---
tag: CX07CWPCDX
kind: concept
name: Partial linear space
msc: ["51A45", "51E14", "05B25"]
specializes: [FGKNAFNHA3]
sources:
  - url: https://en.wikipedia.org/wiki/Partial_linear_space
same_as:
  wikidata: Q7140375
  wikipedia: Partial linear space
---

## Summary

A partial linear space is an [[incidence structure|FGKNAFNHA3]] of points and lines with two rules.
Two distinct points lie on at most one line, and every line has at least two points.
It is the common ground of most point-line geometries: projective and affine planes, [[generalized polygons|PCT2HY9AAZ]] and [[polar spaces|Z7T0V2VAWJ]] are all partial linear spaces.
When any two points lie on exactly one line, it is called a linear space.

## Explanations

### L1 human

In ordinary geometry, two points determine a line.
A partial linear space keeps only half of this rule: two points determine at most one line, and some pairs of points may lie on no common line.
For example, take the nine points of a $3 \times 3$ grid and use only the three rows and three columns as lines.
Two points in the same row or column share exactly one line, and two points in different rows and columns share none.