# Ordinal Sums of Fuzzy Negations: Main Classes and Natural Negations

## TL;DR

This paper investigates how to construct new fuzzy negations through ordinal sums, establishing conditions for their classification and exploring their relationships with natural negations in various fuzzy logic functions.

## Contribution

It provides new conditions for the ordinal sum of fuzzy negations to belong to specific classes and introduces a novel ordinal sum for fuzzy implications.

## Key findings

- Conditions for ordinal sums to be strong, strict, continuous, invertible, and frontier fuzzy negations.
- Relationship between natural negations of ordinal sums and sums of natural negations.
- Introduction of a new ordinal sum method for fuzzy implications.

## Abstract

In the context of fuzzy logic, ordinal sums provide a method for constructing new functions from existing functions, which can be triangular norms, triangular conorms, fuzzy negations, copulas, overlaps, uninorms, fuzzy implications, among others. As our main contribution, we establish conditions for the ordinal sum of a family of fuzzy negations to be a fuzzy negation of a specific class, such as strong, strict, continuous, invertible and frontier. Also, we relate the natural negation of the ordinal sum on families of t-norms, t-conorms and fuzzy implications with the ordinal sum of the natural negations of the respective families of t-norms, t- conorms and fuzzy implications. This motivated us to introduces a new kind of ordinal sum for families of fuzzy implications.