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arXiv:2507.00194 · Asian-European Journal of Mathematics

Critical elements in algebras of numerical events

Dietmar W. Dorninger, Helmut Länger

Published Jan 30, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s1793557126500130

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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.
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 [Formula: see text] be a set of states of a physical system. The probabilities [Formula: see text] of the occurrence of an event when the system is in different states [Formula: see text] define a function from [Formula: see text] to [Formula: see text] called a numerical event or, more accurately, an [Formula: see text]-probability. Sets of [Formula: see text]-probabilities ordered by the partial order of functions give rise to so-called algebras of [Formula: see text]-probabilities, in particular, to the ones that are lattice-ordered. Among these, there are the [Formula: see text]-algebras known from probability theory and the Hilbert-space logics which are important in quantum-mechanics. Any algebra of [Formula: see text]-probabilities can serve as a quantum-logic, and it is of special interest when this logic turns out to be a Boolean algebra because then the observed physical system will be classical. Boolean algebras are in one-to-one correspondence to Boolean rings, and the question arises to find an analogue correspondence for lattice-ordered algebras of [Formula: see text]-probabilities generalizing the correspondence between Boolean algebras and Boolean rings. We answer this question by defining ring-like structures of events (RLSEs). First, the structure of RLSEs is revealed and Boolean rings among RLSEs are characterized. Then we establish how RLSEs correspond to lattice-ordered algebras of numerical events. Further, functions for associating lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs are studied. It is shown that there are only two ways to assign lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs if one restricts the corresponding mappings to term functions over the underlying orthomodular lattice. These term functions are the very functions by which also the Boolean algebras can be assigned to Boolean rings.
A class of ring-like structures of events (RLSEs) is studied that generalizes Boolean rings. Quantum logics represented by orthomodular lattices are characterized within this class and the correspondence between Boolean algebras and Boolean rings is enlarged to orthomodular lattices. The structure of RLSEs and various subclasses is analyzed and classical logics are especially identified. Moreover, sets of numerical events within different contexts of physical problems are described. A numerical event is defined as a function [Formula: see text] from a set [Formula: see text] of states of a physical system to [Formula: see text] such that [Formula: see text] is the probability of the occurrence of an event when the system is in state [Formula: see text]. In particular, the question is answered whether a given (small) set of numerical events will give rise to the assumption that one deals with a classical physical system or a quantum-mechanical one.
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.
Abstract The probability p(s) of the occurrence of an event pertaining to a physical system which is observed in different states s determines a function p from the set S of states of the system to [0, 1]. The function p is called a multidimensional probability or numerical event. Sets of numerical events which are structured either by partially ordering the functions p and considering orthocomplementation or by introducing operations + and · in order to generalize the notion of Boolean rings representing classical event fields are studied with the goal to relate the algebraic operations + and · to the sum and product of real functions and thus to distinguish between classical and quantum mechanical behaviour of the physical system. Necessary and sufficient conditions for this are derived, as well for the case that the functions p can assume any value between 0 and 1 as for the special cases that the values of p are restricted to two or three different outcomes.
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.