Definition:Join Semilattice Ideal

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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

  • Results about join semilattice ideals can be found here.


Sources