What is the difference between maximal element and least upper bound?

Question

Maximal element is given as:

Let $(P,\leq)$ be a partially ordered set and $S\subset P$. Then $m\in S$ is a maximal element of $S$ if for all $s\in S$, $m \leq s$ implies $m = s$.

Least upper bound is given as:

If there exists $p', p' \in A, p' \geq x,\forall x\in A$ satisfies $p' \leq p, \forall p$, p is set of all upper bounds of $A$ then p' is the least upper bound or supremum of $A$

Is there any reconciliation between these two concepts?

Answer 1

Least upper bound and supremum are synonyms which mean the smallest number that is $\geq$ any number in your set; this is well defined for any subset of $\mathbb{R}$. The maximal element (or maximum) is the supremum (or least upper bound) when your set contains it (not every set has a maximum).

sup((0,1))=sup([0,1])=1=max([0,1])

max((0,1)) does not exist, as 1 is the supremum, but $1 \notin (0,1)$.

Answer 2

The $\sup$ of a set is the maximum of a set, if it is in that set. For example $\sup ( (0,1) )=1$ but it is not a maximum. In fact there is no maximum. But $\sup ([0,1])=\max([0,1])=1$.