Axiom:Ideal Axioms (Order Theory)

Definition

Let $\struct {S, \preceq}$ be an ordered set.

Let $I \subseteq S$ be a non-empty subset of $S$.

$I$ is an ideal of $S$ if and only if $I$ satisifes the axioms:

$(1)$	$:$		$I$ is a lower section of $S$:	$\ds \forall x \in I: \forall y \in S:$	$\ds y \preceq x \implies y \in I $
$(2)$	$:$		$I$ is a directed subset of $S$:	$\ds \forall x, y \in I: \exists z \in I:$	$\ds x \preceq z \text{ and } y \preceq z $

These criteria are called the ideal axioms.

\((1)\)	$:$		$I$ is a lower section of $S$:	\(\ds \forall x \in I: \forall y \in S:\)	\(\ds y \preceq x \implies y \in I \)
\((2)\)	$:$		$I$ is a directed subset of $S$:	\(\ds \forall x, y \in I: \exists z \in I:\)	\(\ds x \preceq z \text{ and } y \preceq z \)