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Acta Mathematica Sinica English Series

Conditional Probability on the Kpka’s D-posets

K Ivana

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

Kpka’s D-poset is a very important notion in quantum structures. In this paper the conditional probability on the Kpka’s D-posets is studied. The notion of conditional probability is introduced and the basic properties of conditional probability are proved.

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Abstract In this paper, we propose a type of generalization of triangular norms and quantum logic functions on d-poset algebra. We compare the modified constructions, and properties to the classical properties of the generalized concepts. We show several structures with proof of each type of the proposed generalization. We show several relationships that connect the triangular norms to the quantum logic maps, and also show their relationships to the classical probability space. We provide some explanatory examples that show each structure on d-poset with extra properties that depends on a definition of state.
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Abstract With many physical processes in which quantum mechanical phenomena can occur, it is essential to take into account a decision mechanism based on measurement data. This can be achieved by means of so-called numerical events, which are specified as follows: Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state s∈Ss\in Ss∈S s ∈ S . A function p:S→[0,1]p:S\rightarrow [0,1]p:S→[0,1] p : S → [ 0 , 1 ] is called a numerical event or alternatively, an S-probability. If a set P of S-probabilities is ordered by the order of real functions, it becomes a poset which can be considered as a quantum logic. In case the logic P is a Boolean algebra, this will indicate that the underlying physical system is a classical one. The goal of this paper is to study sets of S-probabilities which are not far from being Boolean algebras by means of the addition and comparison of functions that occur in these sets. In particular, certain classes of so-called Boolean posets of S-probabilities are characterized and related to each other and descriptions based on sets of states are derived.
Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state s in S. Such a function p from S to [0,1] is known as a numerical event or more accurately an S-probability. A set P of numerical events including the constant functions 0 and 1 and 1-p with every p in P becomes a poset when ordered by the order of real functions and can serve as a general setting for quantum logics. We call such a poset P a general set of events (GSE). The thoroughly investigated algebras of S-probabilities (including Hilbert logics), concrete logics and Boolean algebras can all be represented within this setting. In this paper we study various classes of GSEs, in particular those that are orthoposets and their interrelations and connections to known logics. Moreover, we characterize GSEs as posets by means of states and discuss the situation for GSEs to be lattices.