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arXiv:2002.01058 · Advances in Computational Intelligence

On Boolean posets of numerical events

Dietmar W. Dorninger, Helmut Länger

Published Jun 7, 2021• 1 citation
arXivHTMLPDF

Abstract

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 $$s\in S$$ s ∈ S . A function $$p:S\rightarrow [0,1]$$ 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.

DOI
10.1007/s43674-021-00004-w

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The probability [Formula: see text] of the occurrence of an event pertaining to a physical system which is observed in different states [Formula: see text] determines a function [Formula: see text] from the set [Formula: see text] of states of the system to [Formula: see text]. Function [Formula: see text] is called a numerical event or more precisely an [Formula: see text]-probability. When appropriately structured, sets [Formula: see text] of numerical events form so-called algebras of [Formula: see text]-probabilities, which are orthomodular posets that can serve as quantum logics. If one deals with a classical physical system, then [Formula: see text] will be a Boolean algebra. Starting with a supposed quantum logic or logics obtained by individual measurements, newly added numerical events may turn out to be crucial for the assumption of classicality or nonclassicality of the physical system. We will call those numerical events critical and will study their impacts among various classes of algebras of numerical events. Moreover, we will consider the situation of enlarging [Formula: see text] by new states and will describe how elements of an algebra of [Formula: see text]-probabilities contribute to the character of a logic when there exists a certain relation between them and the element that is newly added.

arXiv:2507.00194•Asian-European Journal of Mathematics•Jan 30, 2026•0 citations
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.
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 . If one orders a set P of S -probabilities in respect to the order of functions, further includes the constant functions 0 and 1 and defines p ′ = 1 − p for every p ∈ P , then one obtains a bounded poset of S -probabilities with an antitone involution. We study these posets in respect to various conditions about the existence of the sum of certain functions within the posets and derive properties from these conditions. In particular, questions of relations between different classes of S -probabilities arising this way are settled, algebraic representations are provided and the property that two S -probabilities commute is characterized which is essential for recognizing a classical physical system.
Let S be a set of states of a physical system and p(s) the probability of an occurrence of an event when the system is in state s∈S. The function p from S to [0,1] is called a numerical event, multidimensional probability or, more precisely, S-probability. If a set of numerical events is ordered by the order of real functions one obtains a partial ordered set P in which the sum and difference of S-probabilities are related to their order within P. According to the structure that arises, this further opens up the opportunity to decide whether one deals with a quantum mechanical situation or a classical one. In this paper we focus on the situation that P is generated by a given set of measurements, i.e. S-probabilities, without assuming that these S-probabilities can be complemented by further measurements or are embeddable into Boolean algebras, assumptions that were made in most of the preceding papers. In particular, we study the generation by S-probabilities that can only assume the values 0 and 1, thus dealing with so called concrete logics. We characterize these logics under several suppositions that might occur with measurements and generalize our findings to arbitrary S-probabilities, this way providing a possibility to distinguish between potential classical and quantum situations and the fact that an obtained structure might not be sufficient for an appropriate decision. Moreover, we provide some explanatory examples from physics.
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.
Abstract LetSbe a set of states of a physical system and letp(s) be the probability of an occurrence of an event when the system is in the states∈S. The functionpfromSto [0, 1] is called a numerical event, multidimensional probability or, alternatively,S-probability. Given a set of numerical events which has been obtained by measurements and not supposing any knowledge of the logical structure of the events that appear in the physical system, the question arises which kind of logic is inherent to the system under consideration. In particular, does one deal with a classical situation or a quantum one? In this survey several answers are presented. Starting by associating sets of numerical events to quantum logics we study structures that arise whenS-probabilities are partially ordered by the order of functions and characterize those structures which indicate that one deals with a classical system. In particular, sequences of numerical events are considered that give rise to Bell-like inequalities. At the center of all studies there are so called algebras ofS-probabilities, subsets of these and their generalizations. A crucial feature of these structures is that order theoretic properties can be expressed by the addition and subtraction of real functions entailing simplified algorithmic procedures. The study of numerical events and algebras ofS-probabilities goes back to a cooperation of E. G. Beltrametti and M. J. Mączyński in 1991 and has since then resulted in a series of subsequent papers of physical interest the main results of which will be commented on and put in an appropriate context.
The probability [Formula: see text] of the occurrence of an event pertaining to a physical system which is observed in different states [Formula: see text] determines a function [Formula: see text] from the set [Formula: see text] of states of the system to [Formula: see text]. Function [Formula: see text] is called a numerical event or more precisely an [Formula: see text]-probability. When appropriately structured, sets [Formula: see text] of numerical events form so-called algebras of [Formula: see text]-probabilities, which are orthomodular posets that can serve as quantum logics. If one deals with a classical physical system, then [Formula: see text] will be a Boolean algebra. Starting with a supposed quantum logic or logics obtained by individual measurements, newly added numerical events may turn out to be crucial for the assumption of classicality or nonclassicality of the physical system. We will call those numerical events critical and will study their impacts among various classes of algebras of numerical events. Moreover, we will consider the situation of enlarging [Formula: see text] by new states and will describe how elements of an algebra of [Formula: see text]-probabilities contribute to the character of a logic when there exists a certain relation between them and the element that is newly added.