---
tag: R76EC56V9R
kind: concept
name: Initial algebra
msc: ["18C15", "08C05", "68Q65"]
specializes: [4RB8834SEJ]
sources:
  - url: https://en.wikipedia.org/wiki/Initial_algebra
same_as:
  wikidata: Q6034125
  wikipedia: Initial algebra
---

## Summary

An initial algebra of an endofunctor $F$ is an [[F-algebra|4RB8834SEJ]] from which there is exactly one homomorphism to every other $F$-algebra.
It is unique up to unique isomorphism, and by [[Lambek's lemma|S9FVESJ0CS]] its structure map is an isomorphism, so it is a fixed point of $F$.
Initial algebras give a uniform account of inductive definitions and of definitions by recursion.
The unique homomorphism out of the initial algebra is the function defined by structural recursion.
For a signature, the initial algebra is the term algebra, which is the basis of [[initial algebra semantics|HVF403PQ4H]] for abstract data types.

## Explanations

### L1 human

The natural numbers are built from $0$ by repeatedly adding $1$, and nothing else.
That is why a function on them can be defined by giving its value at $0$ and a rule for passing from $n$ to $n + 1$.
Initiality is the abstract form of this fact.
Lists, finite trees and terms are initial algebras of other functors, and the same recursion principle holds for them.