Spaces with congruence (Q2467999)

The authors consider a space with congruence \((P,{\mathcal L},\equiv)\) not assuming any other geometrical properties. They only assume that the space satisfies the exchange property i.e., the linear space \((P,{\mathcal L})\) satisfies the following exchange condition. Let \(S\subset P\) and let \(x,y\in P\) with \(x\in\overline{S\cup\{y\}}\setminus\overline S\). Then \(y\in\overline{S\cup\{x\}}\), where \(\overline{\phantom{S}}\) is a closure operation for subset \(X\subset P\). They show that for any line \(G\) of a plane \(E\) and any point \(x\in G\) there is a unique perpendicular line through \(x\) in \(E\) and that any line reflection is a motion if the dimension of the space is greater than 2. It follows that for any two points \(b\), \(z\) there exists a unique point \(b'\in\overline{z,b}\setminus\{b\}\) with \((z, b)\equiv(z, b')\) and also point reflections are motions.