Considere the partial order relation defined in $\mathbb R^2$ as follow: $$ (x,y) \mathscr R (x',y') \quad \text{iff} \quad x \leq x' \, \,\, \text{and} \, \, y \leq y'. $$
- How can I determine the minorant, the majorant, the infimum, the supremum?
- Let $A=\{(1,2),(3,1)\}$. Does $A$ possess a largest and smallest?
I have doubts when I say: For majorant and minorant for example $(1,1)$ and $(3,2)$. Infimum and supremum does not exist. The same about smallest and largest; because the order is partial.
Thank in advance for your help.
You're correct that $A$ does not have a largest and smallest, but it's not just because the order is partial - the important part is that the elements of $A$ are incomparable. For example, the set $B = \{(1, 2), (1, 3)\}$ does have a largest and smallest, even though the order itself is still partial.
But you're not right about the infimum and supremum. The infimum, for example, is the greatest lower bound. Suppose $(x,y)$ is some lower bound on $A$. By definition, it must be that $(x,y)\mathscr{R}(1,2)$ and $(x,y)\mathscr{R}(3,1)$. By the definition of $\mathscr{R}$, $x \leq 1$, $y \leq 2$, $x \leq 3$, and $y \leq 1$. In particular, it must be that $x \leq 1$ and $y \leq 1$, so $(x,y)\mathscr{R}(1,1)$. Since $(1,1)$ is already a lower bound, it is therefore the greatest lower bound, and therefore the infimum.