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Geometric Axioms for the Hall Plane
Author(s) -
Room T. G.
Publication year - 1973
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-6.2.351
Subject(s) - axiom , mathematics , plane (geometry) , quadratic equation , combinatorics , point (geometry) , translation (biology) , square (algebra) , pure mathematics , geometry , biochemistry , chemistry , messenger rna , gene
It is proved that the geometric incidence axiom H below provides necessary and sufficient conditions for a plane Π of the following specification to be a Hall plane, namely, (i) Π is a translation plane with axis l ∞ , (ii) Π has a square‐root subplane Δ meeting l ∞ , in a subline, (iii) Δ is a plane over a field (on which axiom H imposes the restriction that it shall contain an irreducible quadratic form). Axiom H. X , Y ∈ l ∞ ∩ Δ, Z ∈ Δ\ l ∞ , A ∈ Δ\{ l ∞ ∪ XZ }, C ∈ Δ\{ ZX ∪ ZY }, are arbitrarily selected points in their respective sets. { S i } = XZ \Δ, T i = AS i ∩ l ∞ , R i = CS t ∩ YZ , B t = R i T i ∩ Δ. (Axiom) (i) { B i } is a single point, B say, (ii) if C ∉ l ∞ then AX , BY , CZ are concurrent.