User:KBlott/Questions/Is the set of protointegers a lattice?

On the following pages, I have used the dual of the de facto standard lattice on a totally ordered set. Although my notation is probably more intuitive, it would probably be best to defer to the majority in order to avoid unnecessary confusion.

$(\mathbb N \times \mathbb N, \wedge)$ is a semilattice. $(\mathbb N \times \mathbb N, \vee)$ is a semilattice. Does the first absorption law hold for the protointegers? Does the second absorption law hold for the protointegers?