Reconstruction of continuous Archimedean t-norms from their section along an increasing curve

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.