Skewaffine spaces in the language of distance spaces (Q1394820)

Let \(X\neq\emptyset\) and \(D\) be sets, and \(d: X\times X\to D\) be a surjective mapping. This structure, abbreviated by \((X,d)\), is called a \(T\)-distance space by the author, provided (0), (T), \((\Delta)\) hold true. (0) For all \(x,y,z\in X\), \(d(z,z)= d(x,y)\) imply \(x= y\). (T) To \(x\in X\) and \(r\in D\) there exists \(z\in X\) with \(r= d(x,z)\). \((\Delta)\) To \(x,y,x',y',z\in X\) satisfying \(d(x,y)= d(x',y')\) there exists \(z'\in X\) with \(d(x,z)= d(x',z')\) and \(d(y,z)= d(y',z')\). The important and impressive result of the present note is that the author identifies the \(T\)-distance spaces exactly with the skewaffine spaces in the sense of \textit{J. André} [see Ann. Univ. Sarav., Ser. Math. 3, No. 1 (1990) and ibid. 4, No. 2, 93-129 (1993; Zbl 0811.51002)], moreover, \textit{J. Pfalzgraf} [J. Geom. 25, 147-163 (1985; Zbl 0581.51008)]. It is now possible to translate, and maybe even to enlarge, the theory of skewaffine spaces, by applying the more fundamental language of distance spaces [see \textit{W. Benz}, `Geometrische Transformationen', BI-Wissenschaftsverlag, Mannheim (1992; Zbl 0754.51005)]. In this direction, the author presents several very interesting facts.