Skolem Arithmetic is the multiplication-flavored cousin of Presburger Arithmetic.
Presburger Arithmetic is a complete theory and listed as an example of a complete theory on the Wikipedia article. I was wondering whether Skolem Arithmetic was complete as well.
This article includes some interesting facts about Skolem Arithmetic and mentions in passing that it's a complete theory, but doesn't contain a proof that it's complete.
The article does mention that Skolem Arithmetic is the theory of the monoid $(M, \cdot)$, but since Peano Arithmetic isn't complete, I don't see a reason to believe a priori that the multiplicative fragment of PA would be either.
Is there a reference for the completeness of Skolem Arithmetic (or if the proof of completeness of SA is simple, that would also be fine)?
As is the case with Presburger arithmetic, Skolem arithmetic satisfies a weak version of quantifier elimination (see e.g. Emil Jerabek's old MO answer), and this can be used to prove completeness. However, I think it's also instructive to at least outline an approach via EF-games, if only because in this case that approach runs into a difficulty which really highlights the efficacy of syntactic (= quantifier-elimination-style) methods.
Suppose I have models $\mathcal{S}_1,\mathcal{S}_2$ of Skolem arithmetic. Let $P_1,P_2$ be their respective sets of primes. In each model $\mathcal{S}_i$, the powers of a given prime $p$ form a model of Presburger arithmetic $\mathcal{A}_p^i$, and the appropriate version of prime factorization ("two numbers divisible by the same powers-of-primes are equal") is provable in Skolem arithmetic, so we can conflate $\mathcal{S}_1$ with a particular substructure of $\prod_{p\in P_1}\mathcal{A}_p^1$ and similarly conflate $\mathcal{S}_2$ with a particular substructure of $\prod_{p\in P_2}\mathcal{A}^2_p$.
Now WLOG (Lowenheim-Skolem!) both $\mathcal{S}_1$ and $\mathcal{S}_2$ are countable, so we can assume that $P_1=\{p_i:i\in\omega\}$ and $P_2=\{q_i: i\in\omega\}$. Moreover, completeness of Presburger arithmetic tells us that for each $n,i$ $\mathsf{Duplicator}$ has a winning strategy $\Sigma_{n,i}$ in the length-$n$ EF-game $\mathsf{EF}_n(\mathcal{A}^1_{p_i}, \mathcal{A}^2_{q_i})$. Fixing $n\in\omega$ there is a natural way to try to paste the $\Sigma_{n,i}$s together to form a winning strategy $\Sigma$ for $\mathsf{Duplicator}$ in $\mathsf{EF}_n(\mathcal{S}_1,\mathcal{S}_2)$: basically, we just "play coordinatewise." E.g. if $n=1$ and player $\mathsf{Spoiler}$ opens with the element $f\in\mathcal{S}_1$, player $\mathsf{Duplicator}$ should respond with the element $g\in\mathcal{S}_2$ satisfying $$g(i)=\Sigma_{1,i}(f(i)).$$
Unfortunately, this doesn't quite work: how do we know that the $g$ this suggests is actually an element of $\mathcal{S}_2$ after all? The issue is supports: one structure may have elements divisible by (externally) infinitely many primes while the other may not.
This can be fixed by showing that each model $\mathcal{S}$ of Skolem arithmetic is elementarily equivalent to its "finite-support substructure," that is, the substructure of $\mathcal{S}$ generated by the powers-of-primes of $\mathcal{S}$ itself. But offhand I don't see a way to do this that doesn't essentially use the same ideas of the quantifier elimination argument. So, while I love EF-games and do think that the picture of decomposing a model of Skolem arithmetic into a bunch of models of Presburger arithmetic is important, in this particular case quantifier-elimination-type results are probably the way to go.