Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 ?

share|cite|improve this question
If they are incomparable, in particular neither is smaller than the other. – Daniel Fischer Jul 19 '14 at 13:26
The definition is technically right, but I agree it is hard to read. I would rather say nobody in $S$ is above it. – André Nicolas Jul 19 '14 at 13:27

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.