---
tag: H7RX5729ZB
kind: concept
name: Partial algebra
msc: [08A55]
specializes: [QJGPT2W0FT]
sources:
  - url: https://en.wikipedia.org/wiki/Partial_algebra
same_as:
  wikidata: Q18386809
  wikipedia: Partial algebra
---

## Summary

A partial algebra is a set with operations that need not be defined for every choice of arguments.
It generalises the [[algebraic structure|QJGPT2W0FT]] of universal algebra, where every operation is total.
Partial algebras model structures such as fields, where the inverse of $0$ is not defined, and categories, where two arrows compose only when they match.

## Explanations

### L1 human

In an ordinary algebra you can combine any elements and always get an answer.
In a partial algebra some combinations are simply not allowed.
A field is the classic example: you can divide by any element except $0$.
Another example is the set of natural numbers with subtraction, where $3 - 5$ has no value in $\N$.
The theory of partial algebras asks which tools of universal algebra still work when operations have gaps.