# Ordinal sums of triangular norms on a bounded lattice

## TL;DR

This paper introduces a new way to construct ordinal sums of triangular norms on bounded lattices, ensuring the result is always a valid triangular norm, unlike previous naive extensions.

## Contribution

It proposes an alternative definition of ordinal sums on bounded lattices that guarantees the resulting operation is a triangular norm.

## Key findings

- The new ordinal sum construction always yields a triangular norm.
- The approach works for countably many subintervals with a chain of endpoints.
- Finiteness removes the need for lattice completeness.

## Abstract

The ordinal sum construction provides a very effective way to generate a new triangular norm on the real unit interval from existing ones. One of the most prominent theorems concerning the ordinal sum of triangular norms on the real unit interval states that a triangular norm is continuous if and only if it is uniquely representable as an ordinal sum of continuous Archimedean triangular norms. However, the ordinal sum of triangular norms on subintervals of a bounded lattice is not always a triangular norm (even if only one summand is involved), if one just extends the ordinal sum construction to a bounded lattice in a na\"{\i}ve way. In the present paper, appropriately dealing with those elements that are incomparable with the endpoints of the given subintervals, we propose an alternative definition of ordinal sum of countably many (finite or countably infinite) triangular norms on subintervals of a complete lattice, where the endpoints of the subintervals constitute a chain. The completeness requirement for the lattice is not needed when considering finitely many triangular norms. The newly proposed ordinal sum is shown to be always a triangular norm. Several illustrative examples are given.