Some characterization theorems for the discrete holometric space (Q1118163)

A discrete plane consists of the lattice points \(z=(q^ mx_ 0,q^ ny_ 0)\), where \(x_ 0,y_ 0,q\) are fixed real numbers such that \(x_ 0,y_ 0\geq 0\) and \(0<q<1\). The distance d between two lattice points \(z_ 1\) and \(z_ 2\) is defined as the number of lattice points of a path joining them. A discrete plane together with a distance is called a holometric space. The author gives a characterization of a domain and shows that domains are invariant under D-isometries. He also studies D- kernels.