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Implication algebras

Ivan Chajda (2006)

Discussiones Mathematicae - General Algebra and Applications

We introduce the concepts of pre-implication algebra and implication algebra based on orthosemilattices which generalize the concepts of implication algebra, orthoimplication algebra defined by J.C. Abbott [2] and orthomodular implication algebra introduced by the author with his collaborators. For our algebras we get new axiom systems compatible with that of an implication algebra. This unified approach enables us to compare the mentioned algebras and apply a unified treatment of congruence properties....

Lattice effect algebras densely embeddable into complete ones

Zdena Riečanová (2011)

Kybernetika

An effect algebraic partial binary operation ø p l u s defined on the underlying set E uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion E ^ of E there exists an effect algebraic partial binary operation ⊕ ^ then ⊕ ^ need not be an extension of ⊕ . Moreover, for an Archimedean atomic lattice effect algebra E we give a necessary and sufficient condition for that ⊕ ^ existing on E ^ is an extension of ⊕ defined on E . Further we show that such ⊕ ^ extending ⊕ exists at most...

M V -test spaces versus M V -algebras

Antonio Di Nola, Anatolij Dvurečenskij (2004)

Czechoslovak Mathematical Journal

In analogy with effect algebras, we introduce the test spaces and M V -test spaces. A test corresponds to a hypothesis on the propositional system, or, equivalently, to a partition of unity. We show that there is a close correspondence between M V -algebras and M V -test spaces.

Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras

Martin Kalina (2010)

Kybernetika

If element z of a lattice effect algebra ( E , ⊕ , 0 , 1 ) is central, then the interval [ 0 , z ] is a lattice effect algebra with the new top element z and with inherited partial binary operation ⊕ . It is a known fact that if the set C ( E ) of central elements of E is an atomic Boolean algebra and the supremum of all atoms of C ( E ) in E equals to the top element of E , then E is isomorphic to a subdirect product of irreducible effect algebras ([18]). This means that if there exists a MacNeille completion E ^ of E which is its extension...

Modular atomic effect algebras and the existence of subadditive states

Zdena Riečanová (2004)

Kybernetika

Lattice effect algebras generalize orthomodular lattices and M V -algebras. We describe all complete modular atomic effect algebras. This allows us to prove the existence of ordercontinuous subadditive states (probabilities) on them. For the separable noncomplete ones we show that the existence of a faithful probability is equivalent to the condition that their MacNeille complete modular effect algebra.

Observables on σ -MV algebras and σ -lattice effect algebras

Anna Jenčová, Sylvia Pulmannová, Elena Vinceková (2011)

Kybernetika

Effect algebras were introduced as abstract models of the set of quantum effects which represent sharp and unsharp properties of physical systems and play a basic role in the foundations of quantum mechanics. In the present paper, observables on lattice ordered σ -effect algebras and their “smearings” with respect to (weak) Markov kernels are studied. It is shown that the range of any observable is contained in a block, which is a σ -MV algebra, and every observable is defined by a smearing of a sharp...

On 0 - 1 measure for projectors

Václav Alda (1980)

Aplikace matematiky

An example of a finite set of projectors in E 3 is exhibited for which no 0-1 measure exists.

Currently displaying 61 – 80 of 157