From Appendix B.2 (relations) of Introduction to Algorithms by Cormen et al:
In a partially ordered set $A$, there may be no single "maximum" element $a$ such that $b R a$ for all $b ∈ A$. Instead, there may several maximal elements a such that for no $b ∈ A$, where $b ≠ a$, is it the case that $a R b$. For example, in a collection of different-sized boxes there may be several maximal boxes that don't fit inside any other box, yet no single "maximum" box into which any other box will fit.
I know the definition of a partial ordered set , but do not get why the author/authors say that "there may be no single "maximum" element" in a partially ordered set ?
Best Answer
Remember that we're dealing with a partial order! This is fundamentally different from a total order like $\leq$ is on numbers.
Because: For all numbers, we have either $a\leq b$ or $b\leq a$, i.e. everything is comparable.
However a partial order is partial because elements need not be comparable. Solely if elements should happen to be comparable, then the partial order imposes some laws like transitivity.
Now take as an example the proper subsets of $\{1,2,3\}$ with the partial order of $\subseteq$. E.g. $\{1\} \subseteq \{1,2\}$, but the sets $\{1,2\}$ and $\{2,3\}$ simply are not comparable because neither includes the other. However there are no "bigger" proper subsets to each of them, so both are maximal.
Now if in turn all elements were comparable, the maximum was unique (hence in total orders it is). But in partial orders, this isn't usually the case.