The problem of Pythagorean triples (or integer-sided right-angled triangles, so called Pythagorean triangles) has been attracting mathematicians for centuries. Although the first general solution to the problem was known already in Ancient Greece, mathematicians continued looking for new parametrisations of the problem. In this vein, one of the authors had previously described a new natural parametrisation of Pythagorean triples, based on the elements of the corresponding right-angled triangle. In the present paper we shall describe an attempt to introduce basic arithmetic operations on the set of all Pythagorean triples or on some of its characteristic subsets.