What does a lattice of the direct power of the two-element chain look like?

Question

In universal algebra, it is known that every finite Boolean lattice is isomorphic to a direct power of the two-element chain.

I am having hard time figuring out what a lattice of the direct power of the two-element chain looks like.

For example, let X={1, 2, 3}. What does a lattice of $2^X$ look like?

Thanks.

Answer 1

Part of this MSE question asks:

Is there is an easy way to visually understand the Lattice product, as defined by:

$$ L_1 \times L_2\times\ ...\times\ L_n = \{(x_1, x_2, ..., x_n)\ \vert \ x_i \in L_i\} $$

Whether this is easy depends on the lattices $L_i$ and the number $n$, but when the lattices are chains and $n\leq 3$, as in your case, the answer is yes, the product is easy to visualize.

In my answer to the question mentioned above, there is a Hasse diagram showing the $2 \times 2 \times 3$ lattice and the $2 \times 3 \times 3$ lattice. The lattice you are asking about, $2\times 2 \times 2$, is a sublattice of each of those diagrams -- it consists of just one cube.