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For example, If I have a function [itex] f(x) [/itex] given by the ordered pairs:

[itex] \{(1,6),(2,4),(3,5),(4,2),(5,3),(6,1) \} [/itex]

We could (arbitrarily) declare that integers in certain sets have certain "properties":

[itex] \{ 1,3\} [/itex] have property [itex] A [/itex]

[itex] \{5,6\} [/itex] have property [itex] B [/itex]

[itex] \{4,2\} [/itex] have property [itex] C [/itex]

[itex] \{1,5,4\} [/itex] have property [itex] X [/itex]

[itex] \{3,6,2\} [/itex] have property [itex] Y [/itex]

With that stipulation, each of the six integers can be given a unique "coordinate" representation as set of properties:

[itex] 1 = [A,X] [/itex]

[itex] 3 = [A,Y] [/itex]

[itex] 5 = [B,X] [/itex]

[itex] 6 = [B,Y] [/itex]

[itex] 4 = [C,X] [/itex]

[itex] 2 = [C,Y] [/itex]

The function [itex] f(x) [/itex] is the "direct product" of the functions given by

[itex] g(x) = \{(A,B), (B,A), (C,C) \} [/itex]

[itex] h(x) = \{(X,Y),(Y,X)\} [/itex]

in the sense that if you apply those functions to the respective "coordinates" of an integer, you determine the integer that [itex] f(x) [/itex] maps it to. For example [itex] (A,X) \rightarrow (B,Y) [/itex] implies [itex] f(x) [/itex] maps [itex] 1 \rightarrow 6 [/itex].

Perhaps this is a generalization of "separation of variables".