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Let [Formula: see text] be a finite group acting on a group [Formula: see text] as a group automorphisms, [Formula: see text] the bar complex, [Formula: see text] the homology of invariant group chains and [Formula: see text] the cohomology invariant, both defined in Knudson’s paper “The homology of invariant group chains”. In this paper, we define the Tate homology of invariants [Formula: see text] and the Tate cohomology of invariants [Formula: see text]. When the coefficient [Formula: see text] is the abelian group of the integers, we proved that these groups are isomorphics, [Formula: see text]. Further, we prove that the homology and cohomology of invariant group chains are duals, [Formula: see text], [Formula: see text].

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.

In 2006, Cameron and Nešetřil introduced the following variant of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely generated substructures of the structure extends to an endomorphism of the structure. In several recent papers homomorphism-homogeneous objects in some well-known classes of algebras have been described (e.g. monounary algebras and lattices), while finite homomorphism-homogeneous groups were described in 1979 under the name of finite quasi-injective groups. In this paper we show that, in general, deciding homomorphism-homogeneity for finite algebras with finitely many fundamental operations and with at least one at least binary fundamental operation is coNP-complete. Therefore, unless P = coNP, there is no feasible characterization of finite homomorphism-homogeneous algebras of this kind.

Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M. In a previous paper, we give a construction G(M) for a graph with prescribed endomorphism monoid M known as a [Formula: see text]-graph. Using this construction, we derived bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. In this paper, we generalize the [Formula: see text]-graph construction and derive several new bounds for monoid classes not handled by our first paper. Among these are the so called "strong semilattices of C-semigroups" where C is one of the following: Groups, Abelian Groups, Rectangular Groups, and completely simple semigroups.

We investigate the interplay between the algebraic structure of a group G and arithmetic properties of its spectrum σ(G) which consists of the eigenvalues of all the inner automorphisms of G. A complex number λ is called an eigenvalue of a group automorphism A:G→G if φ◦A|H=λ·φ for some non-trivial homomorphism [Formula: see text] defined on an A-invariant subgroup H⊂G. It is shown that many properties of a group G (such as the presence of a finitely generated subgroup of infinite rank, nilpotence, periodicity, polycyclicity, etc.) are coded in its spectrum. In this paper the spectra are applied to the investigation of the so-called reversive properties of groups. The paper ends with a list of related open problems.

In this paper we compare the block product of C-varieties which we have defined in a previous paper to the two other constructions which were proposed on the one hand by D. Thérien and on the other hand by P. Jones and S. Pustejovsky. We prove that these three notions are equivalent. However, our definition is built on an explicit block product of two categories and then gives rise to a useful definition of the block product of two C-varieties which is needed for the theory of decompositions of monoids by means of wreath or block products.

This paper is the second of two papers devoted to the study of amalgamated free products of inverse semigroups. We use the characterization of the Schützenberger automata given previously by the author to obtain structural results and preservational properties of lower bounded amalgams. Haataja, Margolis and Meakin have shown that if [S1,S2;U is an amalgam of regular semigroups in which S1∩ S2=U is a full regular subsemigroup of S1 and S2, then the maximal subgroups of the amalgamated free product S1*U S2 may be described by the fundamental groups of certain bipartite graphs of groups. In this paper we show that the maximal subgroups of a lower bounded amalgam [S1,S2;U] are either isomorphic copies of subgroups of S1 and S2 or can be described by the same Bass-Serre theory characterization. It follows, as for the regular case, that if S1 and S2 are combinatorial, then the maximal subgroups of S1*U S2 are free. By studying the endomorphism monoids of the Schützenberger graphs we obtain a number of results concerning when inverse semigroup properties are preserved under the amalgamated free product construction. For example, necessary and sufficient conditions are given for S1*U S2 to be completely semisimple. Under a mild assumption we establish necessary and sufficient conditions for S1*U S2 to have finite ℛ-classes. This enables us to reprove a result of Cherubini, Meakin and Piochi on amalgams of free inverse semigroups. Finally we give sufficient conditions for S1*U S2 to be E-unitary.

In this paper, we give a lower bound of the number of maximal independent sets in a graph [Formula: see text] in terms of the Castelnuovo–Mumford regularity of its edge ideal. We also find two classes of graphs achieving this minimum value.

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 $G=C_q\times C_p\times C_p \times C_{m_1}$.

Let [Formula: see text] be a finite group. The set of all supercharacter theories of [Formula: see text] forms a lattice, where the join operation coincides with the join operation on the lattice of partitions of [Formula: see text], with partial order given by refinement. The meet operation is more complicated however, and seems difficult to describe. In this paper, we outline algorithms for determining the coarsest supercharacter theory whose associated partition is finer than a given partition. One of the primary applications is to compute the supercharacters and superclasses for the meet of two supercharacter theories.

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

International audience; Two finite words [Formula: see text] and [Formula: see text] are [Formula: see text]-binomially equivalent if, for each word [Formula: see text] of length at most [Formula: see text], [Formula: see text] appears the same number of times as a subsequence (i.e., as a scattered subword) of both [Formula: see text] and [Formula: see text]. This notion generalizes abelian equivalence. In this paper, we study the equivalence classes induced by the [Formula: see text]-binomial equivalence. We provide an algorithm generating the [Formula: see text]-binomial equivalence class of a word. For [Formula: see text] and alphabet of [Formula: see text] or more symbols, the language made of lexicographically least elements of every [Formula: see text]-binomial equivalence class and the language of singletons, i.e., the words whose [Formula: see text]-binomial equivalence class is restricted to a single element, are shown to be non-context-free. As a consequence of our discussions, we also prove that the submonoid generated by the generators of the free nil-[Formula: see text] group (also called free nilpotent group of class [Formula: see text]) on [Formula: see text] generators is isomorphic to the quotient of the free monoid [Formula: see text] by the [Formula: see text]-binomial equivalence.

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].

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.

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].

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

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.

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.

This paper concerns several ring theoretic properties related to matrices and polynomials. The basic properties of [Formula: see text]-reversible and power-Armendariz are studied. We provide a method by which one can always construct a power-Armendariz ring but neither symmetric nor Armendariz from given any symmetric ring. We investigate next various interesting relations among ring theoretic properties containing [Formula: see text]-reversibility and power-Armendariz condition.

For any biordered set of idempotents [Formula: see text] there is an initial object [Formula: see text], the free idempotent generated semigroup over[Formula: see text], in the category of semigroups generated by a set of idempotents biorder-isomorphic to [Formula: see text]. Recent research on [Formula: see text] has focused on the behavior of the maximal subgroups. Inspired by an example of Brittenham, Margolis and Meakin, several proofs have been offered that any group occurs as a maximal subgroup of some [Formula: see text], the latest being that of Dolinka and Ruškuc, who show that [Formula: see text] can be taken to be a band. From a result of Easdown, Sapir and Volkov, periodic elements of any [Formula: see text] lie in subgroups. However, little else is known of the “global” properties of [Formula: see text], other than that it need not be regular, even where [Formula: see text] is a semilattice. The aim of this paper is to deepen our understanding of the overall structure of [Formula: see text] in the case where [Formula: see text] is a biordered set with trivial products (for example, the biordered set of a poset) or where [Formula: see text] is the biordered set of a band [Formula: see text]. Since its introduction by Fountain in the late 1970s, the study of abundant and related semigroups has given rise to a deep and fruitful research area. The class of abundant semigroups extends that of regular semigroups in a natural way and itself is contained in the class of weakly abundant semigroups. Our main results show that (1) if [Formula: see text] is a biordered set with trivial products then [Formula: see text] is abundant and (if [Formula: see text] is finite) has solvable word problem, and (2) for any band [Formula: see text], the semigroup [Formula: see text] is weakly abundant and moreover satisfies a natural condition called the congruence condition. Further, [Formula: see text] is abundant for a normal band [Formula: see text] for which [Formula: see text] satisfies a given technical condition, and we give examples of such [Formula: see text]. On the other hand, we give an example of a normal band [Formula: see text] such that [Formula: see text] is not abundant.

Let [Formula: see text] be the moduli space of smooth, genus [Formula: see text] curves over an algebraically closed field [Formula: see text] of zero characteristic. Denote by [Formula: see text] the subset of [Formula: see text] of curves [Formula: see text] such that [Formula: see text] (as a finite nontrivial group) is isomorphic to a subgroup of [Formula: see text] and let [Formula: see text] be the subset of curves [Formula: see text] such that [Formula: see text], where [Formula: see text] is the full automorphism group of [Formula: see text]. Now, for an integer [Formula: see text], let [Formula: see text] be the subset of [Formula: see text] representing smooth, genus [Formula: see text] curves that admit a non-singular plane model of degree [Formula: see text] (in this case, [Formula: see text]) and consider the sets [Formula: see text] and [Formula: see text]. In this paper we first determine, for an arbitrary but a fixed degree [Formula: see text], an algorithm to list the possible values [Formula: see text] for which [Formula: see text] is non-empty, where [Formula: see text] denotes the cyclic group of order [Formula: see text]. In particular, we prove that [Formula: see text] should divide one of the integers: [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Secondly, consider a curve [Formula: see text] with [Formula: see text] such that [Formula: see text] has an element of “very large” order, in the sense that this element is of order [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Then we investigate the groups [Formula: see text] for which [Formula: see text] and also we determine the locus [Formula: see text] in these situations. Moreover, we work with the same question when [Formula: see text] has an element of “large” order: [Formula: see text], [Formula: see text] or [Formula: see text] with [Formula: see text] an integer.

Let [Formula: see text] be a nondegenerate projective integral curve of degree [Formula: see text] which is not linearly normal. In this paper, we continues the study begun in [E. Park, Projective curves of degree=codimension+2, Math. Z. 256 (2007) 685–697] for the minimal free resolution of [Formula: see text]. It is well-known that [Formula: see text] is an isomorphic projection of a rational normal curve [Formula: see text] from a point [Formula: see text]. Our main result is about how the graded Betti numbers of [Formula: see text] are determined by the rank of [Formula: see text] with respect to [Formula: see text], which is a measure of the relative location of [Formula: see text] from [Formula: see text].

Let [Formula: see text] be a smooth hypersurface of degree [Formula: see text] in a projective space [Formula: see text] and take a point [Formula: see text] in [Formula: see text]. Let [Formula: see text] be the relative Hilbert scheme parametrizing zero-dimensional subscheme, of length [Formula: see text], of fibers of the projection morphism [Formula: see text] from [Formula: see text]. In this paper we present an embedding of the relative Hilbert scheme [Formula: see text] into a weighted projective space and describe its defining ideal for general [Formula: see text]. We also study line bundles on the relative Hilbert scheme [Formula: see text] for [Formula: see text] and general [Formula: see text].

Several known results, by Rivin, Calegari-Maher and Sisto, show that an element $\phi_n\in Out(F_r)$, obtained after $n$ steps of a simple random walk on $Out(F_r)$, is fully irreducible with probability tending to 1 as $n\to\infty$. In this paper we construct a natural "train-track directed" random walk $\mathcal W$ on $Out(F_r)$ (where $r\ge 3$). We show that, for the element $\phi_n\in Out(F_r)$, obtained after $n$ steps of this random walk, with asymptotically positive probability the element $\phi_n$ has the following properties: $\phi_n$ is an ageometric fully irreducible, which admits a train-track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that $\phi_n$ has "rotationless index" $\frac{3}{2}-r$ (so that the geometric index of the attracting tree $T_{\phi_n}$ of $\phi_n$ is $2r-3$), has index list $\{\frac{3}{2}-r\}$ and the ideal Whitehead graph being the complete graph on $2r-1$ vertices, and that the axis bundle of $\phi_n$ in the Outer space $CV_r$ consists of a single axis., Comment: Various small updates, incorporating the referee's comments; to appear in the International Journal of Algebra and Computation

In this paper, we describe the automorphism groups of the endomorphism semigroups of free Burnside groups B(m, n) for odd exponents n ≥ 1003. We prove, that the groups Aut (B(m, n)) and Aut ( End (B(m, n))) are canonically isomorphic. In particular, if the groups Aut ( End (B(m, n))) and Aut ( End (B(k, n))) are isomorphic, then m = k.

In this paper, we record a proof that a relative moduli stack of principal G-bundles over a moduli stack of genus g, n-pointed twisted curves is a smooth Artin stack locally of finite type with the universal family. As an application, we reprove that the stack 𝒦g, n(𝒳, β) of Abramovich, Graber and Vistoli is a Deligne–Mumford stack of finite type when 𝒳 is a quotient stack.

It is well known that under mild conditions on a connected topological space $\mathcal X$, connected covers of $\mathcal X$ may be classified via conjugacy classes of subgroups of the fundamental group of $\mathcal X$. In this paper, we extend these results to the study of immersions into 2-dimensional CW-complexes. An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between CW-complexes is a cellular map such that each point $y \in {\mathcal D}$ has a neighborhood $U$ that is mapped homeomorphically onto $f(U)$ by $f$. In order to classify immersions into a 2-dimensional CW-complex $\mathcal C$, we need to replace the fundamental group of $\mathcal C$ by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex.

In [Bardakov and Gongopadhyay, Palindromic width of free nilpotent groups, J. Algebra 402 (2014) 379–391] the authors provided a bound for the palindromic widths of free abelian-by-nilpotent group ANn of rank n and free nilpotent group N n,r of rank n and step r. In the present paper, we study palindromic widths of groups [Formula: see text] and [Formula: see text]. We denote by [Formula: see text] the quotient of the group Gn = 〈x1, …, xn〉, which is free in some variety by the normal subgroup generated by [Formula: see text]. We prove that the palindromic width of the quotient [Formula: see text] is finite and bounded by 3n. We also prove that the palindromic width of the quotient [Formula: see text] is precisely 2(n - 1). As a corollary to this result, we improve the lower bound of the palindromic width of N n,r. We also improve the bound of the palindromic width of a free metabelian group. We prove that the palindromic width of a free metabelian group of rank n is at most 4n - 1.

Let S be a numerical monoid with minimal generating set 〈n1, …, nt〉. For m ∈ S, if [Formula: see text], then [Formula: see text] is called a factorization length of m. We denote by ℒ(m) = {m1, …, mk} (where mi < mi+1 for each 1 ≤ i < k) the set of all possible factorization lengths of m. The Delta set of m is defined by Δ(m) = {mi+1 - mi | 1 ≤ i < k} and the Delta set of S by Δ(S) = ⋃m∈SΔ(m). In this paper, we expand on the study of Δ(S) begun in [C. Bowles, S. T. Chapman, N. Kaplan and D. Reiser, On delta sets of numerical monoids, J. Algebra Appl. 5 (2006) 1–24] in the following manner. Let r1, r2, …, rt be an increasing sequence of positive integers and Mn = 〈n, n + r1, n + r2, …, n + rt〉 a numerical monoid where n is some positive integer. We prove that there exists a positive integer N such that if n > N then |Δ(Mn)| = 1. If t = 2 and r1 and r2 are relatively prime, then we determine a value for N which is sharp.

Let C be a one- or two-sided Kazhdan–Lusztig cell in a Coxeter group (W, S), and let Red (C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Red (C) is regular. In this paper, we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Casselman's.

In this paper, we study the conjugacy problem in polycyclic groups. Our main result is that we construct polycyclic groups Gn whose conjugacy problem is at least as hard as the subset sum problem with n indeterminates. As such, the uniform conjugacy problem over the groups Gn is NP-complete where the parameters of the problem are taken in terms of n and the length of the elements given on input.

Given a family of algebraic curves whose coefficients depend continuously on a parameter t ∈ U ⊂ ℝ (U a union of real intervals) we address the problem of computing the topology types in the family. Under certain conditions, we provide a method to compute a univariate real function R*(t), with the property that the topology of the family stays invariant along every real interval I ⊂ U not containing any real root of R*(t). So, if R*(t) has finitely many real roots the topology types in the family can be computed. We apply this result to provide bounds on the number of topology types of the family in certain cases, including the important case when the coefficients belong to a Pfaffian chain, and to analyze the existence of certain degeneracies in the real part of a surface whose family of level curves corresponds to the type studied here. The results of the paper can be seen as generalizations of previous works [Alcázar, Applications of level curves to some problems on algebraic surfaces, in Contribuciones Científicas en Honor de Mirian Andrés Gòmez, eds. Lambán, Romero and Rubio (Universidad de La Rioja, Servicio de Publicaciones, 2010), pp. 105–122, Alcázar, Schicho and Sendra, A delineability-based method for computing critical sets of algebraic surfaces, J. Symbolic Comput.42 (2007) 678–691] on the topology of families algebraically depending on a parameter t.

Given a collection [Formula: see text] of k2 commutative polynomials in 2k2 variables, the objective is to find a condensed representation for these polynomials in terms of a single non-commutative (nc) polynomial p(X, Y) in two k × k matrix variables X and Y. In this paper, we develop algorithms that will generically determine whether the given family [Formula: see text] has a nc representation and will produce such a representation if it exists. In particular, we determine an open, dense subset of the space of nc polynomials in two variables that satisfies the following property: if a family [Formula: see text] of polynomials admits a nc representation in this subset, then our algorithms will determine this representation.

This paper provides an abstract characterization of the monoids that appear as monoids of subalgebras of the square of finite minimal algebras admitting a majority term. As a tool, a matrix construction similar to the one introduced in [A characterization of the inverse monoid of bi-congruences of certain algebras, Int. J. Algebra Comput.6 (2009) 791–808] is used.

Hassler Whitney's theorem of 1931 reduces the task of finding proper, vertex 4-colorings of triangulations of the 2-sphere to finding such colorings for the class \(\mathfrak H\) of triangulations of the 2-sphere that have a Hamiltonian circuit. This has been used by Whitney and others from 1936 to the present to find equivalent reformulations of the 4 Color Theorem (4CT). Recently there has been activity to try to use some of these reformuations to find a shorter proof of the 4CT. Every triangulation in \(\mathfrak H\) has a dual graph that is a union of two binary trees with the same number of leaves. Elements of a group known as Thompson's group \(F\) are equivalence classes of pairs of binary trees with the same number of leaves. This paper explores this resemblance and finds that some recent reformulations of the 4CT are essentially attempting to color elements of \(\mathfrak H\) using expressions of elements of \(F\) as words in a certain generating set for \(F\). From this, we derive information about not just the colorability of certain elements of \(\mathfrak H\), but also about all possible ways to color these elements. Because of this we raise (and answer some) questions about enumeration. We also bring in an extension \(E\) of the group \(F\) and ask whether certain elements ``parametrize'' the set of all colorings of the elements of \(\mathfrak H\) that use all four colors., Comment: 74 pages, table of contents, index. Revision of V.3

In this paper, we firstly establish Composition-Diamond lemma for $\Omega$-algebras. We give a Gr\"{o}bner-Shirshov basis of the free $L$-algebra as a quotient algebra of a free $\Omega$-algebra, and then the normal form of the free $L$-algebra is obtained. We secondly establish Composition-Diamond lemma for $L$-algebras. As applications, we give Gr\"{o}bner-Shirshov bases of the free dialgebra and the free product of two $L$-algebras, and then we show four embedding theorems of $L$-algebras: 1) Every countably generated $L$-algebra can be embedded into a two-generated $L$-algebra. 2) Every $L$-algebra can be embedded into a simple $L$-algebra. 3) Every countably generated $L$-algebra over a countable field can be embedded into a simple two-generated $L$-algebra. 4) Three arbitrary $L$-algebras $A$, $B$, $C$ over a field $k$ can be embedded into a simple $L$-algebra generated by $B$ and $C$ if $|k|\leq \dim(B*C)$ and $|A|\leq|B*C|$, where $B*C$ is the free product of $B$ and $C$., Comment: 22 pages

The aim of this paper is to show that the weak subalgebra lattice uniquely determines the subalgebra lattice for locally finite algebras of a fixed finite type. However, this algebraic result turns out to be a very particular case of the following hypergraph result (which is interesting itself): A total directed hypergraph D of finite type is uniquely determined, in the class of all the directed hypergraphs of this type, by its skeleton up to the orientation of some pairwise edge-disjoint directed hypercycles and hyperpaths. The skeleton of D is a hypergraph obtained from D by omitting the orientation of all edges.

In a recent paper we assigned to each positive variety [Formula: see text] and each nonnegative integer k the class of all finite unions of finite intersections or Boolean combinations of the languages of the form L0a1L1a2…aℓLℓ, where ℓ ≤ k, a1,…,aℓ are letters and L0,…,Lℓ are in the variety [Formula: see text]. For these polynomial operators on a wide class of varieties we gave a certain algebraic counterpart in terms of identities satisfied by syntactic (ordered) monoids of languages considered. Here we apply our constructions to particular examples of varieties of languages obtaining four hierarchies of (positive) varieties. Two of them have the 3/2 level of the Straubing–Thérien hierarchy as their limits, and two others tend to the level two of this hierarchy. We concentrate here on the existence of finite bases of identities for corresponding pseudovarieties of (ordered) monoids and we are looking for inclusions among those varieties.

In this partly expository paper, we study the set [Formula: see text] of groups of orientation-preserving homeomorphisms of the circle S1 which do not admit non-abelian free subgroups. We use classical results about homeomorphisms of the circle and elementary dynamical methods to derive various new and old results about the groups in [Formula: see text]. Of the known results, we include some results from a family of results of Beklaryan and Malyutin, and we also give a new proof of a theorem of Margulis. Our primary new results include a detailed classification of the solvable subgroups of R. Thompson's group T.

In this paper we mainly study sporadic groups in term of p-local rank, especially the p-local ranks of those groups will be given when p is an odd prime.

The main purpose of this paper is to improve the bound of complexity of the well-known algorithms on polynomial ideals having complexities polynomial in dn, where d is the maximal degree of input polynomials and n is the number of variables. Instead of this bound, we present the more accurate bound max {S, Dn} where S is the size of the input polynomials in dense representation, and D is the arithmetic mean value of the degrees of input polynomials.

We define and give explicit construction of the universal tree-graded space with a given collection of pieces. We apply that to proving uniqueness of asymptotic cones of relatively hyperbolic groups whose peripheral subgroups have unique asymptotic cones. Modulo the Continuum Hypothesis, we show that if an asymptotic cone of a geodesic metric space is homogeneous and has cut points, then it is the universal tree-graded space with pieces - maximal connected subsets without their own cut points. Thus it is completely determined by its collection of pieces., 24 pages; v2: a reference added, we also made some changes suggested to us by Yves de Cornulier; v3: the paper is accepted in IJAC

We introduce the peak normal form for elements of the Baumslag–Solitar groups BS(p,q). This normal form is very close to the length-lexicographical normal form, but more symmetric. Both normal forms are geodesic. This means the normal form of an element u-1v yields the shortest path between u and v in the Cayley graph. For horocyclic elements the peak normal form and the length-lexicographical normal form coincide. The main result of this paper is that we can compute the peak normal form in polynomial time if p divides q. As consequence we can compute geodesic lengths in this case. In particular, this gives a partial answer to Question 1 in [4] For arbitrary p and q it is possible to compute the peak normal form (length-lexicographical normal form resp.) also the for elements in the horocyclic subgroup and, more generally, for elements which we call hills. This approach leads to a linear time reduction of the problem of computing geodesics to the problem of computing geodesics for Britton-reduced words where the t-sequence starts with t-1 and ends with t.

α-Irreducible and α-Strongly Irreducible Ideals of a ring have been characterized in [2] and [4]. A complete lattice which is generated by compact elements is called an algebraic lattice for the simple reason that every such lattice is isomorphic to the lattice of subalgebras of a suitable universal algebra and vice-versa. In this paper, we characterize the irreducible elements and strongly irreducible elements in an algebraic lattice, which extends the results in [4] to arbitrary algebraic lattices. Also we obtain certain necessary and sufficient conditions, in terms of irreducible elements, for an algebraic lattice to satisfy the complete distributivity.

We investigate the descriptive complexity of finite abelian groups. Using Ehrenfeucht–Fraïssé games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let G1 and G2 be a pair of non-isomorphic finite abelian groups. Then there exists a positive integer m that divides one of the two groups' orders such that the following holds: (1) there exists a first-order sentence φ that distinguishes G1 and G2 such that φ is existential, has quantifier depth O(log m), and has at most 5 variables and (2) if φ is a sentence that distinguishes G1 and G2 then φ must have quantifier depth Ω( log m). These results are applied to (1) get bounds on the first-order distinguishability of dihedral groups, (2) to prove that on the class of finite groups both cyclicity and the closure of a single element are not first-order definable, and (3) give a different proof for the first-order undefinability of simplicity, nilpotency, and the normal closure of a single element on the class of finite groups (their undefinability were shown by Koponen and Luosto in an unpublished paper).

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 is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.