Arithmetic on infinite formal prime factorizations By the fundamental theorem of arithmetic, we can identify a natural number $x$ with the sequence $(a_n)$ of exponents in its prime factorization $x=\prod_np_n^{a_n}$, where $p_n$ is the $n$th prime. A sequence obtained in this way will of course have only finitely many nonzero elements.
However, multiplication/$\gcd$/$\operatorname{lcm}$ operations on natural numbers correspond respectively to pointwise addition/$\min$/$\max$ of these sequences, and these operations make sense on arbitrary sequences. So I wonder: How much arithmetic can we do with these sequences, thinking of them as formal infinite products of prime powers?
Specifically:

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*Is there a semiring structure (without $0$) on $R=(\Bbb Z_{\ge 0})^\Bbb N$ where $\times$ is pointwise addition and $+$ coincides with normal addition on finite natural numbers (represented as described above)?

If so, I'm also interested in whether we can get other familiar properties.

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*Does $\gcd(a,a+b)=\gcd(a,b)$ hold (where $\gcd$ = pointwise $\min$)?

*Is $<$, defined by $a<b\iff\exists c.a+c=b$, a total order?

*If we include $0$ and negatives (e.g. by considering equivalence classes of formal differences), does Bézout's lemma hold?

*Can we in fact get a (non-standard) model of arithmetic? Are these the hyperintegers or something?

Edit: Commenters have pointed out that the answer to the last bullet is no, since this structure lacks large primes and includes elements with no largest prime divisor (and both of these properties are inconsistent with the axioms of arithmetic). The latter issue prevents this structure from even being a sub-semiring of a model of arithmetic.
 A: The comments supplied some intuition about why this construction doesn't come close to modeling arithmetic or even supporting addition in a "reasonable" way.
For one thing, we always expect $x$ and $x+1$ to have no common prime divisor, but this forces $\prod p_n+1=1$. This is undesirable since $x+a$ should be infinite whenever $x$ is infinite, and $a+b=a$ should never be true if we're trying to represent positive values.
This essentially answers my question, even though I think a subtraction axiom is needed to turn the argument into an actual contradiction.
I briefly considered avoiding this problem by restricting to sequences that have infinitely many $0$s, but this doesn't work since we still get e.g. $\prod p_{2n}+\prod p_{2n-1}=1$. We could try harder by requiring the density of nonzero entries to be small, but I doubt this will work. In fact, "thinning out" the infinite numbers seems like the opposite of what we need, since the infinite products are already all "highly composite" and therefore too sparse for addition to work. As the comments suggest, if the goal is to model arithmetic, we should add nonstandard primes and consider only finite products (edit: bounded or "internal" products) of them.
