A cartesian product of distributive lattices is distibutive

Question

I have seen a few articles cite this result as coming from the 3rd edition of G Birkhoff's "Lattice theory" but I can't find the result in there myself.

The index does not seem to mention "product"s and the pdf copy I have is not searchable.

If anyone happens to know of a page number for this result in Birkhoff's book or another text I'd really appreciate it.

Thanks!

Answer 1

Found it on page 12:

"Any direct product of distributive lattices is distributive."

This is easy to check from the definitions of meet and join in a product: $$(x_1, x_2) \land (y_1, y_2) = (x_1 \land y_1, x_2 \land y_2)$$ and $$(x_1, x_2) \lor (y_1, y_2) = (x_1 \lor y_1, x_2 \lor y_2)$$ We have \begin{align*} & (x_1, x_2) \land ((y_1, y_2) \lor (z_1, z_2)) \\ &= (x_1,x_2) \land (y_1 \lor z_1, y_2 \lor z_2) \\ &= (x_1 \land (y_1 \lor z_1), x_2 \land (y_2 \lor z_2) ) \\ &= ((x_1 \land y_1) \lor (x_1 \land z_1)), (x_2 \land y_2) \lor (x_2 \land z_2) )\\ &= (x_1 \land y_1, x_2 \land y_2) \lor (x_1 \land z_1, x_2 \land z_2 )\\ &= ( (x_1, x_2) \land (y_1, y_2)) \lor ((x_1, x_2) \land (z_1, z_2)) \end{align*}