On the splitting number and Mazurkiewicz's theorem
On the structure of continuous uninorms
Uninorms were introduced by Yager and Rybalov [13] as a generalization of triangular norms and conorms. We ask about properties of increasing, associative, continuous binary operation in the unit interval with the neutral element . If operation is continuous, then or . So, we consider operations which are continuous in the open unit square. As a result every associative, increasing binary operation with the neutral element , which is continuous in the open unit square may be given in ...
On the structure of numerical event spaces
The probability of the occurrence of an event pertaining to a physical system which is observed in different states determines a function from the set of states of the system to . The function is called a numerical event or multidimensional probability. When appropriately structured, sets of numerical events form so-called algebras of -probabilities. Their main feature is that they are orthomodular partially ordered sets of functions with an inherent full set of states. A classical...
On the structure of perfect sets in various topologies associated with tree forcings
We prove that the Ellentuck, Hechler and dual Ellentuck topologies are perfect isomorphic to one another. This shows that the structure of perfect sets in all these spaces is the same. We prove this by finding homeomorphic embeddings of one space into a perfect subset of another. We prove also that the space corresponding to eventually different forcing cannot contain a perfect subset homeomorphic to any of the spaces above.
On the -conditionality of -power based implications
It is well known that, in forward inference in fuzzy logic, the generalized modus ponens is guaranteed by a functional inequality called the law of -conditionality. In this paper, the -conditionality for -power based implications is deeply studied and the concise necessary and sufficient conditions for a power based implication being -conditional are obtained. Moreover, the sufficient conditions under which a power based implication is -conditional are discussed, this discussions give an...
On the topological complexity of infinitary rational relations
We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [20].
On the Topological Complexity of Infinitary Rational Relations
We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [20].
On the transfer principle in fuzzy theory.
We show in this paper that almost all results proved in many papers about fuzzy algebras can be proved uniformly and immediately by using so-called Transfer Principle.
On tightness in chain-net spaces
On Todorcevic orderings
The Todorcevic ordering 𝕋(X) consists of all finite families of convergent sequences in a given topological space X. Such an ordering was defined for the special case of the real line by S. Todorcevic (1991) as an example of a Borel ordering satisfying ccc that is not σ-finite cc and even need not have the Knaster property. We are interested in properties of 𝕋(X) where the space X is taken as a parameter. Conditions on X are given which ensure the countable chain condition and its stronger versions...
On tolerance analysis
On types of fuzzy numbers under addition
We consider the question whether, for given fuzzy numbers, there are different pairs of -norm such that the resulting membership function within the extension principle under addition are identical. Some examples are given.
On Ulam's problem on families of measures
On unconditionally saturated Banach spaces
We prove a structural property of the class of unconditionally saturated separable Banach spaces. We show, in particular, that for every analytic set 𝓐, in the Effros-Borel space of subspaces of C[0,1], of unconditionally saturated separable Banach spaces, there exists an unconditionally saturated Banach space Y, with a Schauder basis, that contains isomorphic copies of every space X in the class 𝓐.
On universal null and universally measurable sets, II
On Vitali sets and their unions
On weak Normality and Selectivity of Ultrafilters
On weaker forms of choice in second order arithmetic
On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces
We introduce the concept of a family of sets generating another family. Then we prove that if X is a topological space and X has W = {W(x): x ∈ X} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of ω-rank, then X is metaLindelöf as well as a countable product of them. We also prove that if W satisfies ω-rank (F) and, for every x ∈ X, W(x) is of the form W 0(x) ∪ W 1(x), where W 0(x) is Noetherian and W 1(x) consists of neighbourhoods of...
On Weakly Measurable Functions
We show that if T is an uncountable Polish space, 𝓧 is a metrizable space and f:T→ 𝓧 is a weakly Baire measurable function, then we can find a meagre set M ⊆ T such that f[T∖M] is a separable space. We also give an example showing that "metrizable" cannot be replaced by "normal".