## Related pagesIsomorphisms of linear spacesIsomorphisms of projective planes |

In fact, given two partial linear spaces and , the line map belonging to an isomorphism is just the induced map on sets of points.

We have the following lemma.

**Lemma.** Given a bijection
,
the following statements are equivalent.

- is an isomorphism.
- For each subset , we have if and only if .
- For each three points p,q,r, not necessarily distinct, the three points are collinear if and only if their images are collinear.

Automorphisms of a partial linear space are often called collineations, since the third part of our lemma states that a permutation on the set of points is an automorphism if and only if it respects the relation of collinearity.

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