Showing papers in "International Journal of Algebra and Computation in 2010"
TL;DR: The Lie algebra analogue to the Schur multiplier has been investigated in a number of recent articles, and the multipliers of Lie algebras of maximal class are considered.
Abstract: The Lie algebra analogue to the Schur multiplier has been investigated in a number of recent articles. We consider the multipliers of Lie algebras of maximal class, classifying these algebras with ...
58 citations
TL;DR: In this article, the Grobner-Shirshov basis for a dialgebra is defined and the Composition-Diamond lemma for dialgebras is given.
Abstract: In this paper, we define the Grobner–Shirshov basis for a dialgebra. The Composition–Diamond lemma for dialgebras is given then. As a result, we give Grobner–Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the bar extension of a dialgebra, the free product of two dialgebras, and Clifford dialgebra. We obtain some normal forms for algebras mentioned the above.
55 citations
TL;DR: It is proved that if $\pi$ is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are $pi$-groups, and Henckell's decidability of aperiodic pointlikes is obtained.
Abstract: We prove that if π is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are π-groups. In particular, when π is the empty set, we obtain Henckell's decidability of aperiodic pointlikes. Our proof, restricted to the case of aperiodic semigroups, is simpler than the original proof.
31 citations
TL;DR: In this article, it was shown that each countably generated algebra over a countable field k can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebra, associative ǫ-algebraic groups.
Abstract: In this paper, by using Grobner–Shirshov bases, we show that in the following classes, each (respectively, countably generated) algebra can be embedded into a simple (respectively, two-generated) algebra: associative differential algebras, associative Ω-algebras, associative λ-differential algebras. We show that in the following classes, each countably generated algebra over a countable field k can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative Ω-algebras, associative λ-differential algebras. We give another proofs of the well known theorems: each countably generated group (respectively, associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (respectively, associative algebra, semigroup, Lie algebra).
29 citations
TL;DR: Let G be a finite group and Γ(G) of G is the prime graph of G, where Γ are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge.
Abstract: Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there ...
27 citations
TL;DR: Thompson's group V can be thought of as a group of automorphisms of a certain algebra or as a subgroup of the group of self homeomorphisms of the Cantor set.
Abstract: Thompson's group V can be thought of as the group of automorphisms of a certain algebra or as a subgroup of the group of self homeomorphisms of the Cantor set. Thus, the dynamics of an element of V...
25 citations
TL;DR: This article showed that the compressed word problem in a finitely generated fully residually free group is decidable in polynomial time, and used this result to show that the word problem for the automorphism group of an -group is also decidable.
Abstract: We show that the compressed word problem in a finitely generated fully residually free group (-group) is decidable in polynomial time, and use this result to show that the word problem in the automorphism group of an -group is decidable in polynomial time.
22 citations
TL;DR: It is proved that the global operator g : V ↦ gV does not preserve decidability of membership.
Abstract: It is proved that the global operator g : V ↦ gV does not preserve decidability of membership. More specifically, some of Kad'ourek's results on the locality of the pseudovariety DG in combination ...
21 citations
TL;DR: In this paper, a twisted version of Grigorchuk's first group is studied, and the similarities and differences to its model are analyzed, and it is shown that it admits a finite endomorphic presentation, has infinite-rank multiplier, and does not have the congruence property.
Abstract: We study a twisted version of Grigorchuk's first group, and stress its similarities and differences to its model. In particular, we show that it admits a finite endomorphic presentation, has infinite-rank multiplier, and does not have the congruence property.
20 citations
TL;DR: In this paper, it was shown that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup.
Abstract: We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup.
19 citations
TL;DR: The partial transformation semigroup $\mathcal{PT}_n$ is the semigroup of all partial transformations on the finite set n = {1,…, n}.
Abstract: The partial transformation semigroup $\mathcal{PT}_n$ is the semigroup of all partial transformations on the finite set n = {1,…, n}. The transformation semigroup $\mathcal{T}_n\subseteq\mathcal{PT...
TL;DR: This work construct and classify triangle presentations associated to the smallest generalized quadrangle and uses these constructions to find groups acting cocompactly on hyperbolic buildings with n-sided buildings.
Abstract: We construct and classify triangle presentations associated to the smallest generalized quadrangle. We use these constructions to find groups acting cocompactly on hyperbolic buildings with n-sided...
TL;DR: The notion of pseudo- normality is introduced, a concept that provides necessary conditions for a parametrization for being normal and an algorithm for deciding the pseudo-normality is provided, which is shown to be always normal.
Abstract: In this paper we analyze the problem of deciding the normality (ie the surjectivity) of a rational parametrization of a surface The problem can be approached by means of elimination theory techniques, providing a proper close subset where surjectivity needs to be analyzed In general, these direct approaches are unfeasible because is very complicated and its elements computationally hard to manipulate Motivated by this fact, we study ad hoc computational alternative methods that simplifies For this goal, we introduce the notion of pseudo-normality, a concept that provides necessary conditions for a parametrization for being normal Also, we provide an algorithm for deciding the pseudo-normality Finally, we state necessary and sufficient conditions on a pseudo-normal parametrization to be normal As a consequence, certain types of parametrizations are shown to be always normal For instance, pseudo-normal polynomial parametrizations are normal Moreover, for certain class of parametrizations, we derive an algorithm for deciding the normality
TL;DR: It is given a short proof, using profinite techniques, that idempotent pointlikes, stable pairs and triples are decidable for the pseudovariety of aperiodic monoids.
Abstract: We give a short proof, using profinite techniques, that idempotent pointlikes, stable pairs and triples are decidable for the pseudovariety of aperiodic monoids. Stable pairs are also described for the pseudovariety of all finite monoids.
TL;DR: Progress in computing the cabled Jones, HOMFLY and Kauffman polynomial is explained and some group theoretic considerations are applied to the tabulation of low-crossing coefficients.
Abstract: We explain progress in computing the cabled Jones, HOMFLY and Kauffman polynomial. This is applied, first, in combination with some group theoretic considerations, to the tabulation of low-crossing...
TL;DR: This work describes many classes of algebras where the complexity of TERM-SAT is high, and studies the computational complexity of the satisfiability problem of an equation between terms over a finite algebra.
Abstract: We study the computational complexity of the satisfiability problem of an equation between terms over a finite algebra (TERM-SAT). We describe many classes of algebras where the complexity of TERM-...
TL;DR: In this article, the complexity of the monoid generalization Mk, 1 of the Thompson-Higman groups was studied and the computational complexity of both the -order decision problem and the injectiveness problem was characterized.
Abstract: We study the monoid generalization Mk, 1 of the Thompson–Higman groups, and we characterize the - and the -order of Mk, 1. Although Mk, 1 has only one nonzero -class and k-1 nonzero -classes, the - and the -order are complicated; in particular, is dense (even within an -class), and is dense (even within an -class). We study the computational complexity of the - and the -order. When inputs are given by words over a finite generating set of Mk, 1, the - and the -order decision problems are in P. However, over a "circuit-like" generating set the -order decision problem of Mk, 1 is -complete, whereas the -order decision problem is coNP-complete. Similarly, for acyclic circuits the surjectiveness problem is -complete, whereas the injectiveness problem is coNP-complete.
TL;DR: It is shown that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max{|S1|,|S2|}.
Abstract: It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2 whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max{|S1|,|S2|}. Moreover we consider amalgams of finite inverse semigroups respecting the $\mathcal{J}$-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the $\mathcal{R}$-classes to be finite.
TL;DR: In this paper, the authors give an algebraic characterization of tree languages that are defined by logical formulas using certain Lindstrom quantifiers and show that such tree languages are first-order definable.
Abstract: We give an algebraic characterization of the tree languages that are defined by logical formulas using certain Lindstrom quantifiers. An important instance of our result concerns first-order definable tree languages. Our characterization relies on the usage of preclones, an algebraic structure introduced by the authors in a previous paper, and of the block product operation on preclones. Our results generalize analogous results on finite word languages, but it must be noted that, as they stand, they do not yield an algorithm to decide whether a given regular tree language is first-order definable.
TL;DR: The study of prime ideals has been an area of active research and in recent past a considerable work has been done in this direction.
Abstract: The study of prime ideals has been an area of active research. In recent past a considerable work has been done in this direction. Associated prime ideals and minimal prime ideals of certain types ...
TL;DR: This result extends the results of Rowen, Teicher and Vishne to generalized Coxeter groups which have a natural map onto one of the classical Coxetergroups.
Abstract: Let C(T) be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either Bn or Dn. Let CY(T) be a natural quotient of C(T), and if C(T) is simply-laced (which means all the relations between the generators has order 2 or 3), CY(T) is a generalized Coxeter group, too. Let At,n be a group which contains t Abelian groups generated by n elements. The main result in this paper is that CY(T) is isomorphic to At,n ⋊ Bn or At,n ⋊ Dn, depends on whether the signed graph T contains loops or not, or in other words C(T) is simply-laced or not, and t is the number of the cycles in T. This result extends the results of Rowen, Teicher and Vishne to generalized Coxeter groups which have a natural map onto one of the classical Coxeter groups.
TL;DR: The monoid of extensive transformations of a chain of order four is shown to be finitely based and a finite basis for this monoid is given, completing the description of the equational property.
Abstract: The monoid of extensive transformations of a chain of order four is shown to be finitely based and a finite basis for this monoid is given. This completes the description of the equational property...
TL;DR: The inverse semigroup of partial automaton permutations over a finite alphabet is characterized in terms of wreath products and the permutation conjugacy relation in this semigroup and the Green's rel...
Abstract: The inverse semigroup of partial automaton permutations over a finite alphabet is characterized in terms of wreath products. The permutation conjugacy relation in this semigroup and the Green's rel...
TL;DR: The structure of the unit group of the group algebra of Pauli's group over any field of characteristic 2 is established in terms of split extensions of cyclic groups.
Abstract: The structure of the unit group of the group algebra of Pauli's group over any field of characteristic 2 is established in terms of split extensions of cyclic groups.
TL;DR: In this paper, it was shown that the following problems are decidable in a rank 2 free group F2: Does a given finitely generated subgroup H contain primitive elements? And does H meet the orbit of a given word u under the action of G, the group of automorphisms of F2?
Abstract: We show that the following problems are decidable in a rank 2 free group F2: Does a given finitely generated subgroup H contain primitive elements? And does H meet the orbit of a given word u under the action of G, the group of automorphisms of F2? Moreover, decidability subsists if we allow H to be a rational subset of F2, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, the following weaker problem is decidable: given a finitely generated subgroup H, a word u and an integer k, does H contain the image of u by some k-almost bounded automorphism? An automorphism is k-almost bounded if at most one of the letters has an image of length greater than k.
TL;DR: It is proved that each nonzero normal element of a completed group algebra over the special linear group SLn(ℤp) is a unit.
Abstract: By a purely computational approach we prove that each nonzero normal element of a completed group algebra over the special linear group SLn(ℤp) is a unit.
TL;DR: It is proved that the outer automorphism group of Mod(Ng) is cyclic and is equivalent to the mapping class group of Ng.
Abstract: Let Ng be the connected closed nonorientable surface of genus g ≥ 5 and Mod(Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod(Ng) is cyclic.
TL;DR: Thin Lie alagbras are graded Lie algebras with dim Li ≤ 2 for all i satisfying a more stringent but natural narrowness condition modeled on an analogousConditional Lie algebra.
Abstract: Thin Lie algebras are graded Lie algebras $L = \oplus_{i = 1}^{\infty}L_{i}$ with dim Li ≤ 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous cond...
TL;DR: This work considers the family of irreducible cyclically presented groups on n generators whose generating word (in the standard rewrite) has length at most 15 and shows that if 6 <= n <= 100 then the group is non-trivial.
Abstract: We consider the family of irreducible cyclically presented groups on n generators whose generating word (in the standard rewrite) has length at most 15. Using the software packages KBMAG, quotpic and Magma, together with group and number theoretic methods, we show that if 6 <= n <= 100 then the group is non-trivial. In an appendix we list the 47 cases within 2 <= n <= 100 for which we know the group to be trivial, and 27 further cases for which triviality has yet to be determined.