Journal ArticleDOI
On generalized affine planes
TLDR
In this article, the problem of associating an algebraic structure with affine planes has been studied, and it has been shown that one can distinguish the affine plane from the generalized plane by considering these associated near-rings.Abstract:
affine plane. That is, Artin starts with the affine plane axioms, given in terms of points and lines, and proceeds to construct a field associated with a given affine plane. In this paper we initiate the study of generalized affine planes ana consider the problem of associating an algebraic structure with these geometries. (Roughly speaking, a generalized affine plane is a geometry in which it is possible for two points to be incident with more than one line.) In Section 1 we introduce generalized affine planes and investigate some of their basic properties. In Section 2, we adjoin an additional axiom (uniform axiom) and investigate some properties of our new geometry. We now (Theorem 2.2) associate a near-ring to each generalized affine plane. The algebraic structure of this near-ring is considered in Section 3. As a result we find that one can distinguish the affine planes from the generalized affine planes by considering these associated near-rings.read more
Citations
More filters
Posted Content
On the history of ring geometry (with a thematical overview of literature)
TL;DR: In this article, the authors give an historical and at the same time thematical overview of the development of ring geometry from its origin to the current state of the art, and a comprehensive up-to-date list of literature with articles that treat ring geometry within the scope of incidence geometry.
Journal ArticleDOI
Near-rings associated with generalized translation structures
TL;DR: In this paper, the authors studied generalized translation spaces with operators, denoted by GTSO, and showed that every near-ring with identity arises as the kernel of an appropriate GTsO.
Journal ArticleDOI
The structure of dilation groups of generalized affine planes
TL;DR: The concept of a generalized affine plane was introduced in this article, where a group of (bijective) dilations and a subgroup of translations give a nearring of trace preserving quasi-endomorphisms and there is a sub group fo the translations, called the semi-identities, that give an ideal in this near-ring.
References
More filters