Geodesic
Closure planes
Journal of Algebraic Combinatorics, Tome 43 (2016) no. 3, pp. 735-749.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We introduce a simple algebraic method for constructing infinite affine (and projective) planes from an infinite set of finite planes of prime power order stemming from a "root" plane. The construction uses finite fields and infinite extensions of finite fields in a critical way. We obtain a classical-looking result which states that if the construction succeeds over the algebraic closure of a finite field, then both the infinite plane and the original root plane must be Desarguesian. The Lenz-Barlotti types for these planes are then linked to the Lenz-Barlotti type of the root plane. Examples are then given. These show that under suitable conditions, the method can yield infinitely many non-isomorphic infinite planes. These examples are of Lenz-Barlotti types II.1 and V.1.
Classification : 51D20, 51D15, 05B25, 12E05
Keywords: closures of fields, infinite affine planes, planar ternary rings, Lenz-Barlotti classification 
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Coulter, Robert S.; Matthews, Rex W. Closure planes. Journal of Algebraic Combinatorics, Tome 43 (2016) no. 3, pp. 735-749. http://geodesic.mathdoc.fr/item/JAC_2016__43_3_a0/