---
tag: 73DXSAH73B
kind: definition
name: Lattice of clones
msc: [08A40, 06B23]
defines: EX9NCDFYC7
uses: [PTKR703G6A, 769B35T6WG, D7A0ZGZT5S]
sources:
  - url: https://en.wikipedia.org/wiki/Clone_(algebra)
---

## Definition

Let $A$ be a nonempty set and let $\mathcal{L}_A$ be the set of all [[clones|PTKR703G6A]] on $A$.
The *lattice of clones* on $A$ is the partially ordered set $(\mathcal{L}_A, \subseteq)$.
For a family $(C_i)_{i \in I}$ of clones, its meet is $\bigcap_{i \in I} C_i$ and its join is the clone generated by $\bigcup_{i \in I} C_i$.

## Remarks

The intersection of clones is a clone, since each clause of the definition of a clone is kept under intersections, so $\mathcal{L}_A$ is a [[complete lattice|D7A0ZGZT5S]].
Example: on $\set{0, 1}$, the meet of the clones generated by $\wedge$ and by $\vee$ is the clone of [[projections|769B35T6WG]].
Their join is the clone of all monotone operations that preserve both $0$ and $1$.
Check for the meet: the members of the first clone are the meets $x_{i_1} \wedge \cdots \wedge x_{i_k}$, and those of the second are joins.
A meet of $k \ge 2$ distinct variables differs from every join on some input, so only projections are in both.
Non-example: the join is not the union in general.
On $\set{0, 1}$, the union of the clones generated by $\wedge$ and by $\vee$ is not a clone, since it misses the majority operation $(x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$, which lies in their join.
The lattice of clones on any set is an algebraic lattice whose compact elements are the finitely generated clones.