Definition:Join Semilattice Ideal

This page is about Ideal in the context of Join Semilattice. For other uses, see Ideal.

Definition

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 $S$:	\(\ds \forall x, y \in I:\)	\(\ds x \vee y \in I \)

Also see

• Definition:Ideal (Order Theory)
• Join Semilattice Ideal iff Ordered Set Ideal
• Definition:Lattice Ideal
• Equivalence of Definitions of Lattice Ideal
• Definition:Meet Semilattice Filter

• Results about join semilattice ideals can be found here.

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {I}$: Preliminaries, Definition $2.1$