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 4 Next

Displaying 61 – 80 of 85

Showing per page

On the structure of numerical event spaces

Gerhard Dorfer, Dietmar W. Dorninger, Helmut Länger (2010)

Kybernetika

The probability p ( s ) of the occurrence of an event pertaining to a physical system which is observed in different states s determines a function p from the set S of states of the system to [ 0 , 1 ] . The function p is called a numerical event or multidimensional probability. When appropriately structured, sets P of numerical events form so-called algebras of S -probabilities. Their main feature is that they are orthomodular partially ordered sets of functions p with an inherent full set of states. A classical...

On uniform dimensions of finite groups

J. Krempa, A. Sakowicz (2001)

Colloquium Mathematicae

Let G be any finite group and L(G) the lattice of all subgroups of G. If L(G) is strongly balanced (globally permutable) then we observe that the uniform dimension and the strong uniform dimension of L(G) are well defined, and we show how to calculate these dimensions.

Orthocomplemented difference lattices with few generators

Milan Matoušek, Pavel Pták (2011)

Kybernetika

The algebraic theory of quantum logics overlaps in places with certain areas of cybernetics, notably with the field of artificial intelligence (see, e. g., [19, 20]). Recently an effort has been exercised to advance with logics that possess a symmetric difference ([13, 14]) - with so called orthocomplemented difference lattices (ODLs). This paper further contributes to this effort. In [13] the author constructs an ODL that is not set-representable. This example is quite elaborate. A main result...

Orthogonality and complementation in the lattice of subspaces of a finite vector space

Ivan Chajda, Helmut Länger (2022)

Mathematica Bohemica

We investigate the lattice 𝐋 ( 𝐕 ) of subspaces of an m -dimensional vector space 𝐕 over a finite field GF ( q ) with a prime power q = p n together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice 𝐋 ( 𝐕 ) satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when 𝐋 ( 𝐕 ) is orthomodular. For...

Orthomodular lattices and closure operations in ordered vector spaces

Jan Florek (2010)

Banach Center Publications

On a non-trivial partially ordered real vector space (V,≤) the orthogonality relation is defined by incomparability and ζ(V,⊥) is a complete lattice of double orthoclosed sets. We say that A ⊆ V is an orthogonal set when for all a,b ∈ A with a ≠ b, we have a ⊥ b. In our earlier papers we defined an integrally open ordered vector space and two closure operations A → D(A) and A → A ⊥ ⊥ . It was proved that V is integrally open iff D ( A ) = A ⊥ ⊥ for every orthogonal set A ⊆ V. In this paper we generalize this result. We...

Orthomodular lattices that are horizontal sums of Boolean algebras

Ivan Chajda, Helmut Länger (2020)

Commentationes Mathematicae Universitatis Carolinae

The paper deals with orthomodular lattices which are so-called horizontal sums of Boolean algebras. It is elementary that every such orthomodular lattice is simple and its blocks are just these Boolean algebras. Hence, the commutativity relation plays a key role and enables us to classify these orthomodular lattices. Moreover, this relation is closely related to the binary commutator which is a term function. Using the class ℋ of horizontal sums of Boolean algebras, we establish an identity which...

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...

Currently displaying 61 – 80 of 85