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It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper we prove that the set of all solutions of a twisted word equation is an EDT0L language whose specification can be computed in $\mathsf{PSPACE}$. Within the same complexity bound we can decide whether the solution set is empty, finite, or infinite. In the second part of the paper we apply the results for twisted equations to obtain in $\mathsf{PSPACE}$ an EDT0L description of the solution set of equations with rational constraints for finitely generated virtually free groups in standard normal forms with respect to a natural set of generators. If the rational constraints are given by a homomorphism into a fixed (or "small enough") finite monoid, then our algorithms can be implemented in $\mathsf{NSPACE}(n^2\log n)$, that is, in quasi-quadratic nondeterministic space. Our results generalize the work by Lohrey and S\'enizergues (ICALP 2006) and Dahmani and Guirardel (J. of Topology 2010) with respect to both complexity and expressive power. Neither paper gave any concrete complexity bound and the results in these papers are stated for subsets of solutions only, whereas our results concern all solutions., Comment: 72 pages, 14 figures. Small edits to the exposition in Section 15. An extended abstract of a preliminary version of this paper was presented at ICALP 2017

Recent studies of [Formula: see text]-groups of coclass [Formula: see text] concentrate on the coclass graph [Formula: see text]. While the detailed structure of [Formula: see text] is unknown, it is known that its general structure is dominated by the subgraph of ‘skeleton groups’. The original definition of these groups is technical, but some modifications for special cases have been used successfully in the literature. Given their importance, in this paper we define and investigate skeleton groups more rigorously. In particular, we study their isomorphism problem, which is a crucial step towards understanding the skeleton subgraph of [Formula: see text]. During our work we identified erroneous arguments for constructing isomorphisms in a proof of the 2013 paper on [Formula: see text]. We correct these errors here by proving the required results in a more general context.

This paper is the first one in a series of papers where we lay special attention on syntax and semantics in knowledge base theory. The main goal of the present paper is to set up all necessary mathematical background for the definition of a knowledge-base model which will be introduced in the next paper. We should emphasize that in these papers we develop a slightly different approach to the knowledge-base model than in the previous ones.

This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61–93], where we introduced quadratic forms on a module [Formula: see text] over a supertropical semiring [Formula: see text] and analyzed the set of bilinear companions of a quadratic form [Formula: see text] in case the module [Formula: see text] is free, with fairly complete results if [Formula: see text] is a supersemifield. Given such a companion [Formula: see text], we now classify the pairs of vectors in [Formula: see text] in terms of [Formula: see text] This amounts to a kind of tropical trigonometry with a sharp distinction between the cases for which a sort of Cauchy–Schwarz (CS) inequality holds or fails. This distinction is governed by the so-called CS-ratio [Formula: see text] of a pair of anisotropic vectors [Formula: see text] in [Formula: see text]. We apply this to study the supertropicalizations (cf. [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61–93]) of a quadratic form on a free module [Formula: see text] over a field in the simplest cases of interest where [Formula: see text]. In the last part of the paper, we introduce a suitable equivalence relation on [Formula: see text], whose classes we call rays. (It is coarser than usual projective equivalence.) For anisotropic [Formula: see text] the CS-ratio [Formula: see text] depends only on the rays of [Formula: see text] and [Formula: see text]. We develop essential basics for a kind of convex geometry on the ray-space of [Formula: see text], where the CS-ratios play a major role.

The first paper in this series introduced the notion of term to term operation continuity for finite groupoids, and proved that two testable conditions on a finite groupoid imply that it is term continuous (TC). The second presented an evolution inspired algorithm for finding terms for operations, and gave experimental evidence that, in general, it was successful exactly when the groupoid was both idemprimal and TC. In this paper, we describe a new class of algorithms for finding terms which brings these results together. Theorems about idemprimality and term continuity show how each of these two properties impact our algorithms. They lead to a final explanation for the success of our algorithms when the groupoid is both idemprimal and TC.

In this paper, we deal with the problem of computing the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] with [Formula: see text] and [Formula: see text] a finite commutative ring. We completely solve the problem in the case [Formula: see text] and give some results that compute the value of this sum if [Formula: see text] is an arbitrary finite commutative ring for many values of [Formula: see text] and [Formula: see text]. Finally, based on computational evidence and using some technical results proved in this paper, we conjecture that the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] is always [Formula: see text] unless [Formula: see text], [Formula: see text], [Formula: see text] and the only element [Formula: see text] such that [Formula: see text] is idempotent, in which case the sum is [Formula: see text].

In order to study the axiomatization of the if-then-else construct over possibly non-halting programs and tests, this paper introduces the notion of [Formula: see text]-sets by considering the tests from an abstract [Formula: see text]-algebra. When the [Formula: see text]-algebra is an ada, the axiomatization is shown to be complete by obtaining a subdirect representation of [Formula: see text]-sets. Further, this paper considers the equality test with the if-then-else construct and gives a complete axiomatization through the notion of agreeable [Formula: see text]-sets.

A homogeneous pattern is a linear multivariate polynomial without constant term. Bras-Amorós and García-Sánchez introduced the notion of pattern for numerical semigroups, which generalizes the definition of Arf numerical semigroups. The notion of pattern for numerical semigroups is extended in this paper into a family of homogeneous patterns [Formula: see text]. A numerical semigroup admitting a family of homogeneous patterns [Formula: see text] at [Formula: see text]-level is characterized. We pay our attention in this paper to the families with the action of some permutation groups, especially those consisting of certain partition stabilizers. Stable or sensitive patterns are focused on and we characterize them for several specific permutation groups.

In this paper, we prove the claim given in the title. A group [Formula: see text] is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to [Formula: see text] factors through projection to a canonical free subgroup. Higman, in his seminal 1952 paper [Unrestricted free products and varieties of topological groups, J. London Math. Soc. 27 (1952) 73–81], proved that free groups are noncommutatively slender. Such groups were first defined by Eda in [Free [Formula: see text]-products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263]. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few-relator sense are noncommutatively slender.

In this paper, we study exchange rings and clean rings [Formula: see text] with [Formula: see text] (or otherwise). Analogues of a theorem of Camillo and Yu characterizing clean and strongly clean rings with [Formula: see text] are obtained for such rings (as well as for exchange rings) using the viewpoint of exchange equations introduced in a recent paper of the authors. We also study a new class of rings including von Neumann regular rings in which square roots of one (instead of idempotents) can be lifted modulo left ideals, and conjecture that such rings are exchange rings. This conjecture holds for commutative rings, and would hold for all rings if it holds for semiprimitive rings of characteristic [Formula: see text].

Let [Formula: see text] be a commutative ring and [Formula: see text] be an infinite discrete group. The algebraic [Formula: see text]-theory of the group ring [Formula: see text] is an important object of computation in geometric topology and number theory. When the group ring is Noetherian, there is a companion [Formula: see text]-theory of [Formula: see text] which is often easier to compute. However, it is exceptionally rare that the group ring is Noetherian for an infinite group. In this paper, we define a version of [Formula: see text]-theory for any finitely generated discrete group. This construction is based on the coarse geometry of the group. It has some expected properties such as independence from the choice of a word metric. We prove that, whenever [Formula: see text] is a regular Noetherian ring of finite global homological dimension and [Formula: see text] has finite asymptotic dimension and a finite model for the classifying space [Formula: see text], the natural Cartan map from the [Formula: see text]-theory of [Formula: see text] to [Formula: see text]-theory is an equivalence. On the other hand, our [Formula: see text]-theory is indeed better suited for computation as we show in a separate paper. Some results and constructions in this paper might be of independent interest as we learn to construct projective resolutions of finite type for certain modules over group rings.

Denote by $G$ a finite group and by $\psi(G)$ the sum of element orders in $G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order $t$ and write $\psi(t)=\psi(C_t)$. In this paper we proved the following Theorem A: Let $G$ be a non-cyclic group of odd order $n=qm$, where $q$ is the smallest prime divisor of $n$ and $(m,q)=1$. Then the following statements hold. (1) If $q=3$, then $\frac {\psi(G)}{\psi(|G|)}\leq \frac {85}{301}$, and equality holds if and only if $n=3\cdot 7\cdot m_1$ with $(m_1,42)=1$ and $G=(C_7\rtimes C_3)\times C_{m_1}$, with $C_7\rtimes C_3$ non-abelian. (2) If $q>3$, then $\frac {\psi(G)}{\psi(|G|)}\leq \frac {p^4+p^3-p^2+1}{p^5+1}$, where $p$ is the smallest prime bigger than $q$ and equality holds if and only if $n=qp^2m_1$ with $(m_1,p!)=1$ and $G=C_q\times C_p\times C_p \times C_{m_1}$.

In the last years, the sequence of codimensions of PI-algebras has been studied by several authors and the classification of unitary algebras, up to equivalence, with at most cubic codimension growth was given by Giambruno, La Mattina and Petrogradsky in 2007. In this paper, we establish a new approach by studying the possibilities of specific proper codimensions of a unitary algebra with growth [Formula: see text] in order to present a complete list of varieties generated by unitary algebras with polynomial growth [Formula: see text]. Also, we classify, up to PI-equivalence, the unitary algebras with growth [Formula: see text] whose leading coefficient of the polynomial describing the codimension sequence achieves the largest and the smallest possible value.

Under broad finiteness conditions, such as the existence of a greatest or least element, a semidistributive lattice has the two element lattice as a homomorphic image, and so, if it has more than two elements, is not simple. However, the existence of a simple, semidistributive lattice with more than two elements has remained in question. This paper constructs such a lattice.

In this paper, we present new characterizations of several radicals of differential polynomial rings, including the Levitzki radical, strongly prime radical, and uniformly strongly prime radical in terms of the related [Formula: see text]-radical.

In this paper we compute the center of many noncommutative algebras that can be interpreted as skew [Formula: see text] extensions. We show that, under some natural assumptions on the parameters that define the extension, either the center is trivial, or, it is of polynomial type. As an application, we provided new examples of noncommutative algebras that are cancellative.

A good way of parameterizing zero-dimensional schemes in an affine space [Formula: see text] has been developed in the last 20 years using border basis schemes. Given a multiplicity [Formula: see text], they provide an open covering of the Hilbert scheme [Formula: see text] and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent zero-dimensional [Formula: see text]-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley–Bacharach, and strict Cayley–Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by nontrivial, concrete examples.

This paper characterizes the potential behaviors of higher commutators in a simple algebra.

Let [Formula: see text] with [Formula: see text] be [Formula: see text] upper-triangular matrices with rational entries. In the multiplicative semigroup generated by these matrices, we check if there are relations of the form [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] We give algorithms to find relations of the previous form. Our results are extensions of some theorems obtained by Charlier and Honkala in [The freeness problem over matrix semigroups and bounded languages, Inf. Comput. 237 (2014) 243–256]. Our paper is at the interface between algebra, number theory and theoretical computer science. While the main results concern decidability and semigroup theory, the methods for obtaining them come from number theory.

In this paper, we produce the product rules of nonassociative Moufang loops of order 81 by using an analytical approach. We then explore all possible presentations on a suitable set of generators, thereby obtaining a total of five nonisomorphic cases. The result is in agreement with the classification by Nagy and Vojtěchovský using the GAP package LOOPS .

In this paper, we study the manifold structure and the relative hyperbolic structure of right-angled Coxeter groups with planar nerves. We then apply our results to the quasi-isometry problem for this class of right-angled Coxeter groups.

In this paper, it is shown that, for every non-trivial variety [Formula: see text] of groups, the variety [Formula: see text] of all completely regular semigroups all of whose subgroups belong to [Formula: see text] is minimal in its kernel class in the lattice [Formula: see text] of all varieties of completely regular semigroups, and hence it constitutes, in fact, a singleton kernel class in the lattice [Formula: see text]. Even more generally, it is shown that, for every variety [Formula: see text] of completely simple semigroups which does not consist entirely of rectangular groups, the variety [Formula: see text] of all completely regular semigroups all of whose completely simple subsemigroups belong to [Formula: see text] is minimal in its kernel class in the lattice [Formula: see text], and hence it likewise constitutes a singleton kernel class in the mentioned lattice [Formula: see text].

Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalized Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalization of the cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories. © 2020 World Scientific Publishing Company. The first author acknowledges the financial support of the Competitiveness Enhancement Program of Ural Federal University, Russia during the preparation of this paper.

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor defined stability and proved stability is equivalent to convex cocompactness in mapping class groups. Another natural generalization of quasiconvexity is given by the notion of a Morse or strongly quasiconvex subgroup of a finitely generated group, studied recently by Tran and Genevois. In general, a subgroup is stable if and only if the subgroup is Morse and hyperbolic. In this paper, we prove that two properties of being Morse and stable coincide for a subgroup of infinite index in the mapping class group of an oriented, connected, finite type surface with negative Euler characteristic., Comment: 11 pages, v2: changed the title, revised abstract and introduction, corrected typos

A finite group [Formula: see text] is called a [Formula: see text]-group if every proper subgroup of [Formula: see text] is either quasi-normal or self-normal in [Formula: see text]. In this paper, the authors classify the non-[Formula: see text]-groups whose proper subgroups are all [Formula: see text]-groups.

This paper establishes a finite axiomatization of possibly non-halting computer programs and tests, with the if-then-else operation. The model is a two-sorted algebra, with one sort being the programs and the other being the tests. The main operation on programs is composition, and 1 and 0 represent the programs skip and loop (i.e. never halts) respectively. Programs are modeled as partial functions on some state space [Formula: see text], with tests modeled as partial predicates on [Formula: see text]. The operations on the tests are the usual logical connectives ∧, ∨, [Formula: see text], [Formula: see text] and [Formula: see text]. In addition, there is the hybrid operation of if-then-else, and the test-valued operation [Formula: see text] on programs which is true when a program halts, and undefined otherwise. The halting operation [Formula: see text] implies that operations of domain [Formula: see text] and domain join ∨ may also be expressed. When tests are assumed to be possibly non-halting, the evaluation strategy of the logical connectives affects the result. Here we model parallel evaluation, as opposed to the common sequential (or short-circuit) evaluation strategy. For example, we view [Formula: see text] as false if either [Formula: see text] or [Formula: see text] is false, even if the other does not halt.

Let [Formula: see text] be an integral domain, [Formula: see text] be a nonzero torsionless commutative cancellative monoid with quotient group [Formula: see text], and [Formula: see text] be the semigroup ring of [Formula: see text] over [Formula: see text]. In this paper, among other things, we show that if [Formula: see text] (respectively, [Formula: see text], then [Formula: see text] is a weakly factorial domain if and only if [Formula: see text] is a weakly factorial GCD-domain, [Formula: see text] is a weakly factorial GCD-semigroup, and [Formula: see text] is of type [Formula: see text] (respectively, [Formula: see text] except [Formula: see text]).

We know from a result due to Noether–Lefschetz that a very general surface of degree at least 4 in [Formula: see text] contains only curves which are complete intersections with other surfaces. The main goal of this paper is to construct an explicit and smooth compactification of a parameter space for surfaces in [Formula: see text] of degree [Formula: see text] for all sufficiently large [Formula: see text], containing one conic and one line. The construction also applies to surfaces in [Formula: see text] containing one plane curve and one line. As an application, we compute the degree of the locus of surfaces of degree [Formula: see text] containing one conic and one line.

The aim of the paper is to construct, discuss and apply the Galois-type correspondence between subsemigroups of the endomorphism semigroup [Formula: see text] of an algebra [Formula: see text] and sets of logical formulas. Such Galois-type correspondence forms a natural frame for studying algebras by means of actions of different subsemigroups of [Formula: see text] on definable sets over [Formula: see text]. We treat some applications of this Galois correspondence. The first one concerns logic geometry. Namely, it gives a uniform approach to geometries defined by various fragments of the initial language. The next prospective application deals with effective recognition of sets and effective computations with properties that can be defined by formulas from a fragment of the original language. In this way, one can get an effective syntactical expression by semantic tools. Yet another advantage is a common approach to generalizations of the main model theoretic concepts to the sublanguages of the first-order language and revealing new connections between well-known concepts. The fourth application concerns the generalization of the unification theory, or more generally Term Rewriting Theory, to the logic unification theory.

In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups [Formula: see text] are fully characterized in the class of all groups by the set [Formula: see text] of types realized in [Formula: see text]. Furthermore, it turns out that these groups [Formula: see text] are fully characterized already by some particular rather restricted fragments of the types from [Formula: see text]. In particular, every finitely generated nilpotent group is completely defined by its [Formula: see text]-types, while a finitely generated rigid group is completely defined by its [Formula: see text]-types, and a finitely generated metabelian or polycyclic group is completely defined by its [Formula: see text]-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.

Let [Formula: see text] be a finite group and [Formula: see text] be the set of the elements [Formula: see text] of [Formula: see text] such that [Formula: see text] where [Formula: see text]. In this paper, we give strong restrictions on the structure of the quotients [Formula: see text] where [Formula: see text] is a finite group with an automorphism [Formula: see text] such that [Formula: see text] and [Formula: see text] is the largest normal [Formula: see text]-subgroup of [Formula: see text].

In this paper, we characterize the classification of unimodal maps from the plane to the plane with respect to [Formula: see text]-equivalence given by Rieger in terms of invariants. We recall the classification over an algebraically closed field of characteristic [Formula: see text]. On the basis of this characterization, we present an algorithm to compute the type of the unimodal maps from the plane to the plane without computing the normal form and also give its implementation in the computer algebra system Singular.

In 2011, Horváth gave a new proof that the equation solvability problem over finite nilpotent groups and rings is in P. In the same paper, he asked whether his proof can be lifted to nilpotent algebras in general. We show that this is in fact possible for supernilpotent algebras with a Mal’cev term. However, we also describe a class of nilpotent, but not supernilpotent algebras with Mal’cev term that have co-NP-complete identity checking problems and NP-complete equation solvability problems. This proves that the answer to Horváth’s question is negative in general (assuming P[Formula: see text]NP).

Jones introduced unitary representations for the Thompson groups [Formula: see text] and [Formula: see text] from a given subfactor planar algebra. Some interesting subgroups arise as the stabilizer of certain vector, in particular the Jones subgroups [Formula: see text] and [Formula: see text]. Golan and Sapir studied [Formula: see text] and identified it as a copy of the Thompson group [Formula: see text]. In this paper, we completely describe [Formula: see text] and show that [Formula: see text] coincides with its commensurator in [Formula: see text], implying that the corresponding unitary representation is irreducible. We also generalize the notion of the Stallings 2-core for diagram groups to [Formula: see text], showing that [Formula: see text] and [Formula: see text] are not isomorphic, but as annular diagram groups they have very similar presentations.

The main result of this paper shows that if $\mathcal{M}$ is a consistent strong linear Maltsev condition which does not imply the existence of a cube term, then for any finite algebra $\mathbb{A}$ there exists a new finite algebra $\mathbb{A}_\mathcal{M}$ which satisfies the Maltsev condition $\mathcal{M}$, and whose subpower membership problem is at least as hard as the subpower membership problem for $\mathbb{A}$. We characterize consistent strong linear Maltsev conditions which do not imply the existence of a cube term, and show that there are finite algebras in varieties that are congruence distributive and congruence $k$-permutable ($k \geq 3$) whose subpower membership problem is EXPTIME-complete., 13 Pages

In this paper, we prove that certain HNN extensions of finitely generated abelian subgroup separable groups are finitely generated abelian subgroup separable. Using this, we show that certain HNN extensions of finitely generated nilpotent groups are finitely generated abelian subgroup separable.

It is well known [Algebraic properties of plane offset curves, Comput. Aided Geom. Des. 7 (1990) 101–127] that an implicit equation of the offset to a rational planar curve can be computed by removing the extraneous components of the resultant of two certain polynomials computed from the parametrization of the curve. Furthermore, it is also well known that the implicit equation provided by the nonextraneous component of this resultant has at most two irreducible factors [Algebraic analysis of offsets to hypersurfaces, Math. Z. 234 (2000) 697–719]. In this paper, we complete the algebraic description of this resultant by showing that the multiplicity of the factors corresponding to the offset can be computed in advance. In particular, when the parametrization is proper, i.e. when the curve is just traced once by the parametrization, we prove that any factor corresponding to a simple component of the offset has multiplicity 1, while the factor corresponding to the special component, if any, has multiplicity 2. Hence, if the parametrization is proper and there is no special component, the nonextraneous part of the resultant is square-free. In fact, this condition is proven to be also sufficient. Therefore, this result provides a simple test to check whether or not the offset of a given rational curve has a special component, and in turn, whether a given rational curve is the offset of another curve.

L systems generalise context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives in the free group on two generators, the Bridson-Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk's group can be generated by L systems. To complement the result on primitives in free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L and ET0L languages of intermediate growth., Revised following referees suggestions. 21 pages, 2 figures

Let [Formula: see text] be a finite field with the characteristic [Formula: see text] and let [Formula: see text] be the unitary Grassmann algebra generated by an infinite dimensional vector space [Formula: see text] over [Formula: see text]. In this paper, we determine a basis for [Formula: see text]-graded polynomial identities for any [Formula: see text]-grading such that its underlying vector space is homogeneous.

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid $S$, consider the family of "shifted" monoids $M_n$ obtained by adding $n$ to each generator of $S$. In this paper, we examine minimal relations among the generators of $M_n$ when $n$ is sufficiently large, culminating in a description that is periodic in the shift parameter $n$. We explore several applications to computation, combinatorial commutative algebra, and factorization theory., 15 pages, 2 figures

We call a reduced word [Formula: see text] multiplicity-bounding if and only if a finite group on which the word map of [Formula: see text] has a fiber of positive proportion [Formula: see text] can only contain each non-abelian finite simple group [Formula: see text] as a composition factor with multiplicity bounded in terms of [Formula: see text] and [Formula: see text]. In this paper, based on recent work of Nikolov, we present methods to show that a given reduced word is multiplicity-bounding and apply them to give some nontrivial examples of multiplicity-bounding words, such as words of the form [Formula: see text], where [Formula: see text] is a single variable and [Formula: see text] an odd integer.

In this paper, we present an algorithm to give the isomorphism criterion for a subclass of complex filiform Leibniz algebras arising from naturally graded filiform Lie algebras. This subclass appeared as a Leibniz central extension of a linear deformation of filiform Lie algebra. We give the table of multiplication choosing appropriate adapted basis, identify the elementary base changes and describe the behavior of structure constants under these base changes, then combining them the isomorphism criterion is given. The final result of calculations for one particular case also is provided.

This paper treats algebras whose support [Formula: see text]-tilting posets are isomorphic to the poset of the symmetric group with the weak order. We determine such algebras in terms of quivers and relations.

This paper deals with the reducibility property of semidirect products of the form [Formula: see text] relatively to graph equation systems, where D denotes the pseudovariety of definite semigroups. We show that if the pseudovariety [Formula: see text] is reducible with respect to the canonical signature [Formula: see text] consisting of the multiplication and the [Formula: see text]-power, then [Formula: see text] is also reducible with respect to [Formula: see text].

We consider groups [Formula: see text] which have generating sets consisting of pairwise conjugate elements. We introduce a natural condition on such sets, namely path-connectedness, which has strong consequences when [Formula: see text] acts on a real tree. The main result of the paper states that the group of type-preserving automorphisms of a regular simplicial tree has a generating set with this property.

Let [Formula: see text] be a field of characteristic 0 and let [Formula: see text]. The algebra [Formula: see text] admits a natural grading [Formula: see text] by the cyclic group [Formula: see text] of order 2. In this paper, we describe the [Formula: see text]-graded A-identities for [Formula: see text]. Recall that an A-identity for an algebra is a multilinear polynomial identity for that algebra which is a linear combination of the monomials [Formula: see text] where [Formula: see text] runs over all even permutations of [Formula: see text] that is [Formula: see text], the [Formula: see text]th alternating group. We first introduce the notion of an A-identity in the case of graded polynomials, then we describe the graded A-identities for [Formula: see text], and finally we compute the corresponding graded A-codimensions.

Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes for any positive integer k. We show that, for any positive integers n and s, there exist finitely many finite groups G, up to isomorphism, having a normal subgroup N of index n which contains exactly s non-central G-conjugacy classes. Upper bounds for the orders of G and N are obtained; we use these bounds to classify all finite groups with normal subgroups having a small index and few G-classes. We also study the related problems when we consider only the set of G-classes of prime-power order elements contained in a normal subgroup., The results in this paper are part of the third author's Ph.D. thesis at the Jaume I University of Castellon, who is financially supported by a predoctoral grant of the Jaume I University. The research of the first and second authors is supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by the Jaume I University, grant P11B2015-77.

Let [Formula: see text] be a finite abelian group. As a consequence of the results of Di Vincenzo and Nardozza, we have that the generators of the [Formula: see text]-ideal of [Formula: see text]-graded identities of a [Formula: see text]-graded algebra in characteristic 0 and the generators of the [Formula: see text]-ideal of [Formula: see text]-graded identities of its tensor product by the infinite-dimensional Grassmann algebra [Formula: see text] endowed with the canonical grading have pairly the same degree. In this paper, we deal with [Formula: see text]-graded identities of [Formula: see text] over an infinite field of characteristic [Formula: see text], where [Formula: see text] is [Formula: see text] endowed with a specific [Formula: see text]-grading. We find identities of degree [Formula: see text] and [Formula: see text] while the maximal degree of a generator of the [Formula: see text]-graded identities of [Formula: see text] is [Formula: see text] if [Formula: see text]. Moreover, we find a basis of the [Formula: see text]-graded identities of [Formula: see text] and also a basis of multihomogeneous polynomials for the relatively free algebra. Finally, we compute the [Formula: see text]-graded Gelfand–Kirillov (GK) dimension of [Formula: see text].

We study some properties of the generalized local cohomology modules [Formula: see text] of [Formula: see text] with respect to a pair of ideals [Formula: see text] in Serre subcategories. Some results concerning to [Formula: see text]-cofinite modules are also given in this paper.

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Je\.z, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to $\mathsf{NSPACE}(n\log n)$ here., Comment: 38 pages, 3 figures. A conference version of this paper was presented at ICALP 2015, Kyoto (Japan), July 4-10, 2015, see http://arxiv.org/abs/1502.03426