# Definition:Ideal (Order Theory)

## 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 $S$: $\ds \forall x, y \in I: \exists z \in I:$ $\ds x \preceq z \text{ and } y \preceq z$

### Proper Ideal

Let $\II$ be an ideal on $\struct {S, \preccurlyeq}$.

Then:

$\II$ is a proper ideal on $S$
$\II \ne S$

That is, if and only if $\II$ is a proper subset of $S$.

### Join Semilattice

Let $\struct {S, \vee, \preceq}$ be a join semilattice.

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

Then $I$ is a join semilattice ideal of $S$ if and only if $I$ satisifies the join semilattice ideal axioms:

 $(\text {JSI 1})$ $:$ $I$ is a lower section of $S$: $\ds \forall x \in I: \forall y \in S:$ $\ds y \preceq x \implies y \in I$ $(\text {JSI 2})$ $:$ $I$ is a subsemilattice of $S$: $\ds \forall x, y \in I:$ $\ds x \vee y \in I$

### Lattice

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

 $(\text {LI 1})$ $:$ $I$ is a sublattice of $S$: $\ds \forall x, y \in I:$ $\ds x \wedge y, x \vee y \in I$ $(\text {LI 2})$ $:$ $\ds \forall x \in I: \forall a \in S:$ $\ds x \wedge a \in I$