Symmetric incidence groupoids
Let \(X\) be the set of points and lines of the Euclidean (or hyperbolic or elliptic) plane. It is well known that each element \(x\) of \(X\) determines a unique reflection, say \(\tilde x\). Hence one can define a product by putting \(\tilde y\cdot \tilde x:=\tilde y\circ \tilde x\circ \tilde y\). Then \((\tilde X,\cdot)\) is an example of a reflection groupoid [cf. also \textit{H. Hotje}, \textit{M. Marchi} and \textit{S. Pianta}, Discrete Math. 129, No.1-3, 139-147 (1994; Zbl 0804.51018)]. In the present article, reflection groupoids are introduced in an axiomatic way. A reflection groupoid is said to be a symmetric incidence groupoid if its elements are at the same time the points of a linear space such that certain axioms of compatibility between the two structures are satisfied. The authors determine the connection between symmetric incidence groupoids and (commutative) kinematic spaces. Also the automorphisms of such structures are investigated.
scientific article; zbMATH DE number 1982544