Finiteorthoatomistic effect algebras and regular algebraic E-test spaces
-
Joanna Kaleta
and
Grzegorz Bińczak
Abstract
In this article we consider two categories: the category of finiteorthoatomistic effect algebras and the category of regular algebraic E-test spaces. S. Gudder introduced (algebraic) E-test spaces and showed that for each algebraic E-test space (X; 𝒯) there exists an effect algebra Π(X). Moreover Π is a covariant functor from the category of algebraic E-test spaces to the category of effect algebras. We show that the category of regular and algebraic E-test spaces is equivalent to the category of finiteorthoatomistic effect algebras. We proved that atoms in effect algebras corresponding to regular and algebraic E-test spaces are perspectivity classes of characteristic functions of one-element sets. We found an example of algebraic E-test space which is not regular and in corresponding E-test space some perspectivity class of the characteristic function of a singleton is not an atom.
(Communicated by Mirko Navara)
References
[1] Bińczak, G.—Kaleta, J.—Zembrzuski, A.: Matrix representation of finite effect algebras, Kybernetika 59(5) (2023), 737–751.10.14736/kyb-2023-5-0737Search in Google Scholar
[2] Bush, P.—Lahti, P. J.—Mittelstadt, P.: The quantum theory of measurement. In: The Quantum Theory of Measurement, Lecture Notes in Physics Monographs, Vol. 2, Springer, Berlin, Heidelberg, 1991, pp. 27-98.10.1007/978-3-662-13844-1_3Search in Google Scholar
[3] Bush, P.—Grabowski, M.—Lahti, P. J.: Operational Quantum Physics, Springer-Verlag, Berlin, 1995.10.1007/978-3-540-49239-9Search in Google Scholar
[4] Dvurečenskij, A.—Pulmannová , S.: New Trends in Quantum Structures, Kluwer Academic Publ./Ister Science, Dordrecht-Boston-London/Bratislava, 2000.10.1007/978-94-017-2422-7Search in Google Scholar
[5] Foulis, D. J.—Bennett, M. K.: Effect Algebras and Unsharp quantum Logics, Found. Phys. 24(10) (1994), 1331–1351.10.1007/BF02283036Search in Google Scholar
[6] Giuntini, R.—Grueuling, H.: Toward a formal language for unsharp properties, Found. Phys. 19 (1994), 769–780.Search in Google Scholar
[7] Gudder, S.: Effect test spaces and effect algebras, Found. Phys. 27(2) (1997), 287–304.10.1007/BF02550455Search in Google Scholar
[8] Jacobs, B.—Mandemaker, J.: Coreflections in algebraic quantum logic, Found. Phys. 42 (2012), 932–958.10.1007/s10701-012-9654-8Search in Google Scholar
[9] Kôpka, F.—Chovanec, F.: D-posets, Math. Slovaca 44 (1994), 21–34.Search in Google Scholar
[10] Mac Lane, S.: Categories for the Working Mathematician, 2nd ed., Springer, New York, 1971.10.1007/978-1-4612-9839-7Search in Google Scholar
[11] Wei Ji: Characterization of homogeneity in orthocomplete atomic effect algebras, Fuzzy Sets and Systems 236 (2014), 113–121.10.1016/j.fss.2013.06.005Search in Google Scholar
[12] Wiegandt R.: On the general theory of Möbius inversion formula and Möbius product, Acta Sci. Math. 20 (1959), 164–180.Search in Google Scholar
© 2025 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- A new categorical equivalence for stone algebras
- On special classes of prime filters in BL-algebras
- A note on characterized and statistically characterized subgroups of 𝕋 = ℝ/ℤ
- New Young-type integral inequalities using composition schemes
- The structure of pseudo-n-uninorms with continuous underlying functions
- Jensen-type inequalities for a second-order differential inequality condition
- A direct proof of the characterization of the convexity of the discrete Choquet integral
- Envelope of plurifinely plurisubharmonic functions and complex Monge-Ampère type equation
- Fekete-Szegö inequalities for Φ-parametric and β-spirllike mappings of complex order in ℂn
- Entire function sharing two values partially with its derivative and a conjecture of Li and Yang
- Oscillatory properties of third-order semi-canonical dynamic equations on time scales via canonical transformation
- Weighted B-summability and positive linear operators
- Some properties and applications of convolution algebras
- On measures of σ-noncompactess in F-spaces
- On the kolmogorov–feller–gut weak law of large numbers for triangular arrays of rowwise and pairwise negatively dependent random variables
- Intermediately trimmed sums of oppenheim expansions: A strong law
- Novel weighted distribution: Properties, applications and web-tool
- On the q-Gamma distribution: Properties and inference
- Finiteorthoatomistic effect algebras and regular algebraic E-test spaces
- Prof. RNDr. Anatolij Dvurečenskij, DrSc. 75th anniversary
Articles in the same Issue
- A new categorical equivalence for stone algebras
- On special classes of prime filters in BL-algebras
- A note on characterized and statistically characterized subgroups of 𝕋 = ℝ/ℤ
- New Young-type integral inequalities using composition schemes
- The structure of pseudo-n-uninorms with continuous underlying functions
- Jensen-type inequalities for a second-order differential inequality condition
- A direct proof of the characterization of the convexity of the discrete Choquet integral
- Envelope of plurifinely plurisubharmonic functions and complex Monge-Ampère type equation
- Fekete-Szegö inequalities for Φ-parametric and β-spirllike mappings of complex order in ℂn
- Entire function sharing two values partially with its derivative and a conjecture of Li and Yang
- Oscillatory properties of third-order semi-canonical dynamic equations on time scales via canonical transformation
- Weighted B-summability and positive linear operators
- Some properties and applications of convolution algebras
- On measures of σ-noncompactess in F-spaces
- On the kolmogorov–feller–gut weak law of large numbers for triangular arrays of rowwise and pairwise negatively dependent random variables
- Intermediately trimmed sums of oppenheim expansions: A strong law
- Novel weighted distribution: Properties, applications and web-tool
- On the q-Gamma distribution: Properties and inference
- Finiteorthoatomistic effect algebras and regular algebraic E-test spaces
- Prof. RNDr. Anatolij Dvurečenskij, DrSc. 75th anniversary
