Die atomare Struktur topologischer Boolescher Ringe und s-beschränkter Inhalte
Die Operationen in der Klasse komplementärer Verbände
Directoids with an antitone involution
We investigate -directoids which are bounded and equipped by a unary operation which is an antitone involution. Hence, a new operation can be introduced via De Morgan laws. Basic properties of these algebras are established. On every such an algebra a ring-like structure can be derived whose axioms are similar to that of a generalized boolean quasiring. We introduce a concept of symmetrical difference and prove its basic properties. Finally, we study conditions of direct decomposability of directoids...
Discriminator varieties of Boolean algebras with residuated operators
The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolean algebras with a residuated binary operator, abbreviated as r-algebras. More specifically, we show that all finite r-algebras, all integral r-algebras, all unital r-algebras with finitely many elements below the unit, and all commutative residuated...
Disjoint refinement and related topics
Disjoint sequences in Boolean algebras
We deal with the system of all sequential convergences on a Boolean algebra . We prove that if is a sequential convergence on which is generated by a set of disjoint sequences and if is any element of , then the join exists in the partially ordered set . Further we show that each interval of is a Brouwerian lattice.
Dominating analytic families
Let A be an analytic family of sequences of sets of integers. We show that either A is dominated or it contains a continuum of almost disjoint sequences. From this we obtain a theorem by Shelah that a Suslin c.c.c. forcing adds a Cohen real if it adds an unbounded real.
Effect algebras and ring-like structures
The dichotomic physical quantities, also called propositions, can be naturally associated to maps of the set of states into the real interval [0,1]. We show that the structure of effect algebra associated to such maps can be represented by quasiring structures, which are a generalization of Boolean rings, in such a way that the ring operation of addition can be non-associative and the ring multiplication non-distributive with respect to addition. By some natural assumption on the effect algebra,...
Efficient calculation of the Reed-Muller form by means of the Walsh transform
The paper describes a spectral method for combinational logic synthesis using the Walsh transform and the Reed-Muller form. A new algorithm is presented that allows us to obtain the mixed polarity Reed-Muller expansion of Boolean functions. The most popular minimisation (sub-minimisation) criterion of the Reed-Muller form is obtained by the exhaustive search of all the polarity vectors. This paper presents a non-exhaustive method for Reed-Muller expansions. The new method allows us to build the...
Egoroff, σ, and convergence properties in some archimedean vector lattices
An archimedean vector lattice A might have the following properties: (1) the sigma property (σ): For each there are and a ∈ A with λₙaₙ ≤ a for each n; (2) order convergence and relative uniform convergence are equivalent, denoted (OC ⇒ RUC): if aₙ ↓ 0 then aₙ → 0 r.u. The conjunction of these two is called strongly Egoroff. We consider vector lattices of the form D(X) (all extended real continuous functions on the compact space X) showing that (σ) and (OC ⇒ RUC) are equivalent, and equivalent...
Ein Assoziativitätskriterium vom Foulis-Holland-Typ.
Ein neues Axiomensystem für Boolesche Verbände.
Eine Charakterisierung der Boole'schen Algebra zu einer ...-direkten Zerlegung.
Elementary equivalence of lattices of open sets definable in o-minimal expansions of real closed fields
We prove that the boolean algebras of sets definable in elementarily equivalent o-minimal expansions of real closed fields are back-and-forth equivalent, and in particular elementarily equivalent, in the language of boolean algebras with new predicates indicating the dimension, Euler characteristic and open sets. We also show that the boolean algebra of semilinear subsets of [0,1]ⁿ definable in an o-minimal expansion of a real closed field is back-and-forth equivalent to the boolean algebra of definable...
Embedding of free Boolean algebras in complete Boolean algebras
Embeddings into 𝓟(ℕ)/fin and extension of automorphisms
Given a Boolean algebra 𝔹 and an embedding e:𝔹 → 𝓟(ℕ)/fin we consider the possibility of extending each or some automorphism of 𝔹 to the whole 𝓟(ℕ)/fin. Among other things, we show, assuming CH, that for a wide class of Boolean algebras there are embeddings for which no non-trivial automorphism can be extended.
Equational compactness in infinitary algebras
Equational Theories of Varieties defined by P-compatible Identities of Boole'an Algebras
Equimorphy in varieties of distributive double -algebras
Any finitely generated regular variety of distributive double -algebras is finitely determined, meaning that for some finite cardinal , any subclass of algebras with isomorphic endomorphism monoids has fewer than pairwise non-isomorphic members. This result follows from our structural characterization of those finitely generated almost regular varieties which are finitely determined. We conjecture that any finitely generated, finitely determined variety of distributive double -algebras...
Errata : «Facteurs et plans. I : Structure de finesse»