My problem is to know if there exists a lattice isomorphism between $B = (\{A \subset \mathbb{N} : A$ is finite$\}, \cup, \cap)$ and $C = (\mathbb{N},lcm,gcd)$. I think that is posible because have the same cardinality ($\mathbb{N}$), both have a minimum ($\emptyset$ and 1 respectively) and neither have maximum.
I tried to define a function (candidate to isomorphism) from B to A starting sending $\emptyset$ to the 1 and the sets of one element to the primes "in order" ($\{1\}$ to 2, $\{2\}$ to 3, $\{3\}$ to 5, ...) and then I need to define that sends the $\{1,2\}$ to 2*3=6 and so on. But that function isn't a surjection.
I also know that is enough to find a bijective homomorphism.
Wikipedia notes the algebraic and order-theoretic definitions of lattices are equivalent, so we may as well discuss if there is an order isomorphism from $(\mathcal{P}_{\mathrm{fin}}(\mathbb{N}),\subseteq)$ and $(\mathbb{N},\mid )$.
The answer is no, for $(\mathbb{N},\mid )$ has elements that cover one atom but no other atoms (squares of primes) and $(\mathcal{P}_{\mathrm{fin}}(\mathbb{N}),\subseteq)$ doesn't (it's not possible for a set to strictly contain one singleton set but no other singleton sets).
It's kind of like comparing $2^{\mathbb{N}}$ to $\mathbb{N}^{\mathbb{N}}$ with lexicographic orderings, except your two lattices are sublattices of these comprised of functions with finite support.