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