# Extending almost disjoint family preserving the positivity of a member

An almost disjoint family is an infinite collection $\mathcal A$ of infinite subsets of $\omega$ such that for all $A, B \in \mathcal A$, the intersection $A \cap B$ is finite. A mad family is a maximal almost disjoint family.

The free ideal generated by $\mathcal A$ is the collection $\mathcal I(\mathcal A)=\{X \subset \omega: \exists B_1,\dots, B_n \in \mathcal A, m \in \omega(X \subset B_1\cup \dots \cup B_n\cup m)\}$. It’s easy to see that $\mathcal I(\mathcal A)$ is a proper ideal on $\omega$, that is: $\emptyset \in \mathcal I(\mathcal A)$, $\omega \notin \mathcal I(\mathcal A)$ and $\mathcal I(\mathcal A)$ is closed under finite uniions and is closed downwards. This ideal is also free, that is, all finite subsets of $\omega$ belongs to it.

The positive subsets of $\mathcal A$ are defined as the elements of $\mathcal I^*(\mathcal A)=P(\omega)\setminus \mathcal I(\mathcal A)$. Question is: Suppose $X \in \mathcal I^*(\mathcal A)$. Is there a mad family $\mathcal M\supset \mathcal A$ such that $X \in \mathcal I^*(\mathcal M)$?

I tried to apply Zorn’s lemma to define such an $\mathcal M$ but I think it didn’t work.