---
tag: H4XGGXDTKN
kind: definition
name: Clone
msc: [08A40]
defines: PTKR703G6A
uses: [EJ8YJVM9BW, 769B35T6WG, 0THBK9V825, N9M2N27END]
sources:
  - url: https://en.wikipedia.org/wiki/Clone_(algebra)
---

## Definition

Let $A$ be a nonempty set.
For $n \ge 1$, let $O_A^{(n)}$ be the set of all $n$-ary [[operations|EJ8YJVM9BW]] $A^n \to A$, and let $O_A = \bigcup_{n \ge 1} O_A^{(n)}$.
A *clone* on $A$ is a subset $C \subseteq O_A$ such that:

1. $C$ contains every [[projection|769B35T6WG]] $\pi^n_k$ with $1 \le k \le n$.
2. $C$ is closed under [[composition|0THBK9V825]]: if $f \in C$ is $m$-ary and $g_1, \dots, g_m \in C$ are $n$-ary, then the $n$-ary operation
$$h(a_1, \dots, a_n) = f\bigl(g_1(a_1, \dots, a_n), \dots, g_m(a_1, \dots, a_n)\bigr)$$
is in $C$.

## Remarks

Example: the set of all operations $f(x_1, \dots, x_n) = c_1 x_1 + \cdots + c_n x_n$ on $\Z$ with integer coefficients $c_i$ is a clone.
Check: $\pi^n_k$ has $c_k = 1$ and the other $c_i = 0$, and substituting linear forms into a linear form gives a linear form.
Non-example: the set of all operations $f(x_1, \dots, x_n) = x_1 + \cdots + x_n$ on $\Z$, one for each $n \ge 1$, together with all projections, is not a clone.
It contains $f(x_1, x_2) = x_1 + x_2$, so a composition $f(f(\pi^2_1, \pi^2_2), \pi^2_1)$ gives $2x_1 + x_2$, which is not in the set.
The set of all [[term operations|N9M2N27END]] of an algebra $(A, F)$ is the smallest clone containing $F$, the clone generated by $F$.
Some authors also allow nullary operations in clones.