# Ternary and $n$-ary $f$-distributive Structures

## TL;DR

This paper introduces and classifies ternary and higher $n$-ary $f$-distributive structures, develops their extension theory and cohomology, and provides computational examples to illustrate these concepts.

## Contribution

It presents the first classification of ternary $f$-quandles, develops a cohomology theory for $n$-ary $f$-quandles, and explores their extension theory.

## Key findings

- Classification of ternary $f$-quandles in low dimensions
- Development of a cohomology theory for $n$-ary $f$-quandles
- Provision of computational examples

## Abstract

We introduce and study ternary $f$-distributive structures, Ternary $f$-quandles and more generally their higher $n$-ary analogues. A classification of ternary $f$-quandles is provided in low dimensions. Moreover, we study extension theory and introduce a cohomology theory for ternary, and more generally $n$-ary, $f$-quandles. Furthermore, we give some computational examples.