User:KBlott:Questions\Is the set of protointegers a semilattice with respect to join?

$(\mathbb N \times \mathbb N, \vee)$ is a semigroup. $(\mathbb N \times \mathbb N, \vee)$ is abelian since $min(x,y) = min(y,x)$. $(\mathbb N \times \mathbb N, \vee)$ is idempotent since $min(x,x)= x$. Therefore, $(\mathbb N \times \mathbb N, \vee)$ is a semilattice.