---
tag: JFJYSSWSSR
kind: definition
name: Minimal clone
msc: [08A40]
defines: 1V9GKBY29Y
uses: [PTKR703G6A, 769B35T6WG, EX9NCDFYC7]
sources:
  - url: https://en.wikipedia.org/wiki/Clone_(algebra)
---

## Definition

Let $A$ be a set with at least two elements, and let $J_A$ be the clone of all [[projections|769B35T6WG]] on $A$.
A [[clone|PTKR703G6A]] $C$ on $A$ is *minimal* if $C \ne J_A$ and the only clones $D$ with $J_A \subseteq D \subseteq C$ are $J_A$ and $C$.
Equivalently, $C$ is an atom of the [[lattice of clones|EX9NCDFYC7]] on $A$.

## Remarks

A clone $C \ne J_A$ is minimal exactly when every non-projection operation in $C$ generates $C$.
Example: on $\set{0, 1}$, the clone generated by $\wedge$ is minimal.
Check: its $n$-ary members are the operations $x_{i_1} \wedge \cdots \wedge x_{i_r}$, and identifying variables in one with $r \ge 2$ distinct variables gives $x \wedge y$ back.
Non-example: the clone generated by $\wedge$ and $\vee$ on $\set{0, 1}$ is not minimal.
It properly contains the clone generated by $\wedge$, which is not the projection clone.
On $\set{0, 1}$ there are exactly seven minimal clones, generated by the constant $0$, the constant $1$, $\neg$, $\wedge$, $\vee$, the majority operation, and $x + y + z$ modulo $2$.