Definition:Ideal (Order Theory)
Jump to navigation
Jump to search
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 $S$: | \(\ds \forall x, y \in I: \exists z \in I:\) | \(\ds x \preceq z \text{ and } y \preceq z \) |
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 $S$: | \(\ds \forall x, y \in I:\) | \(\ds x \wedge y, x \vee y \in I \) | |||||
\((\text {LI 2})\) | $:$ | \(\ds \forall x \in I: \forall a \in S:\) | \(\ds x \wedge a \in I \) |
Also see
- Definition:Join Semilattice Ideal
- Join Semilattice Ideal iff Ordered Set Ideal
- Definition:Lattice Ideal
- Equivalence of Definitions of Lattice Ideal
- Definition:Filter
- Results about ideals in the context of order theory can be found here.
Sources
- 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {VII}$: Continuous Lattices, Definition $2.1$