All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Previous Page 14 Next

Displaying 261 – 280 of 408

Showing per page

Orthomodular lattices with almost orthogonal sets of atoms

Sylvia Pulmannová, Vladimír Rogalewicz (1991)

Commentationes Mathematicae Universitatis Carolinae

The set A of all atoms of an atomic orthomodular lattice is said to be almost orthogonal if the set { b ∈ A : b ≰ a ' } is finite for every a ∈ A . It is said to be strongly almost orthogonal if, for every a ∈ A , any sequence b 1 , b 2 , ⋯ of atoms such that a ≰ b 1 ' , b 1 ≰ b 2 ' , ⋯ contains at most finitely many distinct elements. We study the relation and consequences of these notions. We show among others that a complete atomic orthomodular lattice is a compact topological one if and only if the set of all its atoms is almost orthogonal.

Orthomodular lattices with fully nontrivial commutators

Milan Matoušek (1992)

Commentationes Mathematicae Universitatis Carolinae

An orthomodular lattice L is said to have fully nontrivial commutator if the commutator of any pair x , y ∈ L is different from zero. In this note we consider the class of all orthomodular lattices with fully nontrivial commutators. We show that this class forms a quasivariety, we describe it in terms of quasiidentities and situate important types of orthomodular lattices (free lattices, Hilbertian lattices, etc.) within this class. We also show that the quasivariety in question is not a variety answering...

Orthomodular lattices with state-separated noncompatible pairs

R. Mayet, Pavel Pták (2000)

Czechoslovak Mathematical Journal

In the logico-algebraic foundation of quantum mechanics one often deals with the orthomodular lattices (OML) which enjoy state-separating properties of noncompatible pairs (see e.g. , and ). These properties usually guarantee reasonable “richness” of the state space—an assumption needed in developing the theory of quantum logics. In this note we consider these classes of OMLs from the universal algebra standpoint, showing, as the main result, that these classes form quasivarieties. We also illustrate...

Orthorings

Ivan Chajda, Helmut Länger (2004)

Discussiones Mathematicae - General Algebra and Applications

Certain ring-like structures, so-called orthorings, are introduced which are in a natural one-to-one correspondence with lattices with 0 every principal ideal of which is an ortholattice. This correspondence generalizes the well-known bijection between Boolean rings and Boolean algebras. It turns out that orthorings have nice congruence and ideal properties.

Partially additive states on orthomodular posets

Josef Tkadlec (1991)

Colloquium Mathematicae

We fix a Boolean subalgebra B of an orthomodular poset P and study the mappings s:P → [0,1] which respect the ordering and the orthocomplementation in P and which are additive on B. We call such functions B-states on P. We first show that every P possesses "enough" two-valued B-states. This improves the main result in [13], where B is the centre of P. Moreover, it allows us to construct a closure-space representation of orthomodular lattices. We do this in the third section. This result may also...

Prime ideal theorem for double Boolean algebras

Léonard Kwuida (2007)

Discussiones Mathematicae - General Algebra and Applications

Double Boolean algebras are algebras (D,⊓,⊔,⊲,⊳,⊥,⊤) of type (2,2,1,1,0,0). They have been introduced to capture the equational theory of the algebra of protoconcepts. A filter (resp. an ideal) of a double Boolean algebra D is an upper set F (resp. down set I) closed under ⊓ (resp. ⊔). A filter F is called primary if F ≠ ∅ and for all x ∈ D we have x ∈ F or x ⊲ ∈ F . In this note we prove that if F is a filter and I an ideal such that F ∩ I = ∅ then there is a primary filter G containing F such that G...

Currently displaying 261 – 280 of 408