Given $\mathbb Z^-=\{x\in \mathbb Z:x<0\}$ and $T = \mathbb Z^-\times \mathbb N$, let the binary relation $\odot$ be defined as follows:
$\begin{aligned} (a,b) \odot (c,d) \Longleftrightarrow a \leq c \land b \mid d \end{aligned}$
Check if the operation $\odot$ sets order, in particular total order and also find minimum, maximum, maximal and minimal elements in $(T, \odot)$ if they exists.
In order for $\odot$ to be a partial order, its reflexivity, antisymmetry and transitivity has to be proved.
Reflexivity
It can be easily shown that $\forall (a,b) \in T$
$\begin{aligned}(a,b) \odot (a,b) \Longleftrightarrow a \leq a \land b \mid b\end{aligned}$
so this relation is reflexive.
Antisymmetry
In order for this property to be true the following condition must be valid $\forall (a,b),(c,d) \in T$:
$\begin{aligned} (a,b) \odot (c,d) \land (c,d) \odot (a,b) \Rightarrow (a,b) = (c,d) \end{aligned}$
The antisymmetry doesn't apply to this relation because while the following is always true:
$\begin{aligned} a \leq c \land c \leq a \Rightarrow a = c \end{aligned}$
we cannot state the same for the following:
$\begin{aligned} b \mid d \land d \mid b \Rightarrow b = d\end{aligned}$
because as $b \neq 0 \land d = 0$ (or similarly $b = 0 \land d \neq 0)$ then
$\begin{aligned}\exists \alpha \in \mathbb Z : \alpha b = d \end{aligned}$ $\begin{aligned}\exists \beta \in \mathbb Z : \beta d = b \end{aligned}$
whereas $\alpha = 0$ but $\nexists \beta \in \mathbb Z : \beta 0 = d$. Hence this relation isn't antisymmetric.
Transitivity
We need to prove $\begin{aligned} (a,b) \odot (c,d) \land (c,d) \odot (e,f) \Rightarrow (a,b) \odot (e,f) \end{aligned}$
it's valid the following: $\begin{aligned} a \leq c \land c \leq e \Rightarrow a \leq e \end{aligned}$
the following is valid as well
$\begin{aligned} a \mid c \land c \mid f \Rightarrow a \mid f \end{aligned}$
as $\exists x \in \mathbb Z : ax = c$ and $\exists y \in \mathbb Z : yc = f$
it's safe to say $yxa=f$.
Conclusion: this relation isn't partial order, so in particular isn't a total order.
Is everything ok with my exercise or did I do anything wrong? When it comes to look for maximum, minimum and minimal or maximal elements I feel lost. So I think there's no maximum or minimum because both $\mathbb Z^-$ and $\mathbb N$ are inifinite. Apparently I don't have a clue on how to spot minimal or maximal elements for $T$. Will you please help me out with that?