Partial orders can be thought of as a set $S$, and a partial relation on that set, $\le$. If a greatest lower bound and least upper bound exist for any subset of $S$, the poset defines a lattice.
The sum operator can be defined on lattices as follows:
$$ L_1 + L_2+...+ L_n = \{ (i, x_i)\ \vert \ x_i \in L_i \backslash \{\bot, \top \}\} \cup \{ \bot, \top\} $$
where $\top$ and $\bot$ are the trivial greatest and least elements, respectively, when dealing with lattices over finite sets. This can be very elegantly visualized as the lattices lined up, with both top and bottom joined at a single point, respectively.
I now have a question in two parts.
1) Is there is an easy way to visually understand the Lattice product, as defined by:
$$ L_1 \times L_2\times\ ...\times\ L_n = \{(x_1, x_2, ..., x_n)\ \vert \ x_i \in L_i\} $$
2) According to defintions, $\le$ and g.u.b and l.u.b are defined point-wise on lattice products.
Computing $\le$ pointwise makes sense to me, but pointwise bounds has my head creaking. Could someone provide some intuition behind this, or otherwise help my understanding?
