We study deterministic one-way communication complexity of functions with Hankel communication matrices. Some structural properties of such matrices are established and applied to the one-way two-party communication complexity of symmetric Boolean functions. It is shown that the number of required communication bits does not depend on the communication direction, provided that neither direction needs maximum complexity. Moreover, in order to obtain an optimal protocol, it is in any case sufficient...

Several open problems posed during FSTA 2006 (Liptovský Ján, Slovakia) are presented. These problems concern the classification of strict triangular norms, Lipschitz t-norms, interval semigroups, copulas, semicopulas and quasi- copulas, fuzzy implications, means, fuzzy relations, MV-algebras and effect algebras.

Eighteen open problems posed during FSTA 2010 (Liptovský Ján, Slovakia) are presented. These problems concern copulas, triangular norms and related aggregation functions. Some open problems concerning effect algebras are also included.

We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is ℵ₂-projective. Previously it was known that this characterization of openly generated Boolean algebras follows from Axiom R. Since FRP is preserved by c.c.c. generic extension, we conclude in particular that this characterization is consistent with any set-theoretic assertion forcable by a c.c.c. poset starting from a model of FRP. A crucial step...

L’espace des configurations de $p$ points distincts de ${\mathbf{R}}^{\infty }$ admet une filtration naturelle qui est induite par les inclusions des ${\mathbf{R}}^n$ dans ${\mathbf{R}}^{\infty }$. Nous caractérisons le type d’homotopie de cette filtration par les propriétés combinatoires d’une structure cellulaire sous-jacente, étroitement liée à la théorie des $E_n$-opérades de May. Cela donne une approche unifiée des différents modèles combinatoires d’espaces de lacets itérés et redémontre les théorèmes d’approximation de Milgram, Smith et Kashiwabara.

It is shown that if α,ζ are ordinals such that 1 ≤ ζ < α < ζω, then there is an operator from $C\left({\omega }^{{\omega }^{\alpha }}\right)$ onto itself such that if Y is a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$ which is isomorphic to $C\left({\omega }^{{\omega }^{\alpha }}\right)$, then the operator is not an isomorphism on Y. This contrasts with a result of J. Bourgain that implies that there are uncountably many ordinals α for which for any operator from $C\left({\omega }^{{\omega }^{\alpha }}\right)$ onto itself there is a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$ which is isomorphic to $C\left({\omega }^{{\omega }^{\alpha }}\right)$ on which the operator is an isomorphism.

Closure $GMV$-algebras are introduced as a commutative generalization of closure $MV$-algebras, which were studied as a natural generalization of topological Boolean algebras.

This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices $𝐋$ easily reduces to the case when $𝐋$ is a subdirect product of two simple lattices ${𝐋}_1$ and ${𝐋}_2$. Our main result claims that such a lattice is locally...

It is proved by an order theoretical and purely algebraic method that any order bounded orthosymmetric bilinear operator $b:E\times E\to F$ where $E$ and $F$ are Archimedean vector lattices is symmetric. This leads to a new and short proof of the commutativity of Archimedean almost $f$-algebras.

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We give some criteria for order boundedness of $E\left(\mu \right)$ in $ba\left(\Re \right)$, in the general case as well as for atomic $\mu$. Order boundedness implies weak compactness of $E\left(\mu \right)$. We show that the converse implication holds under some assumptions on $𝔐$, $\Re$ and $\mu$ or $\mu$ alone, but not in general.