I'll formulate a topic restricted here to the positive rational
numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (**Q2**) plus some related, to which I don't know the answers nor reference. The idea is to understand how general is the notion of the set of all prime numbers:
do we or don't we have other similar objects which I call primary structures.

STANDARD TERMINOLOGY

$\mathbb N := \{1\ 2\ \ldots\}\ $ -- the set of natural numbers;

$\mathbb Z_+:=\mathbb N\cup\{0\}\ $ -- the set on nonnegative integers;

$\mathbb Z\ $ -- the ring of (rational) integers;

$\mathbb Q\ $ -- the field of rational numbers;

$\mathbb Q_{_{>0}} := \{x\in \mathbb Q : x > 0\}\ $ -- the set of positive rational numbers.

$\mathbb P:= (2\ 3\ 5\ 7\ 11\ \ldots)\ $ -- the **sequence** of all primes (i.e. $P_1:=2,\ $ and $\ P_5:= 11,\ $ etc.).

SPECIAL TERMINOLOGY

$\Omega := \{f\in \mathbb Z^\mathbb N:\sum|f|<\infty\}\ $ -- integer sequences with finitely many non-zero values;
$\Lambda := \{f\in \mathbb Z_+^\mathbb N:\sum f<\infty\} = \mathbb Z_+^\mathbb N\cap \Omega;$

$x^f := \prod_{n\in\mathbb N} x_n^{f(n)}\ $ for every sequence $x:=(x_1\ x_2\ \ldots)\ $ of positive rationals, and for $\ f\in \Omega$;

$x^* := \{x^f:f\in\Lambda\}\ $ -- the multiplicative monoid
generated by terms of sequence $x$.

$\mathbb Q(x) := \{x^f:f\in\Omega\}\ $

DEFINITIONS

Let $\ S:=(S_1\ S_2\ \ldots)\in \mathbb Q_{_{>0}}\!^\mathbb N$.

**D1.** $\ $ Sequence $S\ $ has the *unique decomposition property* (*u.d.p.* for short)
$\Leftarrow:\Rightarrow$

$$ \forall_{f\ g\in\Omega}\ \left(S^f = S^g\,\ \Rightarrow\,\ f=g\right) $$

**Note 1.** $\ $ Replacing $\ \Omega\ $ by $\ \Lambda\ $ would not affect the above definition.

**Note 2.** $\ \forall_{x\ y\in S^*}\ \left(x\cdot y=1\,\ \Rightarrow\,\ x=y=1\right)$

There are plenty of sequences with the u.d.p. However one extra condition will narrow the choice drastically:

**D2.** $\ $ A u.d.p. sequence $\ S\ $ is called a *primary structure* $\ \Leftarrow:\Rightarrow\ S^*\ $ is an additive semigroup in $\ Q_{_{>0}}$.

QUESTIONS

Let $\ S\ $ be an arbitrary primary structure. Is it true that:

**Q1:** $\ \forall_{n\in\mathbb N}\ S_n > 1\ ?$

**Q2:** Is $\ S\ $ a permutation of $\ \mathbb P$?

If *NO* to Q2 (just in case :-), we may still wonder about:

**Q3:** If every prime appears in $\ S\ $ is it true that only
primes appear in $\ S\ $ (i.e. that $\ S\ $ is a permutation
of $\ \mathbb P$)?

**Q4.** $\ \mathbb Q(S) = \mathbb Q\ $?

semigroup-- not an additive monoid. Sorry for the typos. $\endgroup$1more comment