I recall learning that we can define linear systems such that any vector in the system can be represented as a weighted sum of basis vectors, as long as we have 'suitable' definitions for addition and multiplication operators (i.e. fulfilling certain properties.) This turned out to be extremely useful, as we could prove general things about linear operations over vector spaces and apply them to a surprisingly wide array of systems with linear properties. One of the interesting things about this was that you could define a series of unique real number coordinates for a given series of independent basis vectors in the system.
It struck me the other day that there is an interesting, albeit slightly different pattern in the natural numbers. Any natural number can be written as the product of natural powers of primes. In a sense, it seems like the primes form a kind of 'basis' for the natural numbers, with the series of powers being a kind of 'coordinate'.
Is there is a name for this pattern?
If so, are there are other kinds of sets that can be decomposed as products of powers in this way, with similar generic results we can deduce for how these 'products of independent factors' behave?
I apologize for the lack of clarity here, but my unfamiliarity with the terminology makes it difficult to describe.
Answer
Here's some terminology that captures the 'multiplicative basis' part of your analogy: the natural numbers are a free commutative monoid on the infinite generating set of prime numbers.
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