---
tag: JQ22QH86DE
kind: concept
name: Quasifield
msc: ["12K99", "51A40", "51A35"]
sources:
  - url: https://en.wikipedia.org/wiki/Quasifield
same_as:
  wikidata: Q1553741
  wikipedia: Quasifield
---

## Summary

A quasifield is a set with an addition and a multiplication that behaves like a [[division ring|0EA5BTA0CX]].
The difference is that multiplication need not be associative and distributes only on one side.
Quasifields are the coordinate systems of [[translation planes|JZ5J6REAMD]]: every quasifield gives a translation plane and every translation plane comes from a quasifield.
[[Semifields|AHH97EJ6VH]], finite [[near-fields|DSE72WQ5A2]] and division rings are special quasifields.

## Explanations

### L1 human

In a field one can add, subtract, multiply, divide, and the usual rules hold.
A quasifield keeps the addition and the possibility to divide, but allows the multiplication to break associativity and one of the two distributive laws.
Such loose arithmetic is still enough to draw a plane in which lines have equations like $y = x \cdot m + k$.
The Hall plane of order $9$ comes from a quasifield with $9$ elements that is not a field.