What lattices are isomorphic to $\mathbb{R}^{N}$ for some $N\in \mathbb{N}$, equipped with the canonical order?

*Remark:*

When I say $\mathbb{R}^N$, I don’t mean it to be a vector space. Instead, I refer to the Cartesian product of $N$ totally ordered sets, ${(A_i, \geq_i)}_{i=1}^{N}$ each of which is isomorphic to $\mathbb{R}$ equipped with its canonical order. Therefore, the canonical order on the Cartesian product of $A_1 \times A_2 \times...A_N$ operates as the following:

For $x=(x_1,x_2,...,x_N)$ and $ y=(y_1,...,y_N)$ both in $A_1 \times A_2 \times...A_N$:

$x \succeq y$ if for all $i$, $x_i \geq_i y_i$.

Also, $x\succ y$ if $x\succeq y$ and there is at least one $i$ such that $x_i >_i y_i$.

For example, we know that the lattice must be distributive, since $(A_1 \times A_2\times ... A_N, \succeq)$ is a distributive lattice. Also, we know that the partial order must be dense. Moreover, the lattice must be unbounded, Dedekind complete, and separable in it's order topology. But I'm looking for the simplest necessary and sufficient conditions. Thanks!

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