Isomorphismen von rechtseitr�umen

Abstract

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation ∥ and whose set of point pairs P2 is equipped with a congruence relation ≡, such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism ϕ of two rectangular spaces (P,G, ∥, ≡) and (P′,G′, ∥′, ≡′) we mean a bijection of the point setP onto P′ which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic ≠ 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, ∥, ≡) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P′ is in fact an isomorphism from (P,G, ∥, ≡) onto (P′,G′, ∥, ≡′) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, ∥, ≡) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, ∥, ≡). By a motion of(P. G,∥, ≡) we mean a bijection ϕ ofP which maps lines onto lines, preserves parallelism and satisfies the condition(ϕ(x), ϕ(y)) ≡ (x,y) for allx, y ∈ P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, ∥, ≡) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).