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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].

The paper is a part of an ongoing program which aims to show that the existential theory in free groups (hyperbolic groups or even toral relatively hyperbolic) is NP-complete. For that we study compression of solutions with straight-line programs (SLPs) as suggested originally by Plandowski and Rytter in the context of a single word equation. We review some basic results on SLPs and give full proofs in order to keep this fundamental part of the program self-contained. Next we study systems of equations with constraints in free groups and more generally in free products of abelian groups. We show how to compress minimal solutions with extended Parikh-constraints. This type of constraints allows to express semi linear conditions as e.g. alphabetic information. The result relies on some combinatorial analysis and has not been shown elsewhere. We show similar compression results for Boolean formula of equations over a torsion-free $\delta$-hyperbolic group. The situation is much more delicate than in free groups. As byproduct we improve the estimation of the "capacity" constant used by Rips and Sela in their paper "Canonical representatives and equations in hyperbolic groups" from a double-exponential bound in $\delta$ to some single-exponential bound. The final section shows compression results for toral relatively hyperbolic group using the work of Dahmani: We show that given a system of equations over a fixed toral relatively hyperbolic group, for every solution of length $N$ there is an SLP for another solution such that the size of the SLP is bounded by some polynomial $p(s+ \log N)$ where $s$ is the size of the system., Comment: 28 pages

This paper contributes to the theory of recursively presented (see Higman [Subgroups of finitely presented groups, Proc. R. Soc. Ser. A 262 (1961) 455–475]) infinitely generated abelian groups with solvable word problem. Mal'cev [On recursive Abelian groups, Dokl. Akad. Nauk SSSR 146 (1962) 1009–1012] and independently Rabin [Computable algebra, general theory and theory of computable fields, Trans. Amer. Math. Soc. 95 (1960) 341–360] initiated the study of such groups in the early 1960's. In this paper, we develop a technique that we call iterated effective embeddings. The significance of our new technique is that it extends existing methods from the realm of iterated 0″ arguments to iterated 0‴ ones. This is a new phenomenon in computable algebra. We use this technique to confirm a 30 year-old conjecture of Ash, Knight and Oates [Recursive abelian p-groups of small length, https://dl.dropbox.com/u/4752353/Homepage/AKO.pdf ]. More specifically, Ash, Knight and Oates [Recursive abelian p-groups of small length. https://dl.dropbox.com/u/4752353/Homepage/AKO.pdf ] conjectured that there exists a computable reduced abelian p-group of Ulm type ω such that its effective invariants, defined using limitwise monotonic functions, cannot be found uniformly. We construct a computable reduced abelian p-group of Ulm type ω where its invariants are at the maximum possible level of non-uniformity. The result confirms the conjecture in a strong way, and it provides us with an explanation of why computable reduced p-groups of Ulm type ω seem hard to classify in general. We also use p-basic trees and their iterated embeddings to solve a problem posed in [W. Calvert, D. Cenzer, V. S. Harizanov and A. Morozov, Effective categoricity of abelian p-groups, Ann. Pure Appl. Logic 159(1–2) (2009) 187–197].

This is the first of two articles studying the structure of left adequate and, more generally, of left Ehresmann monoids. Motivated by a careful analysis of normal forms, we introduce here a concept of proper for a left adequate monoid M. In fact, our notion is that of T-proper, where T is a submonoid of M. We show that any left adequate monoid M has an X*-proper cover for some set X, that is, there is a left adequate monoid [Formula: see text] that is X*-proper, and an idempotent separating surjective morphism [Formula: see text] of the appropriate type. Given this result, we may deduce that the free left adequate monoid on any set X is X*-proper. In a subsequent paper, we show how to construct T-proper left adequate monoids from any monoid T acting via order-preserving maps on a semilattice with identity, and prove that the free left adequate monoid is of this form. An alternative description of the free left adequate monoid will appear in a paper of Kambites. We show how to obtain the labeled trees appearing in his result from our structure theorem. Our results apply to the wider class of left Ehresmann monoids, and we give them in full generality. We also indicate how to obtain some of the analogous results in the two-sided case. This paper and its sequel, and the two of Kambites on free (left) adequate semigroups, demonstrate the rich but accessible structure of (left) adequate semigroups and monoids, introduced with startling insight by Fountain some 30 years ago.

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.

This paper is adjacent to [5] and motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ0 of all finitely generated free algebras in Θ. The notions of geometrically equivalent algebras and weakly geometrically equivalent algebras (cf. [2]) are intimately related to the above problem (the latter notion being related to automorphisms of the category Θ0 is replaced in the present paper by a more natural one: "automorphic equivalence"). Two algebras are said to have the same algebraic geometry if they are automorphically equivalent. If an automorphic equivalence of algebras is provided by an inner automorphism of Θ0, then this equivalence coincides with the geometric equivalence. However, not all automorphisms of Θ0 are inner (see [5, Theorem 2]). In this paper we study the situation when the automorphic equivalence of algebras is provided by a non-inner automorphism. Assuming Θ to be an IBM variety of one-sorted algebras we prove that two algebras H1 and H2 are automorphically equivalent in Θ if and only if H1 is geometrically equivalent to [Formula: see text], where [Formula: see text] is the algebra with the same domain as H2 and with the new operations induced by some system of words W = {wω(x)|ω ∈ Ω, Ω is the signature of algebras in Θ. Thus, the notion of automorphic equivalence is reduced to a more simple one of geometric equivalence.

This paper considers extended finite automata over monoids, in the sense of Dassow and Mitrana. We show that the family of languages accepted by extended finite automata over a monoid K is controlled by the word problem of K in a precisely stated manner. We also point out a critical error in the proof of the main result in the paper by Dassow and Mitrana. However as one consequence of our approach, by analyzing a certain word problem, we obtain a complete proof of this result, namely that the family of languages accepted by extended finite automata over the free group of rank two is exactly the family of context-free languages. We further deduce that along with the free group of rank two, the only finitely generated groups with this property are precisely the groups that have a nonabelian free subgroup of finite index.

In this paper we prove certain Hurwitz equivalence properties of Bn. In particular we prove that for n = 3 every two Artin factorizations of [Formula: see text] of the form Hi1 ⋯ Hi6, Fj1 ⋯ Fj6 (with ik, jk ∈ {1, 2}) where {H1, H2}, {F1, F2} are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti's example). We construct a new invariant based on Hurwitz equivalence of factorizations, to distinguish among diffeomorphic surfaces which are not deformation of each other. The main result of this paper will help us to compute the new invariant.

A commutative residuated lattice, is an ordered algebraic structure [Formula: see text], where (L, ·, e) is a commutative monoid, (L, ∧, ∨) is a lattice, and the operation → satisfies the equivalences [Formula: see text] for a, b, c ∊ L. The class of all commutative residuated lattices, denoted by [Formula: see text], is a finitely based variety of algebras. Historically speaking, our study draws primary inspiration from the work of M. Ward and R. P. Dilworth appearing in a series of important papers [9, 10, 19–22]. In the ensuing decades special examples of commutative, residuated lattices have received considerable attention, but we believe that this is the first time that a comprehensive theory on the structure of residuated lattices has been presented from the viewpoint of universal algebra. In particular, we show that [Formula: see text] is an "ideal variety" in the sense that its congruences correspond to order-convex subalgebras. As a consequence of the general theory, we present an equational basis for the subvariety [Formula: see text] generated by all commutative, residuated chains. We conclude the paper by proving that the congruence lattice of each member of [Formula: see text] is an algebraic, distributive lattice whose meet-prime elements form a root-system (dual tree). This result, together with the main results in [12, 18], will be used in a future publication to analyze the structure of finite members of [Formula: see text]. A comprehensive study of, not necessarily commutative, residuated lattices is presented in [4].

h(a) was studied in [2] from a combinatorial perspective. We found that it could be pictured as a generalization of the generating function for the Zeta Polynomial Z(P, n) of a finite poset (P,≥) [3; §3.11]. In this paper we continue our study of {h(a) | a ∈ S}. But this time we approach it from a more algebraic perspective. For example (i) What is the recurrence formula for {h1(a), h2(a), . . . } and how are these recurrance formulae (as a ∈ S varies) related? (See Proposition 2.4.) (ii) Is there a canonical collection {R0, R1, . . . , Rn−1} of rational functions that forms a basis of SpanQ{h(a) | a ∈ S}? (See Proposition 2.2.) (iii) What role does Q[S] (the monoid algebra) play in the discussion? (See Section 2.) We consider the example S = Mn(Fq) in detail. In a future paper we shall use the results of this paper to obtain explicit calculations for a large class of finite monoids.

The author proves for each of the operations # equalling *, °, **, □ or m, there exist pseudovarieties of finite semigroups [Formula: see text] and [Formula: see text] with decidable membership problems, such that [Formula: see text] has an undecidable membership problem. In addition, if [Formula: see text] denotes the pseudovariety of all finite aperiodic semigroups, [Formula: see text] denotes the pseudovariety of all finite groups, and [Formula: see text](E) denotes the pseudovariety of all finite aperiodic semigroups satisfying the finite number of equations E, then it is proved that there exists E such that [Formula: see text](E) has an undecidable membership problem. Note [Formula: see text] equals all semigroups of complexity ≤1. Section 6 is expanded into a joint paper with B. Steinberg, following this paper.

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.

Given any amalgam [Formula: see text] of inverse semigroups, we show how to construct an amalgam [Formula: see text] such that [Formula: see text] is embedded into [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] and, for any [Formula: see text] and [Formula: see text] with [Formula: see text] in [Formula: see text], where [Formula: see text], there exists [Formula: see text] with [Formula: see text] in [Formula: see text]; that is, [Formula: see text] is a lower bounded subsemigroup of [Formula: see text] and [Formula: see text]. A recent paper by the author describes the Schützenberger automata of [Formula: see text], for an amalgam [Formula: see text] where [Formula: see text] is lower bounded in [Formula: see text] and [Formula: see text], giving conditions for [Formula: see text] to have decidable word problem. Thus we can study [Formula: see text] by considering [Formula: see text]. As an example, we generalize results by Cherubini, Jajcayová, Meakin, Piochi and Rodaro on amalgams of finite inverse semigroups.

Let $G$ be a finite simple graph and $J(G)$ denote its cover ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we show that all symbolic powers of cover ideals of certain vertex decomposable graphs have linear quotients. Using these results, we give various conditions on a subset $S$ of the vertices of $G$ so that all symbolic powers of vertex cover ideals of $G \cup W(S)$, obtained from $G$ by adding a whisker to each vertex in $S$, have linear quotients. For instance, if $S$ is a vertex cover of $G$, then all symbolic powers of $J(G \cup W(S))$ have linear quotients. Moreover, we compute the Castelnuovo-Mumford regularity of symbolic powers of certain cover ideals., Comment: 15 pages, 2 figures

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

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

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 R be a ring, and let S be a strictly ordered monoid. The generalized power series ring R[[S]] is a common generalization of polynomial rings, Laurent polynomial rings, power series rings, Laurent series rings, Mal'cev–Neumann series rings, monoid rings and group rings. In this paper, we examine which conditions on R and S are necessary and which are sufficient for the generalized power series ring R[[S]] to be semilocal right Bézout or semilocal right distributive. In the case where S is a strictly totally ordered monoid we characterize generalized power series rings R[[S]] that are semilocal right distributive or semilocal right Bézout (the latter under the additional assumption that S is not a group).

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.

The study of symmetric rings has important roles in ring theory and module theory. We investigate the structure of ring properties related to symmetric rings and introduce H-symmetric and π-symmetric as generalizations. We construct a non-symmetric reversible ring whose basic structure is infinite-dimensional, comparing with the finite-dimensional such rings of Anderson, Camillo and Marks. The structure of π-reversible rings (with or without identity) of minimal order is completely investigated. The properties of zero-dividing polynomials over IFP rings are studied more to show that polynomial rings over symmetric rings are π-symmetric. It is also proved that all conditions in relation with our arguments in this paper are equivalent for regular or locally finite rings.

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.

Conway hemirings are Conway semirings without a multiplicative unit. We also define iteration hemirings as Conway hemirings satisfying certain identities associated with the finite groups. Iteration hemirings are iteration semirings without a multiplicative unit. We provide an analysis of the relationship between Conway hemirings and (partial) Conway semirings and describe several free constructions. In the second part of the paper we define and study hemimodules of Conway and iteration hemirings, and show their applicability in the analysis of quantitative aspects of the infinitary behavior of weighted transition systems. These include discounted and average computations of weights investigated recently.

In this paper, we study the structure of some Noetherian modules over group rings and deduce some statements regarding the structure of the groups involved. More precisely, we consider a module A over a group ring RG with the following property: A is a Noetherian RH-module for every subgroup H, which is not contained in the centralizer CG(A). If G is some generalized soluble group and R is a locally finite field or some Dedekind domain, we describe the structure of G/CG(A).

Let R = ⨁α∈Γ Rα be a (Γ-)graded integral domain and let H be the multiplicatively closed set of nonzero homogeneous elements of R. In this paper, we introduce the concepts of graded almost GCD-domains (graded AGCD-domain) and graded almost Prüfer v-multiplication domains (graded AP v MD ). Among other things, we show that if R is integrally closed, then (1) H is an almost lcm splitting set of R if and only if R is a graded AGCD-domain and (2) R is a graded AP v MD if and only if R is a P v MD . We also give an example of a (non-integrally closed) graded AGCD-domain (respectively, graded AP v MD ) that is not an almost GCD-domain (respectively, almost Prüfer v-multiplication domain.

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.

In this paper, we generalize the classical Gröbner basis technique to prove the existence and present a method of computation of a dimension polynomial in two variables associated with a finitely generated D-module, that is, a finitely generated module over a Weyl algebra. We also present corresponding algorithms and examples of computation of such polynomials and show that bivariate dimension polynomials contain invariants of a D-module, which are not carried by its Bernstein dimension polynomial. Then we apply the obtained results to the isomorphism problem for D-modules.

This paper examines the computational complexity of determining whether or not an algebra satisfies a certain Mal'Cev condition. First, we define a class of Mal'Cev conditions whose satisfaction can be determined in polynomial time (special cube term satisfying the DCP) when the algebra in question is idempotent and provide an algorithm through which this determination may be made. The aforementioned class notably includes near unanimity terms and edge terms of fixed arity. Second, we define a different class of Mal'Cev conditions whose satisfaction, in general, requires exponential time to determine (Mal'Cev conditions satisfiable by CPB0 operations).

Unitriangular factorization is a presentation of a linear group as a product of unipotent radicals of a Borel subgroup and its opposite. Whether this decomposition is known for Chevalley groups over rings of stable rank 1 and some Dedekind rings of arithmetic type, the case of twisted groups has been studied only over finite fields. In the present paper we give a much simpler proof for twisted groups over finite fields and the field of complex numbers.

This paper presents a new methodology to compute the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.

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 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.

It is well known that in a reductive group, the Borel subgroup is a product of the maximal torus and the one-dimensional positive root subgroups. The purpose of this paper is to find an analog of this result for reductive monoids. Via a study of reductive monoids of semisimple rank 1, we introduce the concept of root semigroups. By analyzing the associated root elements in the Renner monoid, we show that the closure of the Borel subgroup is generated by the maximal torus and positive root semigroups. Along the way we generalize the Jordan decomposition of algebraic groups to reductive monoids.

A limit variety is a variety that is minimal with respect to being nonfinitely based. This paper presents a new infinite series of limit semigroup varieties, each of which is generated by a finite 0-simple semigroup with Abelian subgroups. These varieties exhaust all limit varieties generated by completely 0-simple semigroups with Abelian subgroups.

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).

In this paper, we characterize the set of all ternary algebraic (or polynomial) operations of idempotent algebras that have at least one binary and one ternary algebraic operation depending on every variable, and there exists an integer r > 2 such that there is not any r-ary algebraic operation depending on every variable. We prove that this set forms a finite DeMorgan algebra with a fixed element and then we characterize this DeMorgan algebra.

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.

Thin Lie algebras are graded Lie algebras [Formula: see text] with dim Li ≤ 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous condition for pro-p-groups. The two-dimensional homogeneous components of L, which include L1, are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of classical or of Cartan type depending on the location of the second diamond. The goal of this paper is a description of the initial structure of a thin Lie algebra, up to the second diamond. Specifically, if Lk is the second diamond of L, then the quotient L/Lk is a graded Lie algebras of maximal class. In odd characteristic p, the quotient L/Lk is known to be metabelian, and hence uniquely determined up to isomorphism by its dimension k, which ranges in an explicitly known set of possible values: 3, 5, a power of p, or one less than twice a power of p. However, the quotient L/Lk need not be metabelian in characteristic two. We describe here all the possibilities for L/Lk up to isomorphism. In particular, we prove that k + 1 equals a power of two.

This paper concerns the combinatorics of the orbit Hecke algebra associated with the orbit of a two sided Weyl group action on the Renner monoid of a finite monoid of Lie type, $M$. It is shown by Putcha in \cite{Putcha97} that the Kazhdan-Lusztig involution (\cite{KL79}) can be extended to the orbit Hecke algebra which enables one to define the $R$-polynomials of the intervals contained in a given orbit. Using the $R$-polynomials, we calculate the M\"obius function of the Bruhat-Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group., Comment: 12 pages

In this paper, we verify Dade's invariant conjecture for the general unitary group GU 4(q2), q a power of an odd prime p, in the defining characteristic p. Together with the results in [3], this completes the proof of Dade's invariant conjecture for the group GU 4(q2) in the defining characteristic.

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 of Ore extensions have been characterized. In this paper a relation between completely prime ideals of a ring R and those of R[x; σ, δ] has been given; σ is an automorphisms of R and δ is a σ-derivation of R. It has been proved that if P is a completely prime ideal of R such that σ(P) = P and δ(P) ⊆ P, then P[x; σ, δ] is a completely prime ideal of R[x; σ, δ]. It has also been proved that this type of relation does not hold for strongly prime ideals.

Let us consider the group G = 〈x, y | xm = yn〉 with m and n nonzero integers. The set R(G) of representations of G over SL(2, ℂ) is a four-dimensional algebraic variety which is an invariant of G. In this paper the number of irreducible components of R(G) together with their dimensions are computed. We also study the set of metabelian representations of this family of groups. Finally, the behavior of the projection t : R(G) → X(G), where X(G) is the character variety of the group, and some combinatorial aspects of R(G) are investigated.

Let G be a finite group and π(G) = {p1, p2,…,pk}. For p ∈ π(G), we put deg (p) := |{q ∈ π(G)|p ~ q}|, which is called the degree of p. We also define D(G) := ( deg (p1), deg (p2), …, deg (pk)), where p1 < p2 < ⋯ < pk, which is called the degree pattern of G. Using the classification of finite simple groups, we characterize the projective general linear group PGL(2,q)(q a prime power) by its order and degree pattern in the present paper.

The dimension of a variety V of algebras of a given type was introduced by E. Graczyńska and D. Schweigert in [7] as the cardinality of the set of all derived varieties of V which are properly contained in V. In this paper, we characterize all solid varieties of dimensions 0, 1, and 2; prove that the dimension of a variety of finite type is at most ℵ0; give an example of a variety which has infinite dimension; and show that for every n ∈ ℕ there is a variety with dimension n. Finally, we show that the dimension of a variety is related to the concept of the semantical kernel of a hypersubstitution and apply this connection to calculate the dimension of the class of all algebras of type τ = (n).

This paper provides an abstract characterization of the inverse monoids that appear as monoids of bi-congruences of finite minimal algebras generating arithmetical varieties. As a tool, a matrix construction is introduced which might be of independent interest in inverse semigroup theory. Using this construction as well as Ramsey's theorem, we embed a certain kind of inverse monoid into a factorizable monoid of the same kind. As noticed by M. Lawson, this embedding entails that the embedded finite monoids have finite F-unitary cover.

Differential modes provide examples of modes that do not embed as subreducts into semimodules over commutative semirings. The current paper studies differential modes, so-called Szendrei differential modes, which actually do embed into semimodules. These algebras form a variety. The main result states that the lattice of nontrivial subvarieties is dually isomorphic to the (nonmodular) lattice of congruences of the free commutative monoid on two generators. Consequently, all varieties of Szendrei differential modes are finitely based.

The main tools in the theory of n-ary hyperstructures are the fundamental relations. The fundamental relation on an n-ary hypersemigroup is defined as the smallest equivalence relation so that the quotient would be the n-ary semigroup. In this paper we study neutral elements in n-ary hypersemigroups and introduce a new strongly compatible equivalence relation on an n-ary hypersemigroup so that the quotient is a commutative n-ary semigroup. Also we determine some necessary and sufficient conditions so that this relation is transitive. Finally, we prove that this relation is transitive on an n-ary hypergroup with neutral (identity) element.