Clone

Builds on Finitary operation Projection operation Function composition Term operation

Definition

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

$C$ contains every projection $\pi^n_k$ with $1 \le k \le n$ .
$C$ is closed under composition: 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 of an algebra $(A, F)$ is the smallest clone containing $F$ , the clone generated by $F$ . Some authors also allow nullary operations in clones.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`H4XGGXDTKN`
URI	`m2://H4XGGXDTKN`
Kind	definition
Writing	W0 stub
MSC	08A40
Defines	Clone
Uses	Finitary operation, Projection operation, Function composition, Term operation
Changed	2 November 2026

Clone

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here