Metrics in Cartesian Product 1

The articles [7], [4], [10], [9], [5], [2], [3], [1], [6], and [8] provide the notation and terminology for this paper. We adopt the following rules: X , Y denote non empty metric spaces, x1, y1, z1 denote elements of X , and x2, y2, z2 denote elements of Y . The scheme LambdaMCART deals with non empty sets A , B, C and a 4-ary functor F yielding an element of C , and states that: There exists a function f from [: [:A , B :], [:A , B :] :] into C such that for all elements x1, y1 of A and for all elements x2, y2 of B and for all elements x, y of [:A , B :] if x = 〈x1, x2〉 and y = 〈y1, y2〉, then f (〈x, y〉) = F (x1,y1,x2,y2) for all values of the parameters. Let us consider X , Y . The functor ρX×Y yielding a function from [: [: the carrier of X , the carrier of Y :], [: the carrier of X , the carrier of Y :] :] into R is defined by the condition (Def. 1).