Abstract projective plane

An abstract projective plane is an incidence geometry satisfying the following axioms¹ #m/def/geo

For any two distinct points there exists precisely one line incident with both of them.
For any two distinct lines there exists precisely one point incident with both of them.
Each line of is incident with at least three distinct points of .
Each point of is incident with at least three distinct lines of .

Since ^P1 and ^P2, as well as ^P3 and ^P4, are duals of each other, the dual of any theorem following from these axioms holds. This is known as the principle of duality. The following axioms can replace both ^P3 and ^P4.

(3a.) There are four points of in general position, that is four points no three of which are colinear.
(3b.) At least two lines exist², but no two lines of cover the points of the plane, i.e. for any two lines there is a point of incident with neither line.

Proof of equivalence

Assume satisfies ^P1, ^P2, ^P3, and ^P4. Let . By ^P4 there exist three distinct lines , and by ^P2 and ^P3 there exist distinct points , , and . each distinct from . At least one of , so without loss of generality assume . Then are in general configuration, so ^P3a holds.

Now assume satisfies ^P1, ^P2, and ^P3a, but not ^P3b, so there exist two lines covering the points of the plane. By ^P3a there exist distinct and all different from (otherwise three points would be colinear) but since cannot be on or , the assumption of not ^P3b was invalid. Hence ^P3b holds.

Finally assume satisfies ^P1, ^P2, and ^P3b. ^P4 follows immediately, since for any point there exist lines and and at least one , so . If is any line, there exists , and by ^P4 there are three distinct lines through each of which meet at a different point, hence ^P3 holds.

See Finite projective plane, and the generalizing Abstract projective space.

#state/tidy | #lang/en | #SemBr

2020, Finite geometries, p. 1 ↩
Kiss and Szőnyi leave this out, but I believe without this stipulation it is possible to produce a geometry of one line and one point. ↩

Abstract projective plane

Footnotes