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k -free separable groups with prescribed endomorphism ring

Daniel Herden, Héctor Gabriel Salazar Pedroza (2015)

Fundamenta Mathematicae

We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of k -free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is k -free if every subset of size < k is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of G is contained in a free direct summand of G. Despite the fact that such a module G is...

A Corson compact L-space from a Suslin tree

Peter Nyikos (2015)

Colloquium Mathematicae

The completion of a Suslin tree is shown to be a consistent example of a Corson compact L-space when endowed with the coarse wedge topology. The example has the further properties of being zero-dimensional and monotonically normal.

A new class of weakly K -analytic Banach spaces

Sophocles Mercourakis, E. Stamati (2006)

Commentationes Mathematicae Universitatis Carolinae

In this paper we define and investigate a new subclass of those Banach spaces which are K -analytic in their weak topology; we call them strongly weakly K -analytic (SWKA) Banach spaces. The class of SWKA Banach spaces extends the known class of strongly weakly compactly generated (SWCG) Banach spaces (and their subspaces) and it is related to that in the same way as the familiar classes of weakly K -analytic (WKA) and weakly compactly generated (WCG) Banach spaces are related. We show that: (i) not...

Arhangel'skiĭ sheaf amalgamations in topological groups

Boaz Tsaban, Lyubomyr Zdomskyy (2016)

Fundamenta Mathematicae

We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos’s property α 1 . 5 is equivalent to Arhangel’skiĭ’s formally stronger property α₁. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space X such that the space C p ( X ) of continuous real-valued functions on X with the topology...

Cardinal sequences of length < ω₂ under GCH

István Juhász, Lajos Soukup, William Weiss (2006)

Fundamenta Mathematicae

Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put λ ( α ) = s ( α ) : s ( 0 ) = λ = m i n [ s ( β ) : β < α ] . We show that f ∈ (α) iff for some natural number n there are infinite cardinals λ i > λ > . . . > λ n - 1 and ordinals α , . . . , α n - 1 such that α = α + + α n - 1 and f = f f . . . f n - 1 where each f i λ i ( α i ) . Under GCH we prove that if α < ω₂ then (i) ω ( α ) = s α ω , ω : s ( 0 ) = ω ; (ii) if λ > cf(λ) = ω, λ ( α ) = s α λ , λ : s ( 0 ) = λ , s - 1 λ i s ω - c l o s e d i n α ; (iii) if cf(λ) = ω₁, λ ( α ) = s α λ , λ : s ( 0 ) = λ , s - 1 λ i s ω - c l o s e d a n d s u c c e s s o r - c l o s e d i n α ; (iv) if cf(λ) > ω₁, λ ( α ) = α λ . This yields a complete characterization of the classes (α) for all α < ω₂,...

Certain partitions on a set and their applications to different classes of graded algebras

Antonio J. Calderón Martín, Boubacar Dieme (2021)

Communications in Mathematics

Let ( 𝔄 , ϵ u ) and ( 𝔅 , ϵ b ) be two pointed sets. Given a family of three maps = { f 1 : 𝔄 𝔄 ; f 2 : 𝔄 × 𝔄 𝔄 ; f 3 : 𝔄 × 𝔄 𝔅 } , this family provides an adequate decomposition of 𝔄 { ϵ u } as the orthogonal disjoint union of well-described -invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak H * -algebras.

Completely nonmeasurable unions

Robert Rałowski, Szymon Żeberski (2010)

Open Mathematics

Assume that no cardinal κ < 2ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal of subsets of κ such that the Boolean algebra P(κ)/ satisfies c.c.c.). We show that for a metrizable separable space X and a proper c.c.c. σ-ideal II of subsets of X that has a Borel base, each point-finite cover ⊆ 𝕀 of X contains uncountably many pairwise disjoint subfamilies , with 𝕀 -Bernstein unions ∪ (a subset A ⊆ X is 𝕀 -Bernstein if A and X A meet each Borel 𝕀 -positive subset...

Divergence measure between fuzzy sets using cardinality

Vladimír Kobza (2017)


In this paper we extend the concept of measuring difference between two fuzzy subsets defined on a finite universe. The first main section is devoted to the local divergence measures. We propose a divergence measure based on the scalar cardinalities of fuzzy sets with respect to the basic axioms. In the next step we introduce the divergence based on the generating function and the appropriate distances. The other approach to the divergence measure is motivated by class of the rational similarity...

Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations

Boaz Tsaban, Lubomyr Zdomsky (2012)

Journal of the European Mathematical Society

A classical theorem of Hurewicz characterizes spaces with the Hurewicz covering property as those having bounded continuous images in the Baire space. We give a similar characterization for spaces X which have the Hurewicz property hereditarily. We proceed to consider the class of Arhangel’skii α 1 spaces, for which every sheaf at a point can be amalgamated in a natural way. Let C p ( X ) denote the space of continuous real-valued functions on X with the topology of pointwise convergence. Our main result...

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