Sole minimal element: Why not also the minimum?

A minimal element (any number thereof) of a partially ordered set $S$ is an element that is not greater than any other element in $S$.

The minimum (at most one) of a partially ordered set $S$ is an element that is less than or equal to any other element of $S$.

Let's consider the power set $\mathcal P (\{x,y,z\})$ together with the binary relation $\subseteq$.

The Hasse diagram shows what element(s) we're looking for:

It's easy to see that:

• $\emptyset$ is a minimal element
• $\emptyset$ is the minimum

Now if we remove $\emptyset$ and consider $\mathcal P (\{x,y,z\})\setminus \emptyset$ instead, we get the following:

• $\{x\}$, $\{y\}$ and $\{z\}$ are minimal elements
• there is no minimum

(1) We know that a minimum is unique and it is always the only minimal element.

(2) And from the example above, it seems that, if a sole minimal element exists, it is always the minimum.

But I read that (2) is false. Why?

Solution 1:

The poset suggested by the hasse diagram below has only one minimal element.

Solution 2:

A counterexample to the statement is $\Bbb Z \cup \{c\}$, where $c$ is not comparable to any integer. $c$ is then a minimal element, but there is no minimum because a minimum must be comparable to everything else.