Let , , and . Define the conditional function as

(1)

which, for convenience, we just write as

DEFINITION. A regular aggregation sequence (r.a.s.) is a sequence such that exist fixed tupples and , with , and , such that

(2)

We call the pair a realization of u, which is not unique for each sequence u. The set of all r.a.s. is denoted by .

DEFINITION. The aggregation operation , with and , is the map such that,

LEMMA. For and , the map has the following properties:

(1) It is well-defined, i.e. .

(2) We have

where , , and

is a bijection.

(3) Fix and define by

The function is an homomorphism.

(4) Let , and , then

where .

(5) We have . Moreover, if is a decomposition into prime numbers, then

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