---
tag: 769B35T6WG
kind: concept
name: Projection operation
msc: [08A40, 08A02]
sources:
  - url: https://en.wikipedia.org/wiki/Projection_(mathematics)
---

## Summary

A projection operation on a set $A$ is an operation $A^n \to A$ that returns one fixed argument and ignores the others.
The projections are the operations given by single variables, so they are the simplest term operations of any algebra.
Every [[clone|PTKR703G6A]] contains all projections, and the set of all projections is the smallest clone on $A$.

## Explanations

### L1 human

The projection $\pi^3_2$ takes three inputs and hands back the second one.
For example, $\pi^3_2(5, 7, 9) = 7$.
Such an operation does no real computation; it only selects a position.
Projections matter because composing operations needs a way to feed the same variable into several places, and projections do exactly this bookkeeping.