Maximal element of a poset

Question

I am currently studying Zorn's lemma and its my understanding that the definition of a maximal element is a primitive to the lemma.

From my understanding a maximal element $M$ of a partially ordered set $(S,<)$ is an element $M \in S$ where, if $M \leq x $ for some $ x \in S$, then $x=m$.

This easilty illustrated when the set is {∅, {1}, {2}, {3}, {1, 2}}. Then the maximal elements are {3} and {1,2}.

But what if the partially ordered set i am considering is $(S,R)$. So for example let S=P({a,b,c}) which is the power set of {a,b,c} and the relation $\subseteq$. Now whats is the maximal element in this set?

Thank you in advance.

Answer 1

The only maximal element in the power set $S = P(\{a,b,c\})$ is $\{a,b,c\}$. Clearly $\{a,b,c\}$ is itself maximal and any other maximal subset $M$ of $\{a,b,c\}$ satisfies $M \subseteq \{a,b,c\} = x$ and thus by the maximality has $M = x= \{a,b,c\}$.

Answer 2

In general posets a maximum element $M$ of $S$ is one such that $$\forall x \in S: x \le M$$

(all elements lie under $M$) and a maximal element $M$ of $S$ is one such that

$$\forall x \in S: (M \le x) \to (x=M)$$

(nothing is properly larger than $M$).

A maximum element is always a maximal element, and is always unique.

As you see in your fist example where $\{3\}$ and $\{1,2\}$ are both maximal, in general, we can have many non-comparable maximal elements, and this is typically the kind of poset Zorn is applied in.

In $\mathscr{P}(\{1,2,3\})$ we have a maximum element $\{1,2,3\}$. This is also the only maximal element.