---
tag: EX9NCDFYC7
kind: concept
name: Lattice of clones
msc: [08A40, 06B23]
sources:
  - url: https://en.wikipedia.org/wiki/Clone_(algebra)
---

## Summary

The lattice of clones on a set $A$ is the set of all [[clones|PTKR703G6A]] on $A$, ordered by inclusion.
It is a complete lattice: the meet of a family of clones is its intersection, and the join is the clone generated by the union.
Its least element is the clone of [[projections|769B35T6WG]] and its greatest element is the clone of all finitary operations.
On a two-element set it is countable, and [[Post's lattice|MA20X7JCET]] describes it completely.
On a finite set with at least three elements it has the cardinality of the continuum.

## Explanations

### L1 human

Each clone is a toolbox of operations, and a bigger clone can build more.
Ordering the toolboxes by inclusion gives the lattice of clones.
At the bottom sits the clone that can only copy inputs; at the top sits the clone that can build every operation.
On $\set{0, 1}$ there are countably many clones, while on $\set{0, 1, 2}$ there are already uncountably many.