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arXiv · 2412.04944

Triangular norms on finite lattices

Peng He·Xue-Ping Wang
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Abstract

This article intends to characterize triangular norms on a finite lattice. We first give a method for generating a triangular norm on an atomistic lattice by the values of atoms. Then we prove that every triangular norm on a non-Boolean atomistic lattice is not left-continuous and $T_M$ is the uniquely left-continuous triangular norm on an atomistic Boolean lattice. Furthermore, we show that each atomistic Boolean lattice can be represented by a family of triangular norms on an atomistic lattice with the same number of atoms. Finally, we construct a triangular norm on a finite lattice by restricting a triangular norm on an extended atomistic lattice of the finite lattice to the finite lattice.

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Peng He, Xue-Ping Wang. 2024-12-06. Triangular norms on finite lattices. https://doi.org/10.1016/j.fss.2025.109609

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