functions of amalgamations were studied and the notion of strongly undistorted subgroups was introduced, and conditions under which taking an amalgamation does not increase the dehn function were given, generalizing the combination theorem of Bestvina and Feighn.
Abstract: We study the Dehn functions of amalgamations, introducing the notion of strongly undistorted subgroups. Using this, we give conditions under which taking an amalgamation does not increase the Dehn function, generalizing one aspect of the combination theorem of Bestvina and Feighn. To obtain examples of strongly undistorted subgroups, we define and study the relative Dehn function of pairs of groups. As a result we obtain a new method of constructing examples of pairs of groups that are relatively hyperbolic in the sense of Farb.
TL;DR: This work constructs an explicit realization of the free left self-distributive system on any number of generators in the charged braid group, an extension of Artin's braid groups B∞ with a simple geometrical interpretation.
Abstract: Starting from a certain monoid that describes the geometry of the left self-distributivity identity, we construct an explicit realization of the free left self-distributive system on any number of generators. This realization lives in the charged braid group, an extension of Artin's braid group B∞ with a simple geometrical interpretation.
TL;DR: It is shown that it is undecidable in general whether or not a finitely presented monoid with a polynomial-time decidable word problem has finite derivation type (FDT).
Abstract: By exploiting a new technique for proving undecidability results developed by A. Sattler-Klein in her doctoral dissertation (1996) it is shown that it is undecidable in general whether or not a finitely presented monoid with a polynomial-time decidable word problem has finite derivation type (FDT). This improves upon the undecidability result of R. Cremanns and F. Otto (1996), which was based on the undecidability of the word problem for the finitely presented monoids considered.
TL;DR: This paper will show whether or not certain equations, which are generalizations of the equation studied by Lyndon, have solutions over Z2.
Abstract: Let G be a finite group. A Corollary of a result of Gerstenhaber and Rothaus [3] states that all equations in one variable over G with exponent sum nonzero are solvable over G. Lyndon [6] studied equations in one variable with exponent sum zero over Zm. He showed that there was a relatively simple equation with no solution over Z2. In this paper we will initiate a study of equations with exponent sum zero over Z2. In particular we will show whether or not certain equations, which are generalizations of the equation studied by Lyndon, have solutions over Z2. We are grateful to the referee whose suggestion for the case when p<0
TL;DR: This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternARY algebra L, so that X freely generates L.
Abstract: A ternary algebra is a bounded distributive lattice with additonal operations e and ~ that satisfies (a+b)~=a~b~, a~~=a, e≤a+a~, e~= e and 0~=1. This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternary algebra L, so that X freely generates L. With this characterization, the free ternary algebra on one free generator is displayed. The poset of join irreducibles of finitely generated free ternary algebras is characterized. The uniqueness of the set of free generators and their pseudocomplements is also established.
TL;DR: This paper provides explicit examples of test elements in direct products whose factors are free groups or surface groups and a tool for doing the same for torsion free hyperbolic factors.
Abstract: We characterize test elements in the commutator subgroup of a direct product of certain groups in terms of test elements of the factors. This provides explicit examples of test elements in direct products whose factors are free groups or surface groups and a tool for doing the same for torsion free hyperbolic factors.
TL;DR: It is shown that the theorem of Schutzenberger that a language is star-free if and only if it is recognized by a finite monoid with trivial subgroups is proved.
Abstract: We prove the theorem of Schutzenberger that a language is star-free if and only if it is recognized by a finite monoid with trivial subgroups. The forward direction follows the lines of the proof of Kleene's Theorem which characterizes regular languages as those recognized by finite automata. The converse is a relatively short induction.
TL;DR: This paper introduces extensions of group presentations with a corresponding small cancellation theory and shows how this can be used in order to reduce geometrical and combinatorial problems in certain groups, to the same problems in simpler groups.
Abstract: In this paper we introduce extensions of group presentations with a corresponding small cancellation theory and show how this can be used in order to reduce geometrical and combinatorial problems in certain groups, to the same problems in simpler groups. Let us illustrate this by the following simple example: let P1 = 〈a, b, c, d|R〉 be a presentation of G1, where R = (ab 2c−1dc)7(bd−1c3b)6 and let P2 = 〈x, y|xy〉 be a presentation of G2. We show that since R = A B where A = ab2c−1dc and B = bd−1c3b and since the map φ : F2 → F1 defined by φ(x) = A and φ(y) = B is a monomorphism satisfying R ∈ φ(N2) ⊆ N1, where F2 = 〈x, y|−〉, F1 = 〈a, b, c, d|−〉, N1 is the normal closure of R in F1 and N2 is the normal closure of x y in F2, it follows that G1 is biautomatic, because G2 is biautomatic. Moreover, for example the subgroup H of G1 generated by C and D where C = abdc −2b3 and D = dc−1a5b−7c is a free subgroup of G1 and the natural map φ̂ : G2 → G1 is an embedding. In the above situation we say that P1 is an extension of P2 via φ. (See 1.1 for the precise definition.) Based on the general idea of derived diagrams due to E. Rips, we develop a relative version of small cancellation presentations which is roughly as follows: assume P1 is an extension of P2. Consider φ(N2) as the set of defining relations for G1 and describe the elements of N1 as products of conjugates of elements from φ(N2). Choose a shortest such product. Then P1 is small cancellation relative to P2 if the cancellation between two factors in a shortest product is small. The main results of this work are collected in Theorem 1.1. Its proof is divided into 2 parts. First we prove that if P1 is a small cancellation presentation relative to P2 then all the results of the Main Theorem hold true (Theorem 2.1). Then, we show that under the conditions of the theorem, P1 is small cancellation relative to P2 (Theorem 1.2).
TL;DR: A finite algebra and a relational system, constructed explicitly from the system for , such that and dualizes the variety determined by all normal identities of.
Abstract: Let be a variety of the form where is a finite subdirectly irreducible algebra. We show that if is naturally dualizable (in the sense of D. M. Clark and B. A. Davey, i.e. with respect to the discrete topology) then the variety determined by all normal identities of (the so called nilpotent shift of ) is also naturally dualizable. We give a finite algebra and a relational system , constructed explicitly from the system for , such that and dualizes .
TL;DR: The indecomposable complex linear representations of Φ2 of degree at most 5 are classified, and which ones have an infinite image are determined.
Abstract: Denote by Φ2 the automorphism group of the free group on two generators. We classify the indecomposable complex linear representations of Φ2 of degree at most 5, and determine which ones have an infinite image. As a side-effect, our classification incorporates all the indecomposable representations of GL2(Z) in these degrees.