Compact factors in finally compact products of topological spaces

We present instances of the following phenomenon: if a product of topological spaces satisfies some given compactness property then the factors satisfy a stronger compactness property, except possibly for a small number of factors.The first known result of this kind, a consequence of a theorem by A.H. Stone, asserts that if a product is regular and Lindelöf then all but at most countably many factors are compact. We generalize this result to various forms of final compactness, and extend it to two-cardinal compactness. In addition, our results need no separation axiom.     

A generalization of the Mulholland inequality for continuous Archimedean t-norms

It is well known that dominance between strict t-norms is closely related to the Mulholland inequality, which can be seen as a generalization of the Minkowski inequality. However, strict t-norms constitute only one part of the class of continuous Archimedean t-norms, the basic elements from which all continuous t-norms are composed. In this paper, dominance between continuous Archimedean t-norms is shown to be related to a generalization of the Mulholland inequality. We provide sufficient and necessary conditions for its fulfillment.     

PROPOSITIONAL LOGIC,FRAMES, AND FUZZY ALGEBRA

Abstract

This paper offers a new look at such things as the fuzzy subalgebras and congruences of an algebra, the fuzzy ideals of a ring or a lattice, and similar entities, by exhibiting them as the models, in the chosen frame T of truth values, of naturally corresponding propositional theories. This provides a systematic approach to the study of the partially ordered sets formed by these various entities, and we demonstrate its usefulness by employing it to derive a number of results, some old and some new, concerning these partially ordered sets. In particular, we prove they are complete lattices, algebraic or continuous, depending on whether T is algebraic or continuous, respectively (Proposition 3); they satisfy the same lattice identities for arbitrary T that hold in the case T = 2 (Corollary of Proposition 4); and they are coherent frames for any coherent T whenever this is the case for T = 2 (Proposition 6). In addition we show, generalizing a result by Makamba and Murali [10], that the familiar classical situations where the congruences of an algebra correspond to certain other entities, such as the normal subgroups of a group or the ideals of a ring, extend to the fuzzy case by proving that the corresponding propositional theories are equivalent (Proposition 2). Further, we obtain the result of Gupta and Kantroo [5] that the fuzzy radical ideals of a commutative ring with unit are the meets of fuzzy prime ideals for arbitrary continuous T in place of the unit interval, using basic facts concerning continuous frames (Proposition 7).     

Cofinal types of ultrafilters

We study Tukey types of ultrafilters on ω, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated ultrafilters includes all known ultrafilters that are not Tukey above [ω₁]^<ω. We give a complete characterization of all ultrafilters that are Tukey below a selective. A counterexample showing that Tukey reducibility and RK reducibility can diverge within the class of P-points is also given.     

A combinatorial result related to the consistency of New Foundations

We prove a combinatorial result for models of the 4-fragment of the Simple Theory of Types (TST), TST₄. The result says that if A=〈A₀,A₁,A₂,A₃〉 is a standard transitive and rich model of TST₄, then A satisfies the 〈0,0,n〉-property, for all n≥2. This property has arisen in the context of the consistency problem of the theory New Foundations (NF). The result is a weak form of the combinatorial condition (existence of ω-extendible coherent triples) that was shown in Tzouvaras (2007) [5] to be equivalent to the consistency of NF. Such weak versions were introduced in Tzouvaras (2009) [6] in order to relax the intractability of the original condition. The result strengthens one of the main theorems of Tzouvaras (2007) [5, Theorem 3.6] which is just equivalent to the 〈0,0,2〉-property.     

Nonseparable UHF algebras I: Dixmier's problem

There are three natural ways to define UHF (uniformly hyperfinite) C^∗-algebras, and all three definitions are equivalent for separable algebras. In 1967 Dixmier asked whether the three definitions remain equivalent for not necessarily separable algebras. We give a complete answer to this question. More precisely, we show that in small cardinality two definitions remain equivalent, and give counterexamples in other cases. Our results do not use any additional set-theoretic axioms beyond the usual axioms, namely ZFC.     

On the weak Freese–Nation property of ?(ω)

Continuing [6], [8] and [16], we study the consequences of the weak Freese-Nation property of (?(ω),⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichoń's diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (?(ω),⊆) captures many of the features of Cohen models and hence may be considered as a principle axiomatizing a good portion of the combinatorics available in Cohen models. Received: 7 June 1999 / Revised version: 17 October 1999 /?Published online: 15 June 2001     

LARGE OPEN COVERS AND A POLARIZED PARTITION RELATION

Abstract

We show that a strengthened form of a property of Roth-berger implies that a certain polarized partition relation holds for the ω-covers of a space and that this partition relation implies a strengthened form of a property of Menger.     

Convex combinations of continuous t-norms with the same diagonal function

The problem of whether a non-trivial convex combination of two continuous t-norms with the same diagonal function can be a t-norm is studied. It is shown that in both cases–of two nilpotent and of two strict t-norms–a non-trivial convex combination of t-norms with common diagonal function is associative only if the two t-norms involved coincide. For general continuous t-norms a similar result follows. An example of a convex class of non-continuous t-norms is also included.     

On the Laczkovich-Komjáth property of sigma-ideals

Komjáth in 1984 proved that, for each sequence (An) of analytic subsets of a Polish space X, if lim supn∈HAn is uncountable for every H∈ω[N] then _?n∈GAn is uncountable for some G∈ω[N]. This fact, by our definition, means that the σ-ideal [X]^?ω has property (LK). We prove that every σ-ideal generated by X/E has property (LK), for an equivalence relation E⊂X² of type Fσ with uncountably many equivalence classes. We also show the parametric version of this result. Finally, the invariance of property (LK) with respect to various operations is studied.     

Some remarks on a question of D. H. Fremlin regarding ε-density

We show the relative consistency of ℵ₁ satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω₁ satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω₁, and specifically if this and a stronger property hold under MA + ?CH. Towards this we show that if MA + ?CH holds, then for every family ? of ℵ₁ many infinite subsets of ω₁, one can find a family ? of finite subsets of ω₁ which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?. We then pose some open problems related to the question. Received: 2 June 1999 / Revised version: 2 February 2000 / Published online: 18 July 2001     

Triangular norms which are meet-morphisms in interval-valued fuzzy set theory

In this paper we study t-norms on the lattice of closed subintervals of the unit interval. Unlike for t-norms on a product lattice for which there exists a straightforward characterization of t-norms which are join-morphisms, respectively meet-morphisms, the situation is more complicated for t-norms in interval-valued fuzzy set theory. In previous papers several characterizations were given of t-norms in interval-valued fuzzy set theory which are join-morphisms and which satisfy additional properties, but little attention has been paid to meet-morphisms. Therefore, in this paper, we focus on t-norms which are meet-morphisms. We consider a general class of t-norms and investigate under which conditions t-norms belonging to this class are meet-morphisms. We also characterize the t-norms which are both a join- and a meet-morphism and which satisfy an additional border condition.     

Cascades,order, and ultrafilters

We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. This allows us to prove (ZFC) that the class of strict _Jωω$J_{{\omega }^{\omega }}$-ultrafilters, introduced by J.E. Baumgartner in [2], is empty. We translate the result to the language of _<∞${<}_{\infty }$-sequences under an ultrafilter, investigated by C. Laflamme in [17], and we show that if there is an arbitrary long finite _<∞${<}_{\infty }$-sequence under u, then u   is at least a strict _Jωω+1$J_{{\omega }^{\omega +1}}$-ultrafilter.     

First-order t-norm based fuzzy logics with truth-constants: Distinguished semantics and completeness properties

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms (mainly continuous and weak nilpotent minimum t-norms). We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind , where φ has no additional truth-constants.     

Decomposable Near Polygons

We give a common construction for the product and the glued near polygons by generalizing the glueing construction given in [5]. We call the near polygons arising from this generalized glueing construction decomposable or (again) glued. We will study the geodetically closed sub near polygons of decomposable near polygons. Each decomposable near hexagon has a nice pair of partitions in geodetically closed near polygons. We will give a characterization of the decomposable near polygons using this property.     

A model of Cummings and Foreman revisited

This paper concerns the model of Cummings and Foreman where from ω   supercompact cardinals they obtain the tree property at each _ℵn${\mathrm{\aleph }}_n$ for 2≤n<ω$2\le n<\omega$. We prove some structural facts about this model. We show that the combinatorics at _ℵω+1${\mathrm{\aleph }}_{\omega +1}$ in this model depend strongly on the properties of _ω1${\omega }_1$ in the ground model. From different ground models for the Cummings–Foreman iteration we can obtain either _ℵω+1∈I[_ℵω+1]${\mathrm{\aleph }}_{\omega +1}\in I[{\mathrm{\aleph }}_{\omega +1}]$ and every stationary subset of _ℵω+1${\mathrm{\aleph }}_{\omega +1}$ reflects or there are a bad scale at _ℵω${\mathrm{\aleph }}_{\omega }$ and a non-reflecting stationary subset of _ℵω+1∩cof(_ω1)${\mathrm{\aleph }}_{\omega +1}\cap \mathrm{cof}({\omega }_1)$. We also prove that regardless of the ground model a strong generalization of the tree property holds at each _ℵn${\mathrm{\aleph }}_n$ for n≥2$n\ge 2$.     

Observations on quasi-uniform products

We prove that any product of quotient maps in the category of quasi-uniform spaces and quasi-uniformly continuous maps is a quotient map. We also show that a quasi-uniformly continuous map from a product of quasi-uniform spaces into a quasi-pseudometric T₀-space depends on countably many coordinates.Furthermore we characterize those quasi-uniformities that are unique in their quasi-proximity class and prove that this property is preserved under arbitrary products in the category of quasi-uniform spaces.     

POWERSET OPERATOR BASED FOUNDATION FORPOINT-SET LATTICE-THEORETIC (POSLAT) FUZZY SET THEORIES and TOPOLOGIES

Abstract

This paper sets forth in detail point-set lattice-theoretic or poslat foundations of all mathematical and fuzzy set disciplines in which the operations of taking the image and pre-image of (fuzzy) subsets play a fundamental role; such disciplines include algebra, measure and probability theory, and topology. In particular, those aspects of fuzzy sets, hinging around (crisp) powersets of fuzzy subsets and around powerset operators between such powersets lifted from ordinary functions between the underlying base sets, are examined and characterized using point-set and lattice-theoretic methods. The basic goal is to uniquely derive the powerset operators and not simply stipulate them, and in doing this we explicitly distinguish between the “fixed-basis” case (where the underlying lattice of membership values is fixed for the sets in question) and the “variable-basis” case (where the underlying lattice of membership values is allowed to change). Applications to fuzzy sets/logic include: development and justification/characterization of the Zadeh Extension Principle [36], with applications for fuzzy topology and measure theory; characterizations of ground category isomorphisms; rigorous foundation for fuzzy topology in the poslat sense; and characterization of those fuzzy associative memories in the sense of Kosko [18] which are powerset operators. Some results appeared without proof in [31], some with partial proofs in [32], and some in the fixed-basis case in Johnstone [13] and Manes [22].