# Pivotal decomposition schemes inducing clones of operations

## TL;DR

This paper explores pivotal decomposition schemes and identifies conditions under which classes of pivotally decomposable operations form clones, providing both sufficient and necessary criteria and analyzing their generative properties.

## Contribution

It introduces conditions for pivotal operations that ensure classes of decomposable operations form clones, advancing the theoretical understanding of operation classes.

## Key findings

- Certain pivotal operations guarantee classes are clones.
- Necessary and sufficient conditions for clone formation are established.
- Pivotal operations and constants generate the clone under specific assumptions.

## Abstract

We study pivotal decomposition schemes and investigate classes of pivotally decomposable operations. We provide sufficient conditions on pivotal operations that guarantee that the corresponding classes of pivotally decomposable operations are clones, and show that under certain assumptions these conditions are also necessary. In the latter case, the pivotal operation together with the constant operations generate the corresponding clone.