Affiliation:
1. Institute of Discrete Mathematics and Geometry TU Wien Wiedner Hauptstraβe 8–10 A–1040 Vienna Austria
2. Department of Algebra and Geometry Palacký University Olomouc 17. listopadu 12 CZ–771 46 Olomouc Czech Republic
Abstract
Abstract
Let S be a set of states of a physical system. The probabilities p(s) of the occurrence of an event when the system is in different states s ∈ S define a function from S to [0, 1] called a numerical event or, more precisely, an S-probability. A set of S-probabilities comprising the constant functions 0 and 1 which is structured by means of the addition and order of real functions in such a way that an orthomodular partially ordered set arises is called an algebra of S-probabilities, a structure significant as a quantum-logic with a full set of states. The main goal of this paper is to describe algebraic properties of algebras of S-probabilities through operations with real functions. In particular, we describe lattice characteristics and characterize Boolean features. Moreover, representations by sets are considered and pertinent examples provided.
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