Publisher Summary
The focus of this chapter is on chain geometries. In a projective geometry over a ring R, it makes no sense to intrinsically consider a projective line. Nevertheless, if R is a K-algebra, then on the projective line P1(R) there is an interesting incidence structure, the blocks of which are the K-sublines embedded into P1(R) in a canonical way, and this gives the chain geometry over R. The notion of chain geometries arose from efforts to unify the treatment of such different geometries as the geometry of Mobius (lines and circles of the Euclidean plane), of Laguerre–Lie (spears and cycles), and of Minkowski (the pseudo-Euclidean plane with its hyperbolas as circles). The chain geometry Σ(K, R) is called Laguerre geometry provided the parallel relation is an equivalence relation on P (R) and every chain meets every parallel class of points. A weak chain space is an incidence structure Σ = (P, C). The blocks of Σ are called chains and two different points are called distant provided they are incident with a common chain.

Chapter 14 – Chain Geometries

THE SOLVABILITY OF MIXED PROBLEMS FOR HYPERBOLIC AND PARABOLIC EQUATIONS V A Il'in CONVEX SETS IN RIEMANNIAN SPACES OF NON-NEGATIVE CURVATURE Yu D Burago and V A Zalgaller Two decades of fuzzy topology: basic ideas, notions, and results A P Shostak Singularities of systems of rays Vladimir I Arnol'd NONCOMPACT SEMISIMPLE LIE GROUPS A I Sirota and A S Solodovnikov DISCONTINUOUS SOLUTIONS OF HYPERBOLIC SYSTEMS OF QUASILINEAR EQUATIONS B L Rozhdestvenskii Floquet theory for partial differential equations P A Kuchment Axiomatic relativity theory

https://iopscience.iop.org/article/10.1070/RM1982v037n02ABEH003942/pdf

Axiomatic relativity theory

B . A rtm a n n , G . D o rn , D . D rak e, G . T S rn er: H jelm slev 'sch e In zid en zg eo m etrie u n d v er- w an d te G e b ie te - L iteratu rv erzeich n is. J. G eo m . 7 (1 9 7 6 ), 1 7 5 -1 9 1 . D . Ju n g n ick el: H jelm slev 'sch e In zid en zg eo m etrie u n d v erw an d te G eb iete - L iteratu rv er- zeich n is II. J. G eo m . 1 6 (1 9 8 1 ), 1 3 8 -1 4 7 . A rticles o n reflectio n g eo m etry in th e sen se o f F . B a c h m a n n 's b o o k "A u fb a u d er G eo m etrie au s d em S p ieg elu n g sb eg riff", an d o n ch ain g eo m etries in th e sen se o f W . B en z' "G eo m etrie d er A lg eb ren " are n o t in clu d ed . P ap ers o n g eo m etric alg eb ra o v er rin g s (stru ctu re an d au to m o rp h ism s o f classical g ro u p s o v er rin g s) h av e also b een o m itted w ith so m e ex cep tio n s; fo r sy stem atic listin g s o f su ch p ap ers w e refer to M cD o n ald 76 an d H ah n ; O 'M e a ra 89. T h e au th o rs w ill b e g ratefu l fo r ad d itio n s, co m p letio n s an d co rrectio n s to th is list. A N D R I~ , J. : 69 l)b er H o m o m o rp h ism en p ro jek tiv er E b en en . A bh. M a th . Sere. Univ. H a m b u rg 3 4 (1969), 98-114. 72 U b er v erM lg em ein erte T ran slatio n sstru k tu ren u n d H o m o m o rp h ism en . A rch . M a th . (B a sel) 2 3 (1972), 198-205. AP~CHBOLD, J.W . : 55 P ro jectiv e g eo m etry o v er an alg eb ra. M a th em a tika 2 (1955), 105-115.

Literature on geometry over rings

Near-rings are generalized rings: commutativity of addition is not assumed, and 3=more important — only one distributive law is required. The most famous example is the collection of all maps from an additive group into itself w.r.t. function addition and composition. Compared with a standard class of rings, endomorphism rings of abelian groups, one sees that rings describe “linear” maps on groups, while near-rings handle the general nonlinear case. The theory of near-rings is now a sophisticated theory which has found numerous applications in various areas. In this article, we concentrate on a deep structure theorem for near-rings, the density theorem for primitive near-rings. This result is applied to interpolation theory, group theory and polynomials. Connections between other parts of near-rings (especially near-fields) and geometry come up at several places. Efficient block designs and codes can be constructed from finite near-rings. Finally we mention connections to ring theory, computer science, graph theory and other parts of the “outside world”. Only some easier proofs are given. For the others, references will be given; if these results are available in book form, we cite this book and not the original paper. Up to now, three books on near-rings and one on near-fields have appeared (see [1-4]in the list of references). , , o denote the sets if natural numbers, integers and reals, respectively, and Zn denotes the set of residue classes modulo n.

Near-rings and Near-fields

In this survey paper we 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. A comprehensive up-to-date list of literature is added with articles that treat ring geometry within the scope of incidence geometry.

On the history of ring geometry (with a thematical overview of literature)

Der kleine Satz von Desargues in affinen Hjelmslev-Ebenen

Translationstructures are generalized affine spaces. They can be described algebraically by partitions of groups. For desarguesian affine spaces the group is a vectorspace and the partition is the set of all onedimensional subspaces. In this case each collineation fixing 0 is a regular semilinear mapping, i.e. an automorphism of the vectorspace. In the general case it is a mapping called equivalence. Each equivalence of a partition is an automorphism iff the set of translations of the group is a normal subgroup of the collineationgroup. The translations form a normal subgroup, if the group is finite or abelian. We prove some theorems for the infinite non abelian case.

Werner Seier

Papers

Der kleine Satz von Desargues in affinen Hjelmslev-Ebenen

Kollineationen von Translationsstrukturen

Über Translationen in affinen Hjelmslev-Ebenen

Schwach affine Räume über Skalarensystemen und ihre Kollineationsgruppen

Streckungstransitive affine Hjelmslev-Ebenen