Ideals of $P_3$

Question

This question originates from Pinter's Abstract Algebra Chapter 18.B7.

If $D$ is a set, then the power set of $D$ is the set $P_D$ of all the subsets of $D$. Addition and multiplication are defined as follows: If $A$ and $B$ are elements of $P_D$ (that is, subsets of $D$), then

\begin{align*} A+B &= (A-B) \cup (B-A) \text{ and } AB = A\cap B \end{align*}

Let $P_3$ be $P_D$ where $D=\{a,b,c\}$.

List all the ideals of $P_3$.

($B$ is an ideal of $A$ if and only if $B$ is closed with respect to subtraction and $B$ absorbs products in $A$.)

My attempt:

Based on the multiplication table of $P_3$, there are exactly $8$ ideals of $P_3$. Specifically,

1. $\{\emptyset\}$
2. $\{\emptyset, \{a\}\}$
3. $\{\emptyset, \{b\}\}$
4. $\{\emptyset, \{c\}\}$
5. $\{\emptyset, \{a\}, \{b\}, \{a,b\}\}$
6. $\{\emptyset, \{a\}, \{c\}, \{a,c\}\}$
7. $\{\emptyset, \{b\}, \{c\}, \{b,c\}\}$
8. $P_3$

Is this correct?

Answer 1

Yes. Note that an ideal $I \subseteq P_D$ is closed under unions, since $A+B + AB = A \cup B$. For $A \in I$, every subset $E \subseteq A$ is in $I$, since $E= E\cap A = EA \in I$. Thus we can define ideals just by how many singletons they have, and any ideal with $3$ singletons is $P_3$. Thus the ideals you found are the only ones.