Definition:Ideal (Order Theory)

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This page is about Ideal in the context of Order Theory. For other uses, see Ideal.

Definition

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

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


Then $I$ is an ideal of $S$ if and only if $I$ satisifies 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 $S$:      \(\ds \forall x, y \in I: \exists z \in I:\) \(\ds x \preceq z \text{ and } y \preceq z \)