Complete characterization of associative binary operations generating witness maps

Document Type : Research Paper

Author

Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia

Abstract

Motivated by the study of common measurability in the unsharp observables approach to quantum mechanics, Jen\v{c}a (2011) introduced the notion of a witness map on a partially ordered Abelian group with unit $u$. At the 10th International Conference on Fuzzy Set Theory and Applications (FSTA 2010), Jen\v{c}a and Sarkoci (Open Problem 2.10) asked for a complete characterization of all commutative and associative binary operations on the standard real unit interval $[0,1]$ that generate such witness maps. In this paper, we completely resolve this special-case problem. By translating the discrete combinatorial inclusion-exclusion inequality of the witness map definition into the continuous evaluation of an $n$-dimensional volume, we prove that the witness map condition is algebraically identical to the $n$-increasing property. Consequently, an operation generates a witness map if and only if its $n$-ary extension is a valid $n$-dimensional copula for all $n\geq 2$. Applying Kimberling's Theorem (1974), we establish that a binary operation generates a witness map if and only if it is the minimum t-norm, a strict Archimedean t-norm with a completely monotonic inverse generator, or an ordinal sum of such operations. Several illustrative families, including the product, Clayton, and Gumbel-Hougaard copulas, are discussed in detail, and the explicit form of the corresponding set functions $\beta_O$ on $n$-element subsets is given.

Keywords

Main Subjects

References

[1] P. Busch, M. Grabowski, P. Lahti, Operational quantum physics, Springer-Verlag, Berlin, 1995. https://doi.org/
10.1007/978-3-540-49239-9
[2] P. Busch, P. Lahti, P. Mittelstaedt, The quantum theory of measurement, 2nd Ed., Springer Verlag, 1996. https:
//doi.org/10.1007/978-3-540-37205-9
[3] F. Chovanec, F. Kˆopka, D-posets, Mathematica Slovaca, 44(1) (1994), 21-34. https://doi.org/10.1016/
B978-044452870-4/50031-5
[4] F. Durante, C. Sempi, Principles of copula theory, CRC Press, 2016.
[5] D. J. Foulis, M. K. Bennett, Effect algebras and unsharp quantum logics, Foundations of Physics, 24(10) (1994),
1331-1352. https://doi.org/10.1007/BF02283036
[6] G. P. Geh´er, P. ˇSemrl, Coexistency on Hilbert space effect algebras and a characterisation of its symmetry
transformations, Communications in Mathematical Physics, 379 (2020), 1077-1112. https://doi.org/10.1007/
s00220-020-03873-3
[7] T. Heinosaari, J. Kiukas, D. Reitzner, Coexistence of effects from an algebra of two projections, Journal of Physics A:
Mathematical and Theoretical, 47(22) (2014). https://doi.org/10.1088/1751-8113/47/22/225301
[8] G. Jenˇca, Coexistence in interval effect algebras, Proceedings of the American Mathematical Society, 139(2) (2011),
331-344.
[9] C. H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequationes Mathematicae, 10 (1974),
152-164. https://doi.org/10.1007/BF01832852
[10] E. P. Klemet, R. Mesiar, E. Pap, Triangular norms, Trends in Logic, Vol. 8, Kluwer Academic Publishers, Dordrecht,
2000. https://doi.org/10.1007/978-94-015-9540-7
[11] A. J. McNeil, J. Neˇslehov´a, Multivariate Archimedean copulas, d-monotone functions and ℓ1-norm symmetric
distributions, The Annals of Statistics, 37(5B) (2009), 3059-3097. https://doi.org/10.1214/07-AOS556
[12] R. Mesiar, P. Sarkoci, Open problems posed at the tenth international conference on fuzzy set theory and applications, Kybernetika, 46(4) (2010), 585-599.
[13] R. B. Nelsen, An introduction to copulas, 2nd Ed., Springer, 2006. https://doi.org/10.1007/0-387-28678-0
[14] W. Stulpe, Pairwise coexistence of effects versus coexistence, Journal of Physics: Conference Series, 1638 (2020).
https://doi.org/10.1088/1742-6596/1638/1/012004
Iranian Journal of Fuzzy Systems
Volume 23, Issue 3
May and June 2026
Pages 85-94

Files

Share

How to cite

Statistics