# Finding the order dimension of a set of lattices

Let $L_n$ denote the set of all lattices with $n$ points in SAGE. I want to find for given $n$ (lets say $n$ is smaller than 10 or so) all lattices with order dimension at least 3. For the definition of order dimension see https://www.encyclopediaofmath.org/in.... I can do it for a concrete example, but I am not sure how to obtain a (graphical if possible) output of the result for a given $n$ (that also possibly includes the value of the order dimension in case it is at least 3). Here the concrete example for the boolean lattice with 3 elements:

P=posets.BooleanLattice(3) dimension(P)

I also have two small mathematical questions:

- I hope I see it correctly that the order dimension of a lattice is 1 if and only if this lattice is a chain? In case this is true, this motivates to only look at order dimension at least 3 since then it is clear which lattices have order dimension equal to two (knowing that only chains have order dimension equal to one).
- Are there tables on the internet or the literature on known results of what the order dimension is for lattices coming from combinatorics such as the divisor lattice. Do I see it correcty that the order dimension of the divisor lattice of a natural number n is equal to the number of prime divisors of n?