Why we use ANY in the definition of a maximal element?

Question

I am confused about the following definition: "a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S." I do not understand when it says "maximal is an element of S that is not smaller than ANY other element." We have some elements that are not possible to compare, so why does it use the word ANY if A and B are two different incomparable elements. In this case, how we know if A is not smaller than B ?

Answer 1

There are some partial orderings that have more than one maximal element. As an example, consider the set $\{\emptyset, \{ab\}, \{bc\}, \{cd\}, \{abc\}, \{bcd\} \}$, and we'll say that $A < B$ if $A \subset B$. It can be verified that $\{abc\}$ is greater than every other element except $\{bcd\}$, but that $\{abc\}$ is not smaller than $\{bcd\}$.

So here, $\{abc\}$ and $\{bcd\}$ can't be compared with our partial ordering, but $\{abc\}$ still satisfies the definition of maximal. So does $\{bcd\}$, for that matter.

If we modify our example slightly to be on $\{\emptyset, \{ab\}, \{bc\}, \{cd\}, \{abc\}, \{bcd\}, \{abcd\} \}$, then $\{abcd\}$ is the only maximal set by containment.