Let $\Phi$ denote the set of all identities satisfied by $(\mathbb{N},0,1,+,\times,\mathrm{gcd},\mathrm{lcm}).$
Question. Is $\Phi$ finitely axiomatizable? If so, I'd like to see a list of identities.
Noteworthy elements of $\Phi$:
- $(\mathbb{N},0,1,+,\times)$ is a commutative semiring
- $(\mathbb{N},1,0,\mathrm{gcd},\mathrm{lcm})$ is a bounded distributive lattice (with bottom $1$ and top $0$)
- $\mathrm{gcd}(a,b+a) = \mathrm{gcd}(a,b)$
- $\gcd(a+b,\operatorname{lcm}(a,b))=\gcd(a,b)$
- $\mathrm{gcd}(a,b)\mathrm{lcm}(a,b) = ab$
The second last identity above actually follows from the preceeding one's; see Bill's answer here.