maximal element may not be an upper bound

Question

I got confused by the statement on the topic of Zorn's lemma. maximal element may not be an upper bound kindly help thnx and regards

Answer 1

A maximal element doesn't have to be comparable to all other elements in the set (although the elements it is comparable to must be smaller) ("non-dominated" might be a less confusing term for this, but we are stuck with "maximal", for better or worse). This is in contrast to a greatest element, which needs to be comparable to all elements of the set.

An upper bound also needs to be comparable to all elements of the set, but doesn't itself have to be part of the set. Maximal and greatest elements must both be elements of the set in question.

As an example, take $P(\Bbb N)$, the set of subsets of the natural numbers, ordered by inclusion. Now look at $S\subseteq P(\Bbb N)$ defined the following way: $A\in P(\Bbb N)$ is an element of $S$ iff there is a natural number $k$ such that all elements of $A$ are powers of $k$.

For instance, $\{2, 4, 8, 16\}\in S$ (only powers of $2$), but $\{5, 25, 100\}\notin S$. In that case, $\{1, 2, 4, 8, 16, 32,\ldots\}\in S$ is a maximal element of $S$. However, $S$ doesn't have any greatest element, but $\Bbb N$ itself is an upper bound for $S$ as a subset of $P(\Bbb N)$ (and is, in fact, the least upper bound).