Weak relatively uniform convergences on abelian lattice ordered groups

The notion of relatively uniform convergence has been applied in the theory of vector lattices and in the theory of archimedean lattice ordered groups. Let G be an abelian lattice ordered group. In the present paper we introduce the notion of weak relatively uniform convergence (wru-convergence, for short) on G generated by a system M of regulators. If G is archimedean and M = G ⁺, then this type of convergence coincides with the relative uniform convergence on G. The relation of wru-convergence to the o-convergence is examined. If G has the diagonal property, then the system of all convex ℓ-subgroups of G closed with respect to wru-limits is a complete Brouwerian lattice. The Cauchy completeness with respect to wru-convergence is dealt with. Further, there is established that the system of all wru-convergences on an abelian divisible lattice ordered group G is a complete Brouwerian lattice.     

Weak convergences of probability measures: A uniform principle

We consider a set of probability measures on a locally compact separable metric space. It is shown that a necessary and sufficient condition for (relative) sequential compactness of in various weak topologies (among which the vague, weak and setwise topologies) has the same simple form; i.e. a uniform principle has to hold in . We also extend this uniform principle to some Köthe function spaces.

     

Weak MV-algebras

In a recent paper [CHAJDA, I.—KüHR, J.: A non-associative generalization of MV-algebras, Math. Slovaca 57, (2007), 301–312], authors introduced and studied a non-associative generalization of MV-algebras called NMV-algebras. In contrast to MV-algebras, sections (i.e. principal filters) in NMV-algebras which are proper (i.e. are not MV-algebras), do not admit a structure of an NMV-algebra with respect to the operations defined in a natural way. The aim of the paper is to present a new class of algebras generalizing MV-algebras but sharing the above property. The financial support by the grant of Czech Government MSM 6198959214 is gratefully acknowledged.     

Relatively uniform convergences in archimedean lattice ordered groups

For an archimedean lattice ordered group G let G ^d and G∧ be the divisible hull or the Dedekind completion of G, respectively. Put G ^d ∧ = X. Then X is a vector lattice. In the present paper we deal with the relations between the relatively uniform convergence on X and the relatively uniform convergence on G. We also consider the relations between the o-convergence and the relatively uniform convergence on G. For any nonempty class τ of lattice ordered groups we introduce the notion of τ-radical class; we apply this notion by investigating relative uniform convergences.     

Weak homogeneity and Pierce’s theorem for MV-algebras

In this paper we prove a theorem on weak homogeneity of MV-algebras which generalizes a known result on weak homogeneity of Boolean algebras. Further, we consider a homogeneity condition for MV-algebras which is defined by means of an increasing cardinal property.     

A note on almost sure uniform and complete convergences of a sequence of random variables

The aim of this note is to introduce another way of defining the almost sure uniform convergence, which is necessary when studying some mathematical results on the existence of price bubbles in certain scenarios of trading securities. This mode of convergence of random variables' sequences is intermediate between the uniform and the almost sure ones, and, more specifically, between the uniform and the complete convergences. In this way, this paper presents some mathematical characterizations of both almost sure uniform and complete convergences, and shows that the almost sure uniform convergence is a particular case of complete convergence, when the number of summands in the series defining this mode of convergence is finite. Finally, this paper presents the relation of almost surely uniform convergence with convergence in mean when the random variable limit is integrable. Moreover, almost surely convergence and local boundedness of the sequence of random variables minus its limit are sufficient to derive convergence in mean.     

Weak quasi‐randomness for uniform hypergraphs

We study quasi‐random properties of k‐uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung‐Graham‐Wilson theorem for quasi‐random graphs. Moreover, let K_k be the complete graph on k vertices and M(k) the line graph of the graph of the k‐dimensional hypercube. We will show that the pair of graphs (K_k,M(k)) has the property that if the number of copies of both K_k and M(k) in another graph G are as expected in the random graph of density d, then G is quasi‐random (in the sense of the Chung‐Graham‐Wilson theorem) with density close to d. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Bornological convergences

We study a family of convergences (actually pretopologies) in the hyperspace of a metric space that are generated by covers of the space. This family includes the Attouch-Wets, Fell, and Hausdorff metric topologies as well as the lower Vietoris topology. The unified approach leads to new developments and puts into perspective some classical results.     

Generalized MV-algebras

We generalize the notion of an MV-algebra in the context of residuated lattices to include non-commutative and unbounded structures. We investigate a number of their properties and prove that they can be obtained from lattice-ordered groups via a truncation construction that generalizes the Chang–Mundici Γ functor. This correspondence extends to a categorical equivalence that generalizes the ones established by D. Mundici and A. Dvurečenskij. The decidability of the equational theory of the variety of generalized MV-algebras follows from our analysis.     

State-morphism MV-algebras

We present a stronger variation of state MV-algebras, recently presented by T. Flaminio and F. Montagna, which we call state-morphism MV-algebras. Such structures are MV-algebras with an internal notion, a state-morphism operator. We describe the categorical equivalences of such (state-morphism) state MV-algebras with the category of unital Abelian ?-groups with a fixed state operator and present their basic properties. In addition, in contrast to state MV-algebras, we are able to describe all subdirectly irreducible state-morphism MV-algebras.     

Representation of relatively uniform and order convergence topologies by an inductive limit

Using the concept of a topological affine space, it is proved that a partially ordered topological linear spaces associated with relatively uniform and order convergence can be represented by an inductive limit of its subspaces.     

Weak and strong uniform consistency of kernel regression estimates

Summary We study the estimation of a regression function by the kernel method. Under mild conditions on the window, the bandwidth and the underlying distribution of the bivariate observations {(X _i , Y _i)}, we obtain the weak and strong uniform convergence rates on a bounded interval. These results parallel those of Silverman (1978) on density estimation and extend those of Schuster and Yakowitz (1979) and Collomb (1979) on regression estimation.This research was carried out in part while the authors were guests at the University of Heidelberg, Germany, under the sponsorship of the Sonderforschungsbereich 123 in the summer of 1980     

Cauchy completions of MV-algebras

Abstract. An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We show that such convergences on a given MV-algebra A are exactly the restrictions of the bounded -convergences on the abelian -group in which A appears as the unit interval. Thus the theory of -convergence and Cauchy structures transfers to MV-algebras.?We outline the general theory, and then apply it to three particular MV-convergences and their corresponding Cauchy completions. The Cauchy completion arising from order convergence coincides with the Dedekind-MacNeille completion of an MV-algebra. The Cauchy completion arising from polar convergence allows a tidy proof of the existence and uniqueness of the lateral completion of an MV-algebra. And the Cauchy completion arising from α-convergence gives rise to the cut completion of an MV-algebra. Received August 8, 2001; accepted in final form October 18, 2001.     

Weakly linear quantum MV-algebras

Quantum MV-algebras (QMV-algebras) are a non lattice-theoretic generalization of MV-algebras (multi-valued algebras) and a non-idempotent generalization of orthomodular lattices. In this paper we construct a finite basis for the variety generated by the class of all weakly linear quantum MV-algebras.Dedicated to the memory of Wim BlokReceived October 12, 2000; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Relative subalgebras of MV-algebras

Given an MV-algebra A, with its natural partial ordering, we consider in A the intervals of the form [0, a], where \({a \in A}\). These intervals have a natural structure of MV-algebras and will be called the relative subalgebras of A (in analogy with Boolean algebras). We investigate various properties of relative subalgebras and their relations with the original MV-algebra.     

On implication in MV-algebras

In [1], the authors introduced the notion of a weak implication algebra, which reflects properties of implication in MV-algebras, and demonstrated that the class of weak implication algebras is definitionally equivalent to the class of upper semilattices whose principal filters are compatible MV-algebras. It is easily seen that weak implication algebras are just duals of commutative BCK-algebras. We show here that most results of [1] are, in fact, immediate consequences of two well-known facts: (i) a bounded commutative BCK-algebra possesses a natural upper semilattice structure, (ii) the class of MV-algebras and that of bounded commutative BCK-algebras are definitionally equivalent. Presented by I. Hodkinson. Received November 11, 2005; accepted in final form November 26, 2005.     

Representation and extension of states on MV-algebras

MV-algebras stand for the many-valued Łukasiewicz logic the same as Boolean algebras for the classical logic. States on MV-algebras were first mentioned [20] in probability theory and later also introduced in effort to capture a notion of `an average truth-value of proposition' [15] in Łukasiewicz many-valued logic. In the presented paper, an integral representation theorem for finitely-additive states on semisimple MV-algebra will be proven. Further, we shall prove extension theorems concerning states defined on sub-MV-algebras and normal partitions of unity generalizing in this way the well-known Horn-Tarski theorem for Boolean algebras. The author gratefully acknowledges the support of grant 201/02/1540 of the Grant Agency of the Czech Republic and the partial support by the project 1M6798555601 of the Ministry of Education, Youth and Sports of the Czech Republic.     

State morphism MV-algebras

We present a complete characterization of subdirectly irreducible MV-algebras with internal states (SMV-algebras). This allows us to classify subdirectly irreducible state morphism MV-algebras (SMMV-algebras) and describe single generators of the variety of SMMV-algebras, and show that we have a continuum of varieties of SMMV-algebras.