Constructing left-continuous triangular norms on complete lattices

TL;DR

The work addresses extending left-continuous $t$-norms from $[0,1]$ to general complete lattices by introducing $\mathfrak{f}$-mappings and weak $\mathfrak{f}$-mappings. It develops two construction pathways: weak $\mathfrak{f}$-mappings generate left-continuous $t$-subnorms via $T(x,y)=f(x)\wedge f(y)$, and $\mathfrak{f}$-mappings yield left-continuous $t$-norms when the top element $1$ is completely join-irreducible, with $\mathrm{Im}(f)$ constituting the fixed points. The paper then provides precise necessary and sufficient conditions for building left-continuous $t$-norms on complete lattices that decompose as semi-linear sums of subintervals, using ordinal sums of annihilating binary operators, and highlights structural requirements on the interval components. Overall, the results generalize constructive techniques for $t$-norms to complete lattices, clarifying when such norms exist and how to assemble them from intervalwise operators.

Abstract

This article focuses on the construction of left-continuous t-norms on complete lattices. The concepts of $\mathfrak{f}$-mappings and weak $\mathfrak{f}$-mappings on complete lattices are first introduced, respectively. They are then applied to establish the following key results: weak $\mathfrak{f}$-mappings are used to induce left-continuous t-subnorms; $\mathfrak{f}$-mappings are used to generate left-continuous t-norms whenever the top element $1$ of the complete lattice is a completely join-irreducible element. Finally, some necessary and sufficient conditions are provided for an operator constructed by the ordinal sum of a series of annihilating binary operators being a left-continuous t-norm on a complete lattice.