Structural properties of algebras of S-probabilities
-
Dietmar Dorninger
and
Helmut Länger
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.
Support of the research of the second author by the Austrian Science Fund (FWF), project I 1923-N25, by ÖAD, project CZ 04/2017, as well as by IGA, project PřF 2018 012, is gratefully acknowledged.
References
[1] Beltrametti, E. G.—Cassinelli, G.: The Logic of Quantum Mechanics, Addison-Wesley, Reading, Mass., 1981.Search in Google Scholar
[2] Beltrametti, E. G.—Mączyński, M. J.: On a characterization of classical and nonclassical probabilities, J. Math. Phys. 32 (1991), 1280–1286.10.1063/1.529326Search in Google Scholar
[3] Dorfer, G.—Dorninger, D.—Länger, H.: On algebras of multidimensional probabilities, Math. Slovaca 60 (2010), 571–582.10.2478/s12175-010-0032-8Search in Google Scholar
[4] Dorfer, G.—Dorninger, D.—Länger, H.: On the structure of numerical event spaces, Kybernetika 46 (2010), 971–981.Search in Google Scholar
[5] Dorninger, D.: On the structure of generalized fields of events, Contr. General Algebra 20 (2012), 29–34.Search in Google Scholar
[6] Dorninger, D.—Länger, H.: On a characterization of physical systems by spaces of numerical events, ARGESIM Report 35 (2009), 601–607.Search in Google Scholar
[7] Dorninger, D.—Länger, H.: Testing for classicality of a physical system, Intern. J. Theor. Phys. 52 (2013), 1141–1147.10.1007/s10773-012-1429-7Search in Google Scholar
[8] Dorninger, D.—Länger, H.: Probability measurements characterizing the classicality of a physical system, Rep. Math. Phys. 73 (2014), 127–135.10.1016/S0034-4877(14)60036-7Search in Google Scholar
[9] Godowski, R. M.: Commutativity in orthomodular posets, Rep. Math. Phys. 18 (1980), 347–351.10.1016/0034-4877(80)90095-6Search in Google Scholar
[10] Klukowski, J.: On the representation of Boolean orthomodular partially ordered sets, Demonstratio Math. 8 (1975), 405–423.10.1515/dema-1975-0404Search in Google Scholar
[11] Mackey, G. W.: Mathematical Foundations of Quantum Mechanics, Dover Publ., Mineola, NY, 2004.Search in Google Scholar
[12] Mączyński, M. J.—Traczyk, T.: A characterization of orthomodular partially ordered sets admitting a full set of states, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 3–8.Search in Google Scholar
[13] Navara, M.: On generating finite orthomodular sublattices, Tatra Mt. Math. Publ. 10 (1997), 109–117.Search in Google Scholar
[14] Navara, M.—Pták, P.: Almost Boolean orthomodular posets, J. Pure Appl. Algebra 60 (1989), 105–111.10.1016/0022-4049(89)90108-4Search in Google Scholar
[15] Pták, P.: Concrete quantum logics, Internat. J. Theoret. Phys. 39 (2000), 827–837.10.1023/A:1003626929648Search in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Small-large subgroups of the reals
- Structural properties of algebras of S-probabilities
- Approximation in quantum measure spaces
- On Fibonomial sums identities with special sign functions: analytically q-calculus approach
- On the proximity of multiplicative functions to the number of distinct prime factors function
- Truncated euler polynomials
- Growth series of crossed and two-sided crossed products of cyclic groups
- Weakly U-abundant semigroups with strong Ehresmann transversals
- Iterative learning control with pulse compensation for fractional differential systems
- Rectifiable and nonrectifiable solution curves of half-linear differential systems
- Uniqueness properties of meromorphic functions in the light of three shared sets
- A new family of analytic functions defined by means of Rodrigues type formula
- Perturbation analysis of a nonlinear equation arising in the Schaefer-Schwartz model of interest rates
- On the solutions of a second-order difference equation in terms of generalized Padovan sequences
- Fixed points in C∗-algebra valued b-metric spaces endowed with a graph
- A general fixed point theorem for two pairs of mappings satisfying a mixed implicit relation
- Productively sequential spaces
- Euler classes of vector bundles over iterated suspensions of real projective spaces
- The stationary distribution and ergodicity of a stochastic mutualism model
- Restricted injective dimensions over local homomorphisms
Articles in the same Issue
- Small-large subgroups of the reals
- Structural properties of algebras of S-probabilities
- Approximation in quantum measure spaces
- On Fibonomial sums identities with special sign functions: analytically q-calculus approach
- On the proximity of multiplicative functions to the number of distinct prime factors function
- Truncated euler polynomials
- Growth series of crossed and two-sided crossed products of cyclic groups
- Weakly U-abundant semigroups with strong Ehresmann transversals
- Iterative learning control with pulse compensation for fractional differential systems
- Rectifiable and nonrectifiable solution curves of half-linear differential systems
- Uniqueness properties of meromorphic functions in the light of three shared sets
- A new family of analytic functions defined by means of Rodrigues type formula
- Perturbation analysis of a nonlinear equation arising in the Schaefer-Schwartz model of interest rates
- On the solutions of a second-order difference equation in terms of generalized Padovan sequences
- Fixed points in C∗-algebra valued b-metric spaces endowed with a graph
- A general fixed point theorem for two pairs of mappings satisfying a mixed implicit relation
- Productively sequential spaces
- Euler classes of vector bundles over iterated suspensions of real projective spaces
- The stationary distribution and ergodicity of a stochastic mutualism model
- Restricted injective dimensions over local homomorphisms
