Automorphisms of spine spaces

Prażmowski, Krzysztof; Żynel, Mariusz

doi:10.1007/bf02941665

Cited by 18 publications

(34 citation statements)

References 7 publications

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“…• Using 4.4 we can now strengthen the results of [4,Th. 3.24] in that the extension of an automorphism of a spine space 21 to an automorphism of the underlying space of pencils ^ is unique, provided that the spine space 21 is non-trivial and is not a space of pencils itself.…”

Section: Dilatationssupporting

confidence: 59%

“…Then dimV = 2k, dimW = k, and / is given by a sesqui-linear form on V. By 4.3 we have dim V = 2 which leads to contradiction with the general assumption that 21 is non-trivial. Thus [4,Th. 3.24] gives our claim.…”

Section: Dilatationsmentioning

confidence: 91%

“…The paper is a continuation of the theory of spine spaces originated in [2] and developed in [4] and [3]. It seems that there are two approaches to the geometry of spine spaces, the projective one, with no parallelity relation involved, and the affine, where the parallelity is defined.…”

Section: Introductionmentioning

confidence: 99%

“…It seems that there are two approaches to the geometry of spine spaces, the projective one, with no parallelity relation involved, and the affine, where the parallelity is defined. While [4] deals with projective aspects of spine spaces, and [3] deals with affine aspects of a narrow class of spine spaces of linear complements only, this paper gives an account for general properties of spine spaces common to affine geometry.…”

Section: Introductionmentioning

confidence: 99%

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Affine Geometry of Spine Spaces

Prażmowski¹,

Żynel²

2003

Demonstratio Mathematica

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Abstract. The parallelity relation and the group of dilatations in the geometry of spine spaces are investigated. Fundamental theorems of affine geometry are proved and the analytical representation of dilatations is given. IntroductionThe paper is a continuation of the theory of spine spaces originated in [2] and developed in [4] and [3]. It seems that there are two approaches to the geometry of spine spaces, the projective one, with no parallelity relation involved, and the affine, where the parallelity is defined. While [4] deals with projective aspects of spine spaces, and [3] deals with affine aspects of a narrow class of spine spaces of linear complements only, this paper gives an account for general properties of spine spaces common to affine geometry.Most of results and constructions provided for spine spaces do not make use of the natural parallelity of spine spaces. The geometry of spine spaces with parallelity defined, however, resembles the affine geometry in many aspects. In order to utilize the parallelity it is necessary to make distinction between affine and projective lines. The parallelity is an equivalence relation in the set of affine lines, and two parallel lines which intersect each other coincide. It is only partial, not Euclidean, i.e. directions do not cover the point-set, but affine variants of the Veblen condition hold (3.2), as well as the stronger parallel triangle completion condition (3.3). It also turns out that the geometry of a spine space equipped with the natural parallelity satisfies fundamental theorems of affine geometry, that is Desargues theorem (3.4) holds true, and Pappus axiom holds iff the ground field is commutative (3.5).

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Section: Dilatationssupporting

confidence: 59%

Section: Dilatationsmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Affine Geometry of Spine Spaces

Prażmowski¹,

Żynel²

2003

Demonstratio Mathematica

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show abstract

“…Following [14], for points U 1 , U 2 , we write U 1 α U 2 if U 1 = U 2 , or they can be connected with an α-path, i.e. a polygonal path of α-lines or affine lines.…”

Section: Basicsmentioning

confidence: 99%