Order (包含関係による順序)

• Order (包含関係による順序) #Card
  • Ordered by inclusion $\subset$. The set of all filters on $X$ ordered by inclusion is a complete lattice. - Infimum (交わり): $\mathcal{F}_0 \wedge \mathcal{F}_1 = \mathcal{F}_0 \cap \mathcal{F}_1$ - Supremum (結び): $\mathcal{F}_0 \vee \mathcal{F}_1 = {F_0 \cap F_1 : F_0 \in \mathcal{F}_0, F_1 \in \mathcal{F}_1}$ (proper iff $F_0 \cap F_1 \neq \emptyset$ for all $F_0 \in \mathcal{F}_0, F_1 \in \mathcal{F}_1$)

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Ordered by inclusion $\subset$. The set of all filters on $X$ ordered by inclusion is a complete lattice.

• Infimum (交わり): $\mathcal{F}_0 \wedge \mathcal{F}_1 = \mathcal{F}_0 \cap \mathcal{F}_1$
• Supremum (結び): $\mathcal{F}_0 \vee \mathcal{F}_1 = {F_0 \cap F_1 : F_0 \in \mathcal{F}_0, F_1 \in \mathcal{F}_1}$ (proper iff $F_0 \cap F_1 \neq \emptyset$ for all $F_0 \in \mathcal{F}_0, F_1 \in \mathcal{F}_1$)

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Preorder / PartialOrder（≤）