Document Type : Research Paper

Author

• Adam Šeliga

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

• witness map
• n-dimensional copula
• associative binary operation
• Archimedean t-norm
• complete monotonicity
• effect algebras

Main Subjects

• Approximate Reasoning