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Journal of Physics Conference Series

Qausi-Copulas on Orthomodular Lattice

Ahmed Al-Adilee, Adel Hashem Nouri

Published Mar 1, 2021• 0 citations
Publisher

Abstract

Abstract One of the possible mathematical systems to represent non-compatible random events is a quantum logic. In this paper, we propose and illustrate function called quasi-copula on an orthomodular lattice (briefly QO-copula). Also, we compare such function to its classical definition. Finally, we present several relations and calculations that involve QO-copula and its relationship to a function known by s-map, which has firstly been presented by [4].

DOI
10.1088/1742-6596/1818/1/012138

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Mathematical description of quantum phenomena requires an event structure more general than a Boolean algebra. For this purpose, Birkhoff and von Neumann proposed the notion of an orthomodular lattice. Its typical feature is the existence of so–called non–commuting events, which are not simultaneously observable (like position and momentum, according to Heisenberg’s uncertainty principle). There is an old open problem whether the word problem for orthomodular lattices is solvable. Is it possible to decide whether two formulas are equivalent? In Boolean algebras, an easy positive answer is given by a transformation of the formula to a unique normal form. This requires the commutativity, associativity, and distributivity of the Boolean operations (disjunction and conjunction). In orthomodular lattices, the corresponding lattice operations (join and meet) violate distributivity. This disables the use of normal forms. We looked for alternative approaches. E.g., the join is only one of six orthomodular lattice operations generalizing the disjunction. In the thesis, we study the question whether some of the 96 binary operations in orthomodular lattices admit normal forms similar to the classical logic. The first question was which operations satisfy the associative identity, eventually under the assumption that some variables commute. Then we studied monotonicity because it is related to distributivity over the meet and join. The conclusion is that there is no pair of operations in orthomodular lattices admitting “Boolean–like” normal forms. In the last chapter we study “Moufang–like” identities, which were inspired by the algebras of quaternions and octonions. These identities generalize associativity and may enable further progress. As a by–product, we proved interesting, yet unknown, properties of some orthomodular lattice operations (e.g., the Sasaki projection). These new tools simplify algebraic computations and give a chance to develop algorithms more general than the current specialized software.

•Jan 1, 2015•0 citations
Abstract One of the possible mathematical systems to represent non-compatible random events is a quantum logic. In this paper, we propose and illustrate function called quasi-copula on an orthomodular lattice (briefly QO-copula). Also, we compare such function to its classical definition. Finally, we present several relations and calculations that involve QO-copula and its relationship to a function known by s-map, which has firstly been presented by [4].
In this paper, we study a new type of sub-copulas, which is defined on quantum logic. We review the most common definitions, relations and properties of such functions. We extend the definition of what is recently called 2-dimensional QL-copula to n-dimensional QL-copula with respect to order difference instead of the summation approach, see [5]. We compose this paper to present an alternative interpretation of QL-copula rather thanthe one that we have already defined by AlAdilee and Nanasiova (2009) using an approach depends on definingsub-copulason quantum logic.
The theory of orthomodular lattice and its relationships that depend on s-map and condi-tional states definitions has shown in several types of probabilistic relations. An orthomodularlattice with a conditional states can be applied and defined as a model for non-compatible events.Studying of s-maps or conditional states on an orthomodular lattice helps us to describe suchproperties of random events, which are difficult to be described by Boolean algebra. In a par-ticularly observed quantum or classical system the information, that we cannot access results inour description of the system be coming mixed even if we have perfect initial knowledge.That is, if the system is quantum the conditional states will be given by a state matrix and ifclassical, the conditional states will be given by a probability distribution. We show conditionswhen a quantum system has the same properties as the classical probability space.Acknowledgement. This work was supported by Science and Technology Assistance AgencyunderthecontractNo. APVT-20-003204, VEGA-1/3014/06, VEGA-1/3321/06, VEGA-1/3006/06
Mathematical description of quantum phenomena requires an event structure more general than a Boolean algebra. For this purpose, Birkhoff and von Neumann proposed the notion of an orthomodular lattice. Its typical feature is the existence of so–called non–commuting events, which are not simultaneously observable (like position and momentum, according to Heisenberg’s uncertainty principle). There is an old open problem whether the word problem for orthomodular lattices is solvable. Is it possible to decide whether two formulas are equivalent? In Boolean algebras, an easy positive answer is given by a transformation of the formula to a unique normal form. This requires the commutativity, associativity, and distributivity of the Boolean operations (disjunction and conjunction). In orthomodular lattices, the corresponding lattice operations (join and meet) violate distributivity. This disables the use of normal forms. We looked for alternative approaches. E.g., the join is only one of six orthomodular lattice operations generalizing the disjunction. In the thesis, we study the question whether some of the 96 binary operations in orthomodular lattices admit normal forms similar to the classical logic. The first question was which operations satisfy the associative identity, eventually under the assumption that some variables commute. Then we studied monotonicity because it is related to distributivity over the meet and join. The conclusion is that there is no pair of operations in orthomodular lattices admitting “Boolean–like” normal forms. In the last chapter we study “Moufang–like” identities, which were inspired by the algebras of quaternions and octonions. These identities generalize associativity and may enable further progress. As a by–product, we proved interesting, yet unknown, properties of some orthomodular lattice operations (e.g., the Sasaki projection). These new tools simplify algebraic computations and give a chance to develop algorithms more general than the current specialized software.