Definition:Meet Semilattice Filter
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Definition
Let $\struct {S, \wedge, \preccurlyeq}$ be a meet semilattice.
Let $F \subseteq S$ be a non-empty subset of $S$.
Then $F$ is a meet semilattice ideal of $S$ if and only if $F$ satisifies the meet semilattice filter axioms:
\((\text {MSF 1})\) | $:$ | $F$ is an upper section of $S$: | \(\ds \forall x \in F: \forall y \in S:\) | \(\ds x \preccurlyeq y \implies y \in F \) | |||||
\((\text {MSF 2})\) | $:$ | $F$ is a subsemilattice of $S$: | \(\ds \forall x, y \in F:\) | \(\ds x \wedge y \in F \) |
Also see
- Definition:Filter
- Meet Semilattice Filter iff Ordered Set Filter
- Definition:Lattice (Order Theory)
- Equivalence of Definitions of Lattice Filter
- Definition:Join Semilattice Ideal
- Results about meet semilattice filters can be found here.
Sources
- 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {I}$: Preliminaries, Definition $2.2$