[Paper Review] Affine planes, ternary rings, and examples of non-Desarguesian planes

Nikolai V. Ivanov|arXiv (Cornell University)|Apr 18, 2016

Mathematics and Applications4 references4 citations

TL;DR

This paper presents a self-contained exposition of affine planes and their coordinate systems using ternary rings, introducing a novel notation ⟨ax+b⟩ for the ternary operation. It constructs non-Desarguesian planes by demonstrating that left or right quasi-fields with non-associative or non-distributive multiplication—such as left André quasi-fields—yield affine planes not isomorphic to any skew-field-defined plane, thereby providing explicit examples of non-Desarguesian geometry.

ABSTRACT

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.

Motivation & Objective

• To provide a comprehensive, self-contained exposition of affine planes and their algebraic coordinatization using ternary rings.
• To bridge the gap between combinatorial geometry and abstract algebra by showing how non-Desarguesian planes arise from non-skew-field coordinate systems.
• To introduce a new, more intuitive notation ⟨ax+b⟩ for the ternary operation in ternary rings, enhancing accessibility.
• To construct explicit examples of non-Desarguesian affine planes using left and right quasi-fields with non-associative or non-distributive multiplication.
• To demonstrate that such planes cannot be coordinatized by any skew-field, thus proving their non-Desarguesian nature.

Proposed method

• Defines affine planes axiomatically and establishes that every affine plane admits a coordinate system over a ternary ring.
• Introduces a novel notation ⟨ax+b⟩ for the ternary operation, replacing the standard (a,x,b)↦x·a∘b, to improve clarity and intuition.
• Establishes isomorphism and isotopism conditions between ternary rings and their corresponding affine planes.
• Uses Galois theory to construct field extensions with finite automorphism groups, enabling the construction of non-associative and non-distributive quasi-fields.
• Applies Theorems 12, 17, and 19 to show that affine planes over left or right quasi-fields with defective algebraic properties are not isomorphic to any skew-field-defined plane.
• Constructs left André quasi-fields via automorphisms and a homomorphism φ from the multiplicative group to the automorphism group, ensuring non-associativity or failure of right distributivity.

Experimental results

Research questions

• RQ1 How can affine planes be coordinatized using ternary rings, and what algebraic properties of the ring correspond to geometric axioms?
• RQ2 What conditions on a ternary ring ensure that the corresponding affine plane satisfies the Desargues axiom?
• RQ3 Can non-Desarguesian affine planes be explicitly constructed using algebraic structures other than skew-fields?
• RQ4 What role do automorphisms and Galois-theoretic constructions play in generating non-associative or non-distributive quasi-fields?
• RQ5 Under what conditions is a plane defined over a quasi-field not isomorphic to any plane defined over a skew-field?

Key findings

• Affine planes defined over a skew-field satisfy the Desargues axiom, and conversely, a plane satisfying Desargues' axiom admits a skew-field coordinate system.
• The novel notation ⟨ax+b⟩ for the ternary operation improves readability and accessibility of the theory of ternary rings.
• Left André quasi-fields with non-associative multiplication yield affine planes that are not isomorphic to any plane over a skew-field, thus being non-Desarguesian.
• Planes over left quasi-fields where right distributivity fails are also non-Desarguesian, as they cannot be coordinatized by any skew-field.
• Right quasi-fields with failing left distributivity also produce non-Desarguesian planes, as shown via Theorem 17 and the non-isomorphism to left quasi-field planes.
• Explicit constructions of such planes are made possible through Galois-theoretic choices of automorphism groups and homomorphisms φ, enabling non-associative or non-distributive algebras.