---
tag: 1V9GKBY29Y
kind: concept
name: Minimal clone
msc: [08A40]
specializes: [PTKR703G6A]
sources:
  - url: https://en.wikipedia.org/wiki/Clone_(algebra)
---

## Summary

A minimal clone on a set $A$ is a clone that lies just above the clone of [[projections|769B35T6WG]] in the [[lattice of clones|EX9NCDFYC7]].
It is generated by any one of its non-projection operations.
On a finite set, every clone other than the projection clone contains a minimal clone, and there are only finitely many minimal clones.
Minimal clones are the atoms of the clone lattice, dual to the maximal clones.

## Explanations

### L1 human

The smallest clone can only copy inputs.
A minimal clone adds the least possible amount of power to it.
For example, on $\set{0, 1}$ the clone generated by negation $\neg$ is minimal.
Every operation in it is a projection or a negated projection, and each negated projection generates $\neg$ back.