Existence, Uniqueness and Counterexamples · Hard problem

For every $x$ there is a $y$ with $y^2 = x$.

There is a $y$ such that for every $x$, $x \le y$.

For all $x$ and $y$ there is a $z$ with $z^2 = xy$.

For every $x$ there is a $y$ with $y^3 = x$.