7.1 Ideals
Definition 7.1.1 (Ideal).label Let $X$ be a set and $\sigma \subset 2^{X}$, then $\sigma$ is an ideal over $X$ if:
- (I1)
For any $E \in \sigma$ and $F \subset E$, $F \in \sigma$.
- (I2)
For any $E, F \in \sigma$, $E \cup F \in \sigma$.
Definition 7.1.2 (Covering).label Let $X$ be a set and $\sigma \subset 2^{X}$, then $\sigma$ covers $X$/is covering if $\bigcup_{E \in \sigma}E = X$.
Definition 7.1.3 (Fundamental).label Let $X$ be a set, $\sigma \subset 2^{X}$ be an ideal, and $\tau \subset \sigma$, then $\tau$ is fundamental with respect to $\sigma$ if for any $E \in \sigma$, there exists $F \in \tau$ such that $E \subset F$.
Proposition 7.1.4.label Let $X$ be a set and $\tau \subset 2^{X}$, then the following are equivalent:
- (1)
For any $E, F \in \tau$, there exists $G \in \tau$ such that $E \cup F \subset G$.
- (2)
There exists an ideal $\sigma \subset 2^{X}$ such that $\tau$ is fundamental with respect to $\sigma$.
If the above holds, then the ideal $\sigma$ in (2) is the ideal generated by $\tau$.
Proof. (1) $\Rightarrow$ (2): Let
then $\sigma$ satisfies (I1) by definition. For any $E, F \in \sigma$, there exists $E_{0}, F_{0} \in \tau$ such that $E \subset E_{0}$ and $F \subset F_{0}$. By assumption, there exists $G \in \tau$ such that
so $\sigma$ satisfies (I2), and is an ideal.
(2) $\Rightarrow$ (1): Let $E, F \in \tau$, then $E \cup F \in \sigma$. Since $\tau$ is fundamental, there exists $G \in \tau$ such that $E \cup F \subset G$.$\square$
Definition 7.1.5 (Product Ideal).label Let $X, Y$ be sets, $\sigma \subset 2^{X}$ and $\tau \subset 2^{Y}$ be ideals, and
then there exists a unique ideal $\sigma \times \tau$ such that $\beta$ is fundamental with respect to $\sigma$. The ideal $\sigma \otimes \tau$ is the product of $\sigma$ and $\tau$.
Proof. For each $A_{1}, A_{2} \in \sigma$ and $B_{1}, B_{2} \in \tau$,
By Proposition 7.1.4, there exists an ideal $\sigma \otimes \tau$ such that $\beta$ is fundamental with respect to $\sigma \otimes \tau$.$\square$
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