In the set of positive rational numbers, consider partially order relation, defined as follows : $$\frac{x}{y} \prec \frac{a}{b} \iff \frac{x}{y},\frac{a}{b} \text{are non-reduced fractions}\ \land \ y=b \ \land \frac{x}{y} \le \frac{a}{b}.$$
According to me, fraction that has a numerator equal to zero is minimum element. I also think, that there are no maximum elements, because we can always find larger fraction.
UPDATE: I came to conclusion, that there are no minimum and maximum elements.