On the Fundamental Theorem of Affine Geometry

1962 ◽  
Vol 5 (1) ◽  
pp. 67-69 ◽  
Author(s):  
P. Scherk
Keyword(s):  
Linear Dependence ◽  
Geometric Algebra ◽  
Fundamental Theorem ◽  
Linear Independence ◽  
Direct Proof ◽  
Affine Geometry ◽  
Skew Field ◽  
Straight Lines ◽  
Preserve Linear ◽  
Linear Vector Space

The fundamental theorem of affine geometry is an easy corollary of the corresponding projective theorem 2.26 in Artin's Geometric Algebra. However, a simple direct proof based on Lipman's paper [this Bulletin, 4, 265−278] and his axioms 1 and 2 may be of some interest.Lipman's [desarguian] affine geometry G determined a left linear vector space L={a, b,…} over a skew field F. We wish to construct 1−1 transformations γ of G onto itself such that γ and γ-1 map straight lines onto straight lines preserving parallelism. Designate any point 0 as the origin of G. Multiplying γ with a suitable translation, we may assume γ0=0. Thus γ will then be equivalent to a 1−1 transformation Γ of L onto itself which preserves linear dependence. Since Γ-1 will have the same properties, Γ must also preserve linear independence.

On the Fundamental Theorem of Geometric Algebra Over SF-Rings

2008 ◽  
Vol 36 (9) ◽  
pp. 3564-3573 ◽  
Author(s):  
A. Lashkhi ◽  
T. Kvirikashvili
Keyword(s):  
Geometric Algebra ◽  
Fundamental Theorem

scholarly journals Galois Theory with Infinitely Many Idempotents

1969 ◽  
Vol 35 ◽  
pp. 83-98 ◽  
Author(s):  
O.E. Villamayor ◽  
D. Zelinsky
Keyword(s):  
Finite Group ◽  
Fundamental Theorem ◽  
Galois Theory ◽  
Linear Independence ◽  
Commutative Rings ◽  
Classical Field ◽  
Fixed Field ◽  
Noncommutative Rings ◽  
Real Change ◽  
A Finite Group

In 1942 Artin proved the linear independence, over a field S, of distinct automorphism of S; in other words if G is a finite group of automorphisms of S and R is the fixed field, then Horn^S, S) is a free S-module with G as basis. Since then, this last condition (“S is G-Galois”) or its equivalents have been used as a postulate in all the Galois theories of rings that are not fields, for example by Dieudonné, Jacobson, Azumaya and Nakayama for noncommutative rings and then in [AG, Appendix] and [CUR] for commutative rings. When S has no idempotents but 0 and 1, [CHR] proves that the ordinary fundamental theorem of Galois theory holds with no real change from the classical, field case.

scholarly journals On Ordered Geometries

1963 ◽  
Vol 6 (1) ◽  
pp. 27-36 ◽  
Author(s):  
P. Scherk
Keyword(s):  
Geometric Algebra ◽  
Plane Geometry ◽  
Skew Field ◽  
Ordered Field ◽  
Weak Ordering

In Theorem 2.20 of his Geometric Algebra, Artin shows that any ordering of a plane geometry is equivalent to a weak ordering of its skew field. Referring to his Theorem 1. 16 that every weakly ordered field with more than two elements is ordered, he deduces his Theorem 2.21 that any ordering of a Desarguian plane with more than four points is (canonically) equivalent to an ordering of its field. We should like to present another proof of this theorem stimulated by Lipman's paper [this Bulletin, vol.4, 3, pp. 265-278]. Our proof seems to bypass Artin's Theorem 1. 16; cf. the postscript.

A direct proof of a fundamental theorem of renewal theory

1953 ◽  
Vol 1953 (sup1) ◽  
pp. 139-150 ◽  
Author(s):  
D. R. Cox ◽  
Walter L. Smith
Keyword(s):  
Fundamental Theorem ◽  
Direct Proof ◽  
Renewal Theory

scholarly journals Order in Affine and Projective Geometry

1963 ◽  
Vol 6 (1) ◽  
pp. 37-43 ◽  
Author(s):  
Joe Lipman
Keyword(s):  
Projective Geometry ◽  
Affine Geometry ◽  
Skew Field ◽  
Ordered Fields ◽  
Skew Fields

In this note we give a characterisation of ordered skew fields by certain properties of the negative elements.Properties of negative elements of a skew field K are then interpreted as statements about "betweenness" in any affine geometry over K, or as statements about "separation" in any projective geometry over K.In this way, our axioms for ordered fields permit of immediate translation into postulates of order for affine or projective geometry (over a field). The postulates so obtained seem simple enough to be of interest, (cf. [4], p. 22)

scholarly journals Linear independence of translations

1995 ◽  
Vol 59 (1) ◽  
pp. 131-133 ◽  
Author(s):  
Joseph Rosenblatt
Keyword(s):  
Linear Dependence ◽  
Linear Independence ◽  
Dependence Relation ◽  
Linearly Independent ◽  
Sharp Result

AbstractIt was shown by Edgar and Rosenblatt that f ∈ Lp (ℝn), 1 ≤ p < 2n/ (n-1), and f ≠0, then f has linearly independent translates. Using a result of Hömander, it is shown here that the same theorem holds if p = 2n / (n−1). This gives a sharp result because for n ≥2, there exists f ∈C0 (ℝn), f ≠0, which is simultaneously in all Lp (ℝn), p > 2n/(n−1), that has a linear dependence relation among its translates. References and some discussion are included.

scholarly journals Definition of Affine Geometry by a Group of Transformations

1961 ◽  
Vol 4 (3) ◽  
pp. 265-278 ◽  
Author(s):  
Joe Lipman
Keyword(s):  
New York ◽  
Geometric Algebra ◽  
Affine Plane ◽  
Linear Equations ◽  
Point Of View ◽  
Affine Geometry ◽  
Geometric Point ◽  
Definition Of

In his Geometric Algebra (New York, 1957) E. Artin poses the problem of co-ordinatizing an affine plane in the following terms.How little do we have to assume, from a geometric point of view, about an affine plane, in order to be able to describe its points by pairs of elements of a field, and its lines by linear equations?

Geometric Mappings on Geometric Lattices

1971 ◽  
Vol 23 (1) ◽  
pp. 22-35 ◽  
Author(s):  
David Sachs
Keyword(s):  
Vector Space ◽  
Linear Algebra ◽  
Classical Result ◽  
Linear Independence ◽  
Closure Operator ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Algebraic Methods ◽  
Linear Transformations ◽  
Geometric Lattices

It is a classical result of mathematics that there is an intimate connection between linear algebra and projective or affine geometry. Thus, many algebraic results can be given a geometric interpretation, and geometric theorems can quite often be proved more easily by algebraic methods. In this paper we apply topological ideas to geometric lattices, structures which provide the framework for the study of abstract linear independence, and obtain affine geometry from the mappings that preserve the closure operator that is associated with these lattices. These mappings are closely connected with semi-linear transformations on a vector space, and thus linear algebra and affine geometry are derived from the study of a certain closure operator and mappings which preserve it, even if the “space” is finite.

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