A family is constructed of cardinality equal to the continuum, whose members are totally incomparable hereditarily indecomposable Banach spaces.

Any given increasing ${[0,1]}^2\to [0,1]$ function is completely determined by its contour lines. In this paper we show how each individual uninorm property can be translated into a property of contour lines. In particular, we describe commutativity in terms of orthosymmetry and we link associativity to the portation law and the exchange principle. Contrapositivity and rotation invariance are used to characterize uninorms that have a continuous contour line.

We compare the forcing-related properties of a complete Boolean algebra $𝔹$ with the properties of the convergences ${\lambda }_{\mathrm{s}}$ (the algebraic convergence) and ${\lambda }_{\mathrm{ls}}$ on $𝔹$ generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that ${\lambda }_{\mathrm{ls}}$ is a topological convergence iff forcing by $𝔹$ does not produce new reals and that ${\lambda }_{\mathrm{ls}}$ is weakly topological if $𝔹$ satisfies condition $\left(\hslash \right)$ (implied by the $𝔱$-cc). On the other hand, if ${\lambda }_{\mathrm{ls}}$ is a weakly topological convergence, then $𝔹$ is a $2^{𝔥}$-cc algebra...

By a nearlattice is meant a join-semilattice where every principal filter is a lattice with respect to the induced order. The aim of our paper is to show for which nearlattice $𝒮$ and its element $c$ the mapping ${\varphi }_c\left(x\right)=\langle x\vee c,x{\wedge }_{p}c\rangle$ is a (surjective, injective) homomorphism of $𝒮$ into $\left[c\right)\times \left(c\right]$.

$BL$-algebras, introduced by P. Hájek, form an algebraic counterpart of the basic fuzzy logic. In the paper it is shown that $BL$-algebras are the duals of bounded representable $DRl$-monoids. This duality enables us to describe some structure properties of $BL$-algebras.

We establish categorical dualities between varieties of isotropic median algebras and suitable categories of operational and relational topological structures. We follow a general duality theory of B.A. Davey and H. Werner. The duality results are used to describe free isotropic median algebras. If the number of free generators is less than five, the description is detailed.

We construct a totally ordered set Γ of positive infinite germs (i.e. germs of positive real-valued functions that tend to +∞), with order type being the lexicographic product ℵ1 × ℤ2. We show that Γ admits $$2^{{\aleph }_1}$$ order preserving automorphisms of pairwise distinct growth rates.

Let $K$ be a field with a Krull valuation $\left|\right|$ and value group $G\ne \left\{1\right\}$, and let $B_K$ be the valuation ring. Theories about spaces of countable type and Hilbert-like spaces in [1] and spaces of continuous linear operators in [2] require that all absolutely convex subsets of the base field $K$ should be countably generated as $B_K$-modules.By [1] Prop. 1.4.1, the field $K$ is metrizable if and only if the value group $G$ has a cofinal sequence. We prove that for any fixed cardinality ${\aleph }_{\kappa }$, there exists a metrizable field $K$...

It was proved by Juhász and Weiss that for every ordinal α with $0<\alpha <{\omega }_2$ there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that ${\kappa }^{<\kappa }=\kappa$ and α is an ordinal such that $0<\alpha <{\kappa }^{++}$, then there is a cardinal-preserving partial order that forces the existence of a superatomic Boolean algebra of height α and width κ. Furthermore, iterating this forcing through all $\alpha <{\kappa }^{++}$, we obtain a notion of forcing that preserves cardinals and such that in the corresponding generic...