On rings whose number of centralizers of ideals is finite

LetR be a ring. For the setF of all nonzero ideals ofR, we introduce an equivalence relation inF as follows: For idealsI andJ, I～J if and only ifV _R (I)=V _R(J), whereV _R() is the centralizer inR. LetI _R=F/～. Then we can see thatn(I _R), the cardinality ofI _R, is 1 if and only ifR is either a prime ring or a commutative ring (Theorem 1.1). An idealI ofR is said to be a commutator ideal ifI is generated by{st?ts; s∈S, t∈T} for subsetS andT ofR, andR is said to be a ring with (N) if any commutator ideal contains no nonzero nilpotent ideals. Then we have the following main theorem: LetR be a ring with (N). Thenn(I _R) is finite if and only ifR is isomorphic to an irredundant subdirect sum ofS⊕Z whereS is a finite direct sum of non commutative prime rings andZ is a commutative ring (Theorem 2.1). Finally, we show that the existence of a ringR such thatn(I _R)=m for any given natural numberm.     

Modules over group rings of solvable groups with rank restrictions on subgroups

Let A be an R G-module over a commutative ring R, where G is a group of infinite section p-rank (0-rank), C _G (A) = 1, A is not a Noetherian R-module, and the quotient A/C _A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (0-rank). We describe the structure of solvable groups G of this type.     

Regular principal factors in free objects in varieties in the interval [B 2 ,NB 2 ∨G n ]

The author has shown previously how to associate a completely 0-simple semigroup with a connected bipartite graph containing labelled edges and how to describe the regular principal factors in the free objects in the Rees-Sushkevich varieties RS _n generated by all completely 0-simple semigroups over groups from the Burnside variety G _n of groups of exponent dividing a positive integer n by employing this graphical construction. Here we consider the analogous problem for varieties containing the variety B ₂ , generated by the five element Brandt semigroup B ₂, and contained in the variety NB ₂ ∨G _n where NB ₂ is the variety generated by all left and right zero semigroups together with B ₂. The interval [NB ₂ ,NB ₂ ∨G _n ] is of particular interest as it is an important interval, consisting entirely of varieties generated by completely 0-simple semigroups, in the lattice of subvarieties of RS _n .     

Finitely Related Clones and Algebras with Cube Terms

Aichinger et al. (2011) have proved that every finite algebra with a cube-term (equivalently, with a parallelogram-term; equivalently, having few subpowers) is finitely related. Thus finite algebras with cube terms are inherently finitely related??every expansion of the algebra by adding more operations is finitely related. In this paper, we show that conversely, if A is a finite idempotent algebra and every idempotent expansion of A is finitely related, then A has a cube-term. We present further characterizations of the class of finite idempotent algebras having cube-terms, one of which yields, for idempotent algebras with finitely many basic operations and a fixed finite universe A, a polynomial-time algorithm for determining if the algebra has a cube-term. We also determine the maximal non-finitely related idempotent clones over A. The number of these clones is finite.     

The structure of finite monoids satisfying the relation ℛ = ℋ

It is shown that any finite monoid S on which Green’s relations R and H coincide divides the monoid of all upper triangular row-monomial matrices over a finite group. The proof is constructive; given the monoid S, the corresponding group and the order of matrices can be effectively found. The obtained result is used to identify the pseudovariety generated by all finite monoids satisfying R = H with the semidirect product of the pseudovariety of all finite groups and the pseudovariety of all finite R-trivial monoids.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

The free idempotent generated locally inverse semigroup

In this paper we construct a model for the free idempotent generated locally inverse semigroup on a set X. The elements of this model are special vertex-labeled bipartite trees with a pair of distinguished vertices. To describe this model, we need first to introduce a variation of a model for the free pseudosemilattice on a set X presented in Auinger and Oliveira (On the variety of strict pseudosemilattices. Stud Sci Math Hungarica 50:207–241, 2013). A construction of a graph associated with a regular semigroup was presented in Brittenham et al. (Subgroups of free idempotent generated semigroups need not be free. J Algebra 321:3026–3042, 2009) in order to give a first example of a free regular idempotent generated semigroup on a biordered set E with non-free maximal subgroups. If G is the graph associated with the free pseudosemilattice on X, we shall see that the models we present for the free pseudosemilattice on X and for the free idempotent generated locally inverse semigroup on X are closely related with the graph G.     

Contraction mappings in interval analysis

Under certain conditions, the contraction mapping fixed point theorem guarantees the convergence of the iterationx _i+1=f(x _i ) toward a fixed point of the functionf:R ⁿ →R ⁿ. When an interval extensionF off is used in a similar iteration scheme to obtain a sequence of interval vectors these conditions need not provide convergence to a degenerate interval vector representing the fixed point, even if the width of the initial interval vector is chosen arbitrarily small. We give a sufficient condition on the extensionF in order that the convergence is guaranteed. The centered form of Moore satisfies this condition.     

Reducibility of pointlike problems

We show that the pointlike and the idempotent pointlike problems are reducible with respect to natural signatures in the following cases: the pseudovariety of all finite semigroups in which the order of every subgroup is a product of elements of a fixed set \(\pi \) of primes; the pseudovariety of all finite semigroups in which every regular \(\mathcal J\)-class is the product of a rectangular band by a group from a fixed pseudovariety of groups that is reducible for the pointlike problem, respectively graph reducible. Allowing only trivial groups, we obtain \(\omega \)-reducibility of the pointlike and idempotent pointlike problems, respectively for the pseudovarieties of all finite aperiodic semigroups (\(\mathsf{A}\)) and of all finite semigroups in which all regular elements are idempotents (\(\mathsf{DA}\)).     

On characterization of multivariate stable distributions via random linear statistics

LetX ₁,X ₂, ...,X _n be independent and identically distributed random vectors inR ^d , and letY=(Y ₁,Y ₂, ...,Y _n )′ be a random coefficient vector inR ⁿ , independent ofX _j ^/′ . We characterize the multivariate stable distributions by considering the independence of the random linear statistic $$U = Y_1 X_1 + Y_2 X_2 + \cdot \cdot \cdot + Y_n X_n $$ and the random coefficient vectorY.     

Pointlike sets with respect to R and J

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety of all finite R-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute -pointlike sets, where denotes the pseudovariety of all finite J-trivial semigroups. We finally show that, in contrast with the situation for , the natural adaptation of Henckell’s algorithm to computes pointlike sets, but not all of them.     

E-local pseudovarieties

Generalizing a property of the pseudovariety of all aperiodic semigroups observed by Tilson, we call E -local a pseudovariety V which satisfies the following property: for a finite semigroup, the subsemigroup generated by its idempotents belongs to V if and only if so do the subsemigroups generated by the idempotents in each of its regular $\mathcal{D}$ -classes. In this paper, we present several sufficient or necessary conditions for a pseudovariety to be E-local or for a pseudoidentity to define an E-local pseudovariety. We also determine several examples of the smallest E-local pseudovariety containing a given pseudovariety.     

Nilpotent and solvable radicals in locally finite congruence modular varieties

The Loewy rank of a modular latticeL of finite height is defined as the leastn for which there exista ₀=0t, < ... r=1 inL such that each interval I[a_i, a_i+1] is a complemented lattice. In this paper, a generalized notion of Loewy rank is applied to obtain new results in the commutator theory of locally finite congruence modular varieties. LetV be a finitely generated congruence modular variety. We prove that every algebra inV has a largest nilpotent congruence and a largest solvable congruence. Moreover, there exist first order formulas which define these special congruences in every algebra ofV.     

Modular and lower-modular elements of lattices of semigroup varieties

The paper contains three main results. First, we show that if a commutative semigroup variety is a modular element of the lattice Com of all commutative semigroup varieties then it is either the variety $\mathcal{COM}$ of all commutative semigroups or a nilvariety or the join of a nilvariety with the variety of semilattices. Second, we prove that if a commutative nilvariety is a modular element of Com then it may be given within $\mathcal{COM}$ by 0-reduced and substitutive identities only. Third, we completely classify all lower-modular elements of Com. As a corollary, we prove that an element of Com is modular whenever it is lower-modular. All these results are precise analogues of results concerning modular and lower-modular elements of the lattice of all semigroup varieties obtained earlier by Je?ek, McKenzie, Vernikov, and the author. As an application of a technique developed in this paper, we provide new proofs of the ??prototypes?? of the first and the third our results.     

The Discrepancy of the Sequence (nα + (log n)β)

In this paper we give an upper bound for the discrepancy of the sequence (nα + (log n)β) where α = (α₁, ..., α _s ) R ^s , which satisfies that 1, α₁, ..., α _s are linearly independent over Z, is of finite type η or is of constant type.     

MV-closures of Wajsberg hoops and applications

In this paper we construct, given a Wajsberg hoop A, an MV-algebra MV(A) such that the underlying set A of A is a maximal filter of MV(A) and the quotient MV(A)/A is the two element chain. As an application we provide a topological duality for locally finite Wajsberg hoops based on a previously known duality for locally finite MV-algebras. We also give another duality for k-valued Wajsberg hoops based on a different representation of k-valued MV-algebras and show the relation to the first duality. We also apply this construction to give a topological representation for free k-valued Wajsberg hoops.     

The subvariety of commutative residuated lattices represented by twist-products

Given an integral commutative residuated lattice L, the product L × L can be endowed with the structure of a commutative residuated lattice with involution that we call a twist-product. In the present paper, we study the subvariety ${\mathbb{K}}$ of commutative residuated lattices that can be represented by twist-products. We give an equational characterization of ${\mathbb{K}}$ , a categorical interpretation of the relation among the algebraic categories of commutative integral residuated lattices and the elements in ${\mathbb{K}}$ , and we analyze the subvariety of representable algebras in ${\mathbb{K}}$ . Finally, we consider some specific class of bounded integral commutative residuated lattices ${\mathbb{G}}$ , and for each fixed element ${{\bf L} \in \mathbb{G}}$ , we characterize the subalgebras of the twist-product whose negative cone is L in terms of some lattice filters of L, generalizing a result by Odintsov for generalized Heyting algebras.