Commuting is an important property in any two-step information merging procedure where the results should not depend on the order in which the single steps are performed. In the case of bisymmetric aggregation operators with the neutral elements, Saminger, Mesiar and Dubois, already reduced characterization of commuting $n$-ary operators to resolving the unary distributive functional equations. And then the full characterizations of these equations are obtained under the assumption that the unary...

The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, $\theta ={\left(2^{cf\left(\mu \right)}\right)}^{+}$ and $2^{\mu }={\mu }^{+}$ then there are Boolean algebras ${𝔹}_1,{𝔹}_2$ such that $c\left({𝔹}_1\right)=\mu ,c\left({𝔹}_2\right)<\theta butc({𝔹}_1*{𝔹}_2)={\mu }^{+}$. Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if $𝔹$ is a ccc Boolean algebra and ${\mu }^{{\beth }_{\omega }}\le \lambda =cf\left(\lambda \right)\le 2^{\mu }$ then $𝔹$ satisfies the λ-Knaster condition (using the “revised GCH theorem”).

Let $(𝔄,{ϵ}_u)$ and $(𝔅,{ϵ}_b)$ be two pointed sets. Given a family of three maps $ℱ=\{f_1:𝔄\to 𝔄;f_2:𝔄\times 𝔄\to 𝔄;f_3:𝔄\times 𝔄\to 𝔅\}$, this family provides an adequate decomposition of $𝔄\setminus \left\{{ϵ}_u\right\}$ as the orthogonal disjoint union of well-described $ℱ$-invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak $H^{*}$-algebras.

We show the consistency of CH and the statement “no ccc forcing has the Sacks property” and derive some consequences for ccc ${\omega }^{\omega }$-bounding forcing notions.

We present two ${\mathbb{P}}_{max}$ varations which create maximal models relative to certain counterexamples to Martin’s Axiom, in hope of separating certain classical statements which fall between MA and Suslin’s Hypothesis. One of these models is taken from [19], in which we maximize relative to the existence of a certain type of Suslin tree, and then force with that tree. In the resulting model, all Aronszajn trees are special and Knaster’s forcing axiom ₃ fails. Of particular interest is the still open question...

We give an affirmative answer to problem DJ from Fremlin’s list [8] which asks whether $M{A}_{{\omega }_1}$ implies that every uncountable Boolean algebra has an uncountable set of pairwise incomparable elements.

In the present paper, fuzzy order relations on a real vector space are characterized by fuzzy cones. It is well-known that there is one-to-one correspondence between order relations, that a real vector space with the order relation is an ordered vector space, and pointed convex cones. We show that there is one-to-one correspondence between fuzzy order relations with some properties, which are fuzzification of the order relations, and fuzzy pointed convex cones, which are fuzzification of the pointed...