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On the Existence of Free Ultrafilters on ω and on Russell-sets in ZF

Eleftherios Tachtsis (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

In ZF (i.e. Zermelo-Fraenkel set theory without the Axiom of Choice AC), we investigate the relationship between UF(ω) (there exists a free ultrafilter on ω) and the statements "there exists a free ultrafilter on every Russell-set" and "there exists a Russell-set A and a free ultrafilter ℱ on A". We establish the following results: 1. UF(ω) implies that there exists a free ultrafilter on every Russell-set. The implication is not reversible in ZF. 2. The statement...

On the existence of group localizations under large-cardinal axioms.

Carles Casacuberta, Dirk Scevenels (2001)

RACSAM

Uno de los problemas abiertos más antiguos de la teoría de grupos categórica es si todo par ortogonal (formado por una clase de grupos y una clase de homomorfismos que se determinan mutuamente por ortogonalidad en el sentido de Freyd-Kelly), se halla asociado a un funtor de localización. Se sabe que esto es cierto si se acepta la validez de un cierto axioma de cardinales grandes (el principio de Vopenka), pero no se conoce ninguna demostración mediante los axiomas ordinarios (ZFC) de la teoría de...

On the existence of prime ideals in Boolean algebras

Jörg Flum (1999)

Banach Center Publications

Rasiowa and Sikorski [5] showed that in any Boolean algebra there is an ultrafilter preserving countably many given infima. In [3] we proved an extension of this fact and gave some applications. Here, besides further remarks, we present some of these results in a more general setting.

On the existence of super-decomposable pure-injective modules over strongly simply connected algebras of non-polynomial growth

Stanisław Kasjan, Grzegorz Pastuszak (2014)

Colloquium Mathematicae

Assume that k is a field of characteristic different from 2. We show that if Γ is a strongly simply connected k-algebra of non-polynomial growth, then there exists a special family of pointed Γ-modules, called an independent pair of dense chains of pointed modules. Then it follows by a result of Ziegler that Γ admits a super-decomposable pure-injective module if k is a countable field.

On the extensibility of closed filters in T 1 spaces and the existence of well orderable filter bases

Kyriakos Keremedis, Eleftherios Tachtsis (1999)

Commentationes Mathematicae Universitatis Carolinae

We show that the statement CCFC = “the character of a maximal free filter F of closed sets in a T 1 space ( X , T ) is not countable” is equivalent to the Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE 0 = “closed filters in a T 0 space ( X , T ) extend to maximal closed filters”. We also show that AC is equivalent to each of the assertions: “every closed filter ℱ in a T 1 space ( X , T ) extends to a maximal closed filter with a well orderable filter base”, “for every set A ≠ ∅ ,...

On the extent of separable, locally compact, selectively (a)-spaces

Samuel G. da Silva (2015)

Colloquium Mathematicae

The author has recently shown (2014) that separable, selectively (a)-spaces cannot include closed discrete subsets of size . It follows that, assuming CH, separable selectively (a)-spaces necessarily have countable extent. However, in the same paper it is shown that the weaker hypothesis " 2 ℵ ₀ < 2 ℵ ₁ " is not enough to ensure the countability of all closed discrete subsets of such spaces. In this paper we show that if one adds the hypothesis of local compactness, a specific effective (i.e., Borel) parametrized...

On the first homology of Peano continua

Gregory R. Conner, Samuel M. Corson (2016)

Fundamenta Mathematicae

We show that the first homology group of a locally connected compact metric space is either uncountable or finitely generated. This is related to Shelah's well-known result (1988) which shows that the fundamental group of such a space satisfies a similar condition. We give an example of such a space whose fundamental group is uncountable but whose first homology is trivial, showing that our result does not follow from Shelah's. We clarify a claim made by Pawlikowski (1998) and offer a proof of the...

On the fundamentals of fuzzy sets.

Robert Lowen (1984)

Stochastica

A considerable amount of research has been done on the notions of pseudo complement, intersection and union of fuzzy sets [1], [4], [11]. Most of this work consists of generalizations or alternatives of the basic concepts introduced by L. A. Zadeh in his famous paper [13]: generalization of the unit interval to arbitrary complete and completely distributive lattices or to Boolean algebras [2]; alternatives to union and intersection using the concept of t-norms [3], [10]; alternative complements...

On the generators of T-indistinguishability operator.

Joan Jacas (1988)

Stochastica

The structure of the generators' set of a T-indistinguishability operator is analyzed. A suitable characterization of such generators is given. T-indistinguishability operators generated by a single fuzzy set, in the sense of the representation problem, are studied.

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