Another example: there are 3x3 matrices which, when applied to a vector representing a pythagorean triple, produce other pythagorean triples. I think it is even a way to
produce all primitive pythagorean triples from (3,4,5).
In general, you are looking for a finite number of operations which produce through
composition a number of objects, and it seems that you want no overlap, i.e. if
h and k are two composition series for which h(obj)=k(obj) for some initial object
obj, then h=k as composition series, or in other words they are built up the same way.
Universal algebra has some means for the study of the generation of objects through
operation composition. In particular, the operations form a clone (semigroup if you look at just the operations which take one argument) which, given the uniqueness requirement
above, is relatively free on the set of generators, as there will be no nontrivial
equations satisfied by the compostions.
There are other areas and examples as well, but until the question becomes more specific,
I will stop here.
Gerhard "Ask Me About System Design" Paseman, 2010.06.08
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