Elsevier

Fuzzy Sets and Systems

Volume 159, Issue 9, 1 May 2008, Pages 1116-1122
Fuzzy Sets and Systems

Effect algebraic extensions of generalized effect algebras and two-valued states

https://doi.org/10.1016/j.fss.2007.12.014Get rights and content

Abstract

D-posets and effect algebras are very natural structures to be carriers of probabilities for events with fuzziness, uncertainty or unsharpness. In spite of that there are effect algebras admitting no states and probabilities. We give necessary and sufficient conditions for existence of two-valued states on Archimedean atomic effect algebras. Consequently a construction of states, probabilities from bounded orthogonally additive measures existing on some ideal sub-generalized effect algebras is shown.

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    Citation Excerpt :

    Different kinds of states have been deeply investigated within the quantum logical program not only because of their importance in order to understand quantum mechanics [11, 12, 25, 28], but also because they provide different representations of the event structure of quantum systems [21, 32, 33]. Recently, several authors have paid attention to the study of states over extended algebraic structures, directly or indirectly related to quantum mechanics, as orthomodular posets [5, 26], MV-algebras [7, 15, 16, 22, 27] or effect algebras [9, 29, 30]. Common open problems of these structures are the characterization of classes of algebras admitting some special types of states [10, 20] and the internalization in an algebraic structure of the concept of state [6, 17].

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