Let $(\mathbb N^* \times \mathbb N^*, \varphi)$ be a poset defined as follows:
$\begin{aligned} (a,b)\varphi(c,d)\Leftrightarrow ab
Check if $\varphi$ is a total order and determine maximum, minimum, maximal and minimal elements.
It's easy to prove that $\varphi$ is not total because as we consider $(a,b),(c,d) \in \mathbb N^* \times \mathbb N^* : a = d \text{ and } b = c$ then
$\begin{aligned} ab \nless cd \text{, } cd \nless ab \text{ and } (a,b) \neq (c,d)\end{aligned}$
What I am having a very hard time with is spotting maximum, minimum, maximal and minimal elements. In my opinion the poset doesn't have any maximum or maximal elements as $\mathbb N^* \times \mathbb N^*$ is infinite, but it does have a minimal element which is $(1,1)$. Given that $\nexists (\varepsilon, \varepsilon') : 1 = \varepsilon' \text{ and } 1 = \varepsilon$ and $(1,1) \neq (\varepsilon, \varepsilon')$ then $(1,1)$ is comparable to all elements in $\mathbb N^* \times \mathbb N^*$, does this make it also the minimum?