Equivalence between two definitions of a compact element

Question

According to Wikipedia, the two following definitions of a compact in a partially ordered set $(P,\le)$ are equivalent:

1. An element $c$ is called compact if for every directed subset $D$ of $P$, if $D$ has a supremum $sup \; D$ and $c \le sup \; D$ then $c \le d$ for some element $d \in D$.

2. An element $c$ is called compact if for every ideal $I$ of $P$, if $I$ has a supremum $sup \; I$ and $c \le sup \; I$ then $c$ is an element of $I$.

Question: When $c$ satisfies the second definition, why does it also satisfy the first one?

My thoughts: In the other direction it is obvious by using the fact that an ideal is directed by definition. But given a directed set $S$, I cannot see how to derive an ideal allowing to prove the statement.

Answer 1

It's just a matter of using the concept of ideal generated by a subset, in this case, a directed subset.

Suppose $D$ is a directed subset with $d = \sup D$.
Let $I$ be the ideal generated by $D$, that is, $$I = \{ p \in P : \exists x \in D\; (p \leq x) \}.$$ Then $d = \sup I$.
Suppose $c \leq d$. By the second condition, $c \in I$, that is, there exists $x \in D$ such that $c \leq x$.