Abstract projective plane

An abstract projective plane $Π$ is an incidence geometry $(P, E, I)$ satisfying the following axioms¹ #m/def/geo

For any two distinct points $𝐴, 𝐵 \in P$ there exists precisely one line $𝐴 𝐵 \in E$ incident with both of them.
For any two distinct lines $𝑒, 𝑓 \in E$ there exists precisely one point $𝑒 \cap 𝑓 \in P$ incident with both of them.
Each line of $E$ is incident with at least three distinct points of $P$ .
Each point of $P$ is incident with at least three distinct lines of $E$ .

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

Proof of equivalence

Assume $Π$ satisfies ^P1, ^P2, ^P3, and ^P4. Let $𝐴 \in P$ . By ^P4 there exist three distinct lines $𝑒, 𝑓, 𝑔 I 𝐴$ , and by ^P2 and ^P3 there exist distinct points $𝐸 1, 𝐸 2 I 𝑒$ , $𝐹 I 𝑓$ , and $𝐺 I 𝑔$ . each distinct from $𝐴$ . At least one of $𝐸 1, 𝐸 2 ⧸ I 𝐹 𝐺$ , so without loss of generality assume $𝐸 1 ⧸ I 𝐹 𝐺$ . Then $𝐴, 𝐹, 𝐺, 𝐸 1$ are in general configuration, so ^P3a holds.

Now assume $Π$ satisfies ^P1, ^P2, and ^P3a, but not ^P3b, so there exist two lines $𝑒, 𝑓 \in E$ covering the points of the plane. By ^P3a there exist distinct $𝐸 1, 𝐸 2 I 𝑒$ and $𝐹 1, 𝐹 2 I 𝑓$ all different from $𝑒 \cap 𝑓$ (otherwise three points would be colinear) but since $𝐸 1 𝐹 1 \cap 𝐸 2 𝐹 2$ cannot be on $𝑒$ or $𝑓$ , the assumption of not ^P3b was invalid. Hence ^P3b holds.

Finally assume $Π$ satisfies ^P1, ^P2, and ^P3b. ^P4 follows immediately, since for any point $𝐴 \in P$ there exist lines $𝑒, 𝑓 I 𝐴$ and and at least one $𝐵 ⧸ I 𝑒, 𝑓$ , so $𝐴 𝐵, 𝑒, 𝑓 I 𝐴$ . If $𝑒$ is any line, there exists $𝐴 ⧸ I 𝑒$ , and by ^P4 there are three distinct lines through $𝐴$ each of which meet $𝑒$ 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