Summary
This paper studies how much can be known about a special kind of mathematical operation called a continuous Archimedean t-norm from knowing its values just along a curve in a square. The authors show how to reconstruct the whole operation if you have some information from a curve defined by a function that goes up continuously. While the reconstruction is not always unique, they describe how to find all possible completions and under what conditions a unique solution emerges. They also combine these findings with previous results to pinpoint cases where two types of partial information together identify a t-norm exactly.
What this means in practice
- •For fuzzy logic engineers: Design and verify fuzzy reasoning systems where t-norms are partially known from system constraints along certain input patterns.
- •For mathematical software developers: Implement tools that reconstruct continuous t-norms from partial data given on curves, improving flexible modeling in uncertainty quantification.
A theory result. No direct application yet.
Abstract
The completion problem for continuous and Archimedean triangular norms (t-norms for short) consists in describing all the operators that take predefined values on a given subset of the unit square, and it reveals how much of a t-norm is determined by partial information. In this paper, we focus on the case where this subset is a parametric curve. We first formulate the problem in terms of the functional equation satisfied by an additive generator, and we then restrict our attention to curves of the form $(x,p(x))$ with $p$ continuous and increasing, which include the classical diagonal and horizontal section cases. We determine when the known values are compatible with a continuous and Archimedean t-norm, and we show that every completion is obtained by choosing the additive generator on a subinterval and reconstructing it elsewhere from the values on the curve. This description is recursive in general and becomes a closed formula under additional conditions, recovering the two classical results as particular cases. The completions are not unique, and we give examples with infinitely many t-norms sharing the same section along a curve. Finally, we combine our results with the reconstruction of a t-norm from its values above a level curve, which determines exactly the additive generator on that subinterval, so that the two pieces of information are complementary and together determine a unique completion.