# Products of Lattices

Lattice elements can be combined into finite or infinite-dimensional vectors, and the result is again a lattice.

Lemma: Let be partial orders. Define by

For define

Then is a partial order. We denote by

Moreover, if for each , is a lattice, then is also a lattice.

Note: for a function with , is isomorphic to an ordered pair . We denote the product by .

Example: Let denote set of values. Let . Let

and let

Then we can define the product . In this product, iff: and . The original partial orders were lattices, so the product is also a lattice. For example, we have

Collecting Semantics as example of products, fixpoints.