Recent developments on absolute geometries and algebraization by K-loops

Let (P,L,α) be an ordered space. A spatial version of Pasch's assertion is proved, with that a short proof is given for the fact that (P,L) is an exchange space and the concepts h-parallel, one sided h-parallel and hyperbolic incidence structure are introduced (Section 2). An ordered space with hyperbolic incidence structure can be embedded in an ordered projective space (Pp,Lp,τ) of the same dimension such that P is projectively convex and projectively open (cf. Property 3.2). Then spaces with congruence (P,L,≡) are introduced and those are characterized in which point reflections do exist (Section 4). Incidence, congruence and order are joined together by assuming a compatibility axiom (ZK) (Section 5). If (P,L,α,≡) is an absolute space, if o∈P is fixed and if for x∈P,x′ denotes the midpoint of o and x and x the point reflection in x then the map o: P→J; x→xo≔x′ satisfies the conditions (B1) and (B2) of Section 6, and if one sets a+b≔ao∘0o(b) then (P,+) becomes a K-loop (cf. Theorem 6.1) and the J of all lines through o forms an incidence fibrtion in the sense of Zizioli consisting of commutative subgroups of (P,+) (cf. Property 7.1). Therefore K-loops can be used for an algebraization of absolute spaces; in this way Ruoff's proportionality Theorem 8.4 for hyperbolic spaces is presented.