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

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 $\wedge: \mathbb N^2 \times \mathbb N^2 \to \mathbb N^2$ such that $n \wedge m = (max(n_0,m_0), max(n_1,m_1))$. Then $l \wedge (n \wedge m) = l \wedge (max(n_0,m_0),max(n_1,m_1))) =(max(l_0,max(n_0,m_0)),max(l_1,max(n_1,m_1))) = (max(max(l_0,n_0),m_0),max(max(l_1,n_1),m_1)))$
 * $= (max(l_0,n_0), max(l_1,n_1)) \wedge m = (l \wedge n) \wedge m$.

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