Cartesian Products of Graphs and Metric Spaces

We prove uniqueness of decomposition of a finite metric space into a product of metric spaces for a wide class of product operations. In particular, this gives the positive answer to the long-standing question of S. Ulam: ‘IfU × U ≃ V × V with U,V compact metric spaces, will then U and V be isometric?’ in the case of finite metric spaces. In the proof we use uniqueness of cartesian decomposition of connected graphs; a known fact to which we give a new proof which is shorter and more transparent than existing ones.