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

10.22111/ijfs.2026.55004.9752

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.

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References

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Iranian Journal of Fuzzy Systems
Volume 23, Issue 3
May and June 2026
Pages 85-94

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