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

Let $\mathbb N^2 =\mathbb N \times \mathbb N$ be the set of protointegers. Let $n = (n_0, n_1) \in \mathbb N^2$. Let $m = (m_0, m_1) \in \mathbb N^2$. Let $l = (l_0, l_1) \in \mathbb N^2$. Let $\vee: \mathbb N^2 \times \mathbb N^2 \to \mathbb N^2$ such that $n \vee m = (min(n_0,m_0), min(n_1,m_1))$. Then $l \vee (n \vee m) = l \vee (min(n_0,m_0),min(n_1,m_1))) =(min(l_0,min(n_0,m_0)),min(l_1,min(n_1,m_1))) = (min(min(l_0,n_0),m_0),min(min(l_1,n_1),m_1)))$
 * $= (min(l_0,n_0), min(l_1,n_1)) \vee m = (l \vee n) \vee m$.

Therefore, $(\mathbb N \times \mathbb N, \vee)$ is a semigroup.