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Relatively additive states on quantum logics

Pavel Pták, Hans Weber (2005)

Commentationes Mathematicae Universitatis Carolinae

In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous...

Remarks on effect-tribes

Sylvia Pulmannová, Elena Vinceková (2015)

Kybernetika

We show that an effect tribe of fuzzy sets 𝒯 [ 0 , 1 ] X with the property that every f 𝒯 is 0 ( 𝒯 ) -measurable, where 0 ( 𝒯 ) is the family of subsets of X whose characteristic functions are central elements in 𝒯 , is a tribe. Moreover, a monotone σ -complete effect algebra with RDP with a Loomis-Sikorski representation ( X , 𝒯 , h ) , where the tribe 𝒯 has the property that every f 𝒯 is 0 ( 𝒯 ) -measurable, is a σ -MV-algebra.

Representation theorem for convex effect algebras

Stanley P. Gudder, Sylvia Pulmannová (1998)

Commentationes Mathematicae Universitatis Carolinae

Effect algebras have important applications in the foundations of quantum mechanics and in fuzzy probability theory. An effect algebra that possesses a convex structure is called a convex effect algebra. Our main result shows that any convex effect algebra admits a representation as a generating initial interval of an ordered linear space. This result is analogous to a classical representation theorem for convex structures due to M.H. Stone.

Ring-like structures with unique symmetric difference related to quantum logic

Dietmar Dorninger, Helmut Länger, Maciej Maczyński (2001)

Discussiones Mathematicae - General Algebra and Applications

Ring-like quantum structures generalizing Boolean rings and having the property that the terms corresponding to the two normal forms of the symmetric difference in Boolean algebras coincide are investigated. Subclasses of these structures are algebraically characterized and related to quantum logic. In particular, a physical interpretation of the proposed model following Mackey's approach to axiomatic quantum mechanics is given.

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