maximum and maximal elements

Question

Let B = {(2,4), (4,0), (4,3), (7,3)}. B has the order like the product order of $\mathbb{N} \times \mathbb{N}$ . Then, why B hasn't maximum and why the maximal elements are (2,4) and (7,3)?

Answer 1

The product order (meaning $(a,b) \leq (c,d)$ iff $a \leq c$ and $b \leq d$) gives the following cover relations for $B$:

\begin{eqnarray*} (4,0) &\leq& (4,3) \\ (4,3) &\leq& (7,3) \\ \end{eqnarray*}

So the maximal elements are $(2,4)$ and $(7,3)$. E.g. $(2,4)$ is maximal because there is no element $(a,b) \in B$ such that $(2,4) \leq (a,b)$.

There is no maximum because none of the elements is such that all other elements are $\leq$ it. E.g. $(2,4)$ is maximal - but it is not a maximum because, for example, $(7,3) \not\leq (2,4)$.