publication . Preprint . 2016

Affine planes, ternary rings, and examples of non-Desarguesian planes

Ivanov, Nikolai V.;
Open Access English
  • Published: 17 Apr 2016
The paper is devoted to a detailed self-contained exposition of a part of the theory of affine planes leading to a construction of affine (or, equivalently, projective) planes not satisfying the Desarques axiom. It is intended to complement the introductory expositions of the theory of affine and projective planes. A novelty of our exposition is a new notation for the ternary operation in a ternary ring, much more suggestive than the standard one.
free text keywords: Mathematics - Combinatorics, 51E15, 51A35 (Primary), 12K99 (Secondary)
Download from

A. A. Albert, Non-associative algebras: I. Fundamental concepts and isotopy, Annals of Math., Vol. 43, No. 4. (1942), 685-707.

E. Artin, Geometric algebra, Wiley, Hoboken, NJ, 1988; Reprint of the 1st edition, Interscience Publishers, New York, 1957.

M. Hall, Jr., Theory of groups, AMS Chelsea Publishing, 1999; 1st edition: Macmillam, New York, 1959.

M. Hall, Jr., Combinatorial theory, Wiley, Hoboken, NJ, 1986.

R. Harstshorne, Foundations of projective geometry, W.A. Benjamin, 1967.

D. R. Hughes, F. C. Piper, Projective planes, Graduate Texts in Mathematics, V. 6, Springer-Verlag, 1973.

D. Knuth, Finite semifields and projective planes, J. Algebra, V. 2, No. 2 (1965), 182-217.

[VW] O. Veblen and J. Wedderburn, Non-Desarguesian and non-Pascalian geometries, Trans. AMS, V. 8 (1907), 379-388. [OpenAIRE]

Ch. Weibel, Survey of non-Desarguesian planes, Notices of the AMS, V. 54. No 10 (2007), 1294-1303.

Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue