Almost universal varieties of monoids

The constant mappings onto the unit form a zero subcategory of any category of monoid homomorphisms; a varietyV of monoids isalmost universal if every category of algebras is isomorphic to a class of all nonzero homomorphisms between members ofV. Almost universal monoid varieties are shown to be exactly those varieties containing all commutative monoids in which the identity xⁿyⁿ=(xy)ⁿ fails for every n>1. Almost universal varieties of monoids can also be characterized categorically as the varieties containing all groups with zero as one-object full subcategories.Presented by B. M. Schein.The support of NSERC is gratefully acknowledged.     

Cryptic Semigroup Varieties

We describe all minimal noncryptic periodic semigroup [monoid] varieties. We prove that there are exactly three distinct maximal cryptic semigroup [monoid] varieties contained in the variety determined by xⁿ ≈ x ^n+m, n ≥ 2, m ≥ 2. Analogous results are obtained for pseudovarieties: there are exactly three maximal cryptic pseudovarieties of semigroups [monoids]. It is shown that if a cryptic variety or pseudovariety of monoids contains a nonabelian group, then it consists of bands of groups only. Several characterizations are given of the cryptic overcommutative semigroup [monoid] varieties.     

Crossed products and blocks with normal defect groups

Abstract

Let D be an integral domain. A multiplicative set S of D is an almost splitting set if for each 0 ≠ d ∈ D, there exists an n = n(d) with d ⁿ  = st where s ∈ S and t is v-coprime to each element of S. An integral domain D is an almost GCD (AGCD) domain if for every x, y ∈ D, there exists a positive integer n = n(x, y) such that x ⁿ D ∩ y ⁿ D is a principal ideal. We prove that the polynomial ring D[X] is an AGCD domain if and only if D is an AGCD domain and D[X] ? D′[X] is a root extension, where D′ is the integral closure of D. We also show that D + XD _S [X] is an AGCD domain if and only if D and D _S [X] are AGCD domains and S is an almost splitting set.     

On structure and commutativity of certain periodic near rings

In the present paper we first establish decomposition theorems for near rings satisfying either of the properties xy = x^my^pxⁿ or xy = y^mx^pyⁿ, where m≥1, n≥1, p≥1 are positive integers depending on the pair of near ring elements x,y; and further, we investigate commutativity of such near rings. Moreover, it is also shown that under some additional hypotheses, such nearrings turn out to be commutative rings.     

Posner and herstein theorems for derivations of 3-prime near-rings

The analog of Posner's theorem on the composition of two derivations in prime rings is proved for 3-prime near-rings. It is shown that if d is a nonzero derivation of a 2-torsionfree 3-prime near-ring N and an element a ? N is such that ax^d = x^da for all x ? N, then a is a central element. As a consequence it is shown that if d\ and d₂ are nonzero derivations of a 2-torsionfree 3-prime near-ring N and x^d1y^d2 = y^d2x^d1 for all x, y ? N, then N is a commutative ring. Thus two theorems of Herstein are generalized     

Asymptotic Elasticity in Atomic Monoids

Let M be a commutative atomic monoid (i.e. every nonzero nonunit of M can be factored as a product of irreducible elements). Let ρ(x) denote the elasticity of x ∈ M, R(M) = {ρ(x) | x ∈ M} the set of elasticities of elements in M, and ρ(M) = sup R(M) the elasticity of M. Define \overline{ρ}(x) = lim_n→∞ ρ(xⁿ) to be the asymptotic elasticity of x. We determine some basic properties of the function \overline{ρ} and determine its image for certain block monoids.     

Diversity in inside factorial monoids

In a recent paper (Diversity in Monoids, Czech. Math. J. 62 (2012), 795–809), the last two authors introduced and developed the monoid invariant “diversity” and related properties “homogeneity” and “strong homogeneity”. We investigate these properties within the context of inside factorial monoids, in which the diversity of an element counts the number of its different almost primary components. Inside factorial monoids are characterized via diversity and strong homogeneity. A new invariant complementary to diversity, height, is introduced. These two invariants are connected with the well-known invariant of elasticity.     

Ein Einschliessungsverfahren für Lösungen Fehlerbehafteter Linearer Gleichungen

In a partially ordered space, the method x_n+1 = L⁺x _n ⁺ – N⁺x _n ^- – L^–y⁺ + N^– y _n ^- + r, y_n+1 = N⁺y⁺ – L⁺y _n ^- – N^–x _n ⁺ + L^–x^– + t of successive approximation is developed in order to enclose the solutions of a set of linear fixed point equations monotonously. The method works if only the inequalities x₀ y₀, x₀ x₁, y₁ y₀ related to the starting elements are satisfied. In finite-dimensional spaces suitable starting vectors can be computed if a sufficiently good approximation for the fixed points is known.

     

Monoids and Groups of <Emphasis Type="Italic">I</Emphasis>-Type

A monoid S generated by {x₁,. . .,x_n} is said to be of (left) I-type if there exists a map v from the free Abelian monoid FaM_n of rank n generated by {u₁,. . .,u_n} to S so that for all a∈FaM_n one has {v(u₁a),. . .,v(u_na)}={x₁v(a),. . .,x_nv(a)}. Then S has a group of fractions, which is called a group of (left) I-type. These monoids first appeared in the work of Gateva-Ivanova and Van den Bergh, inspired by earlier work of Tate and Van den Bergh. In this paper we show that monoids and groups of left I-type can be characterized as natural submonoids and groups of semidirect products of the free Abelian group Fa_n and the symmetric group of degree n. It follows that these notions are left–right symmetric. As a consequence we determine many aspects of the algebraic structure of such monoids and groups. In particular, they can often be decomposed as products of monoids and groups of the same type but on less generators and many such groups are poly-infinite cyclic. We also prove that the minimal prime ideals of a monoid S of I-type, and of the corresponding monoid algebra, are principal and generated by a normal element. Further, via left–right divisibility, we show that all semiprime ideals of S can be described. The latter yields an ideal chain of S with factors that are semigroups of matrix type over cancellative semigroups. In memory of Paul Wauters Mathematics Subject Classifications (2000) 20F05, 20M05; 16S34, 16S36, 20F16. The authors were supported in part by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish–Polish bilateral agreement BIL 01/31, and KBN research grant 2P03A 033 25 (Poland).     

Global center manifolds in singular systems

The problem of existence of aglobal center manifold for a system of O.D.E. like (*) $$\left\{ {\begin{array}{*{20}c} {\dot x = A(y)x + F(x,y)} \\ {\dot y = G(x,y), (x,y) \in \mathbb{R}^n \times \mathbb{R}^m ,} \\ \end{array} } \right.$$ is considered. We give conditions onA(y), F(x, y), G(x, y) in order that a functionH: ? ^m →? ⁿ , with the same smoothness asA(y), F(x, y), G(x, y), exists and is such that the manifoldC={(x,y)∈? ⁿ ×? ^m ∣x=H(y),y∈? ^m } is an invariant manifold for (*), and there exists ρ>0 such that any solution of (*) satisfying sup _t∈?∣x(t)∣ <ρ must belong toC. This is why we callC global center manifold. Applications are given to the problem of existence of heteroclinic orbits in singular systems.     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

A combinatorial condition on a certain variety of groups

Let n be an integer and B_n \mathcal B_n be the variety defined by the law [xⁿ,y][x,yⁿ]^-1 = 1.¶ Let B_n^* \mathcal B_n^* be the class of groups in which for any infinite subsets X, Y there exist x ? X x \in X and y ? Y y \in Y such that [xⁿ,y][x,yⁿ]^-1 = 1. For $ n \in {\pm 2, 3\} $ n \in {\pm 2, 3\} we prove that¶ B_n^* = B_n èF \mathcal B_n^* = \mathcal B_n \cup \mathcal F , F \mathcal F being the class of finite groups. Also for $ n \in {- 3, 4\} $ n \in {- 3, 4\} and an infinite group G which has finitely many elements of order 2 or 3 we prove that G ? B_n^* G \in \mathcal B_n^* if and only if G ? B_n G \in \mathcal B_n .     

Localization of a Corner-Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Let VV be a convex neighborhood of the origin contained in the square Q = { (x,y) ? \mathbbR² : |x| ￡ p, |y| ￡ p} Q = \{ (x,y) \in \mathbb{R}^2 : |x| \leq \pi, |y| \leq \pi \} . Let AA be a 'sector' of VV bounded by two convex or concave curves intersecting at the origin. Let ff be an integrable function on QQ, smooth on AA and on V  AV \ A but having a jump discontinuity at the origin, whose coordinate sections (i.e., the restrictions of ff to x = const x = \textit{const} , resp.\ y = const y = \textit{const} ) have uniformly bounded variation. Under essentially these conditions the partial sums S_n,n S_{n,n} of the Fourier series of ff display for n ? ￥ n \rightarrow \infty at the origin a corner point Gibbs phenomenon with an overshoot of up to 37,4% of half the jump size. This Gibbs phenomenon manifests itself in the pointwise convergence of S_n,n ( \fracxn, \fracyn ; f)S_{n,n} ( \frac{x}{n}, \frac{y}{n} ; f) as n ? ￥ n \rightarrow \infty for all (x,y) ? \mathbbR² (x,y) \in \mathbb{R}^2 to a non-constant limiting function only depending on the slopes of the boundary curves of AA at the origin and on the jump of f (x,y) f (x,y) as (x,y) (x,y) approaches (0,0)(0,0) within AA resp.V  A V \ A.     

A note on intersections of free submonoids of a free monoid

According to a theorem of Tilson [6] any intersection of free submonoids of a free monoid is free. Here we consider intersections of the form {x, y}^* ∩ {u, v}^*, where x, y, u and v are words in a finitely generated free monoid Σ^*, and show that if both the monoids {x, y}^* and {u, v}^* are of the rank two, then the intersection is a free monoid generated either by (at most) two words or by a regular language of the form β₀ + β((γ(1+ δ + ... δ^t))^*ε for some words β⁰, β, γ, δ and ε, and some integer t≥0. An example is given showing that the latter possibility may occur for each t≥0 with nonempty values of the words.     

Divisibility theory in commutative rings: Bezout monoids

A ubiquitous class of lattice ordered semigroups introduced by Bosbach, which we call Bezout monoids, seems to be the appropriate structure for the study of divisibility in various classical rings like GCD domains (including UFD??s), rings of low dimension (including semi-hereditary rings), as well as certain subdirect products of such rings and certain factors of such subdirect products. A Bezout monoid is a commutative monoid S with 0 such that under the natural partial order (for a, b ?? S, a ?? b ?? S ? bS ? aS), S is a distributive lattice, multiplication is distributive over both meets and joins, and for any x, y ?? S, if d = x ?? y and dx ₁ = x then there is a y ₁ ?? S with dy ₁ = y and x ₁ ?? y ₁ = 1. We investigate Bezout monoids by using filters and m-prime filters, and describe all homorphisms between them. We also prove analogues of the Pierce and the Grothendieck sheaf representations of rings for Bezout monoids. The question whether Bezout monoids describe divisibility in Bezout rings (rings whose finitely generated ideals are principal) is still open.     

Non-Unique Factorization of Polynomials Over Residue Class Rings of the Integers

We investigate non-unique factorization of polynomials in ? _{p ⁿ} [x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, ? _{p ⁿ} [x] is atomic. We reduce the question of factoring arbitrary nonzero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of ? _{p ⁿ} [x] is a direct sum of monoids corresponding to irreducible polynomials in ? _p [x], and we show that each of these monoids has infinite elasticity. Moreover, for every m ∈ ?, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles.     

Factorisation dans un Ordre Non Maximal d'un Corps Quadratique

Let O_f be an order of index f in a quadratic field. We denote A_f the set of elements of O_f whose norm is relatively prime to f. An element v ∈ A_f is called k-prime if for x, y ∈ A_f, v|xy implies v|x^k or v|y^k where k is the exponent of the group O_f. We prove that the k-th powers of the elements of A_f have a unique representation as a product of elements which are irreductible and k-prime. One criteria for resolution of some diophantine equations is an illustration from it.     

Commutativity for a certain class of rings

We discuss the commutativity of certain rings with unity 1 and one-sideds-unital rings under each of the following conditions:x ^r [x ^s ,y]=±[x,y ^t ]x ⁿ x ^r [x ^s ,y]=±x ⁿ [x,y ^t ]x ^r [x ^s ,y]=±[x,y ^t ]y ^m , andx ^r [x ^s ,y]=±y ^m [x,y ^t ], wherer, n, andm are non-negative integers andt>1,s are positive integers such that eithers, t are relatively prime ors[x,y]=0 implies [x,y]=0. Further, we improve the result of [6, Theorem 3] and reprove several recent results.     

On sofic monoids

We investigate a notion of soficity for monoids. A group is sofic as a group if and only if it is sofic as a monoid. All finite monoids, all commutative monoids, all free monoids, all cancellative one-sided amenable monoids, all multiplicative monoids of matrices over a field, and all monoids obtained by adjoining an identity element to a semigroup are sofic. On the other hand, although the question of the existence of a non-sofic group remains open, we prove that the bicyclic monoid is not sofic. This shows that there exist finitely presented amenable inverse monoids that are non-sofic.