Is $\mathfrak b_a$ a new cardinal characteristic of the continuum?

Question

By a partial function from $\omega$ to $\omega$ we understand a function $f:dom(f)\to\omega$ defined on an infinite subset of $\omega$.

A family $\mathfrak F$ of partial functions from $\omega$ to $\omega$ is called domain almost disjoint (briefly, DAD) if the family $(dom(f))_{f\in\mathcal F}$ is almost disjoint in the sense that for any distinct partial functions $f,g\in\mathcal F$ the intersection $dom(f)\cap dom(g)$ is finite.

Let $\mathfrak b_a$ be the smallest cardinality of a DAD family $\mathfrak F$ of partial functions from $\omega$ to $\omega$ such that for every function $g:\omega\to\omega$ there exists a partial function $f\in\mathcal F$ such that $f(n)\not\le g(n)$ for infinitely many numbers $n\in dom(f)$.

It is easy to see that $\mathfrak b\le\mathfrak b_a\le \mathfrak d$.

Question. Is the strict inequality $\mathfrak b<\mathfrak b_a$ consistent?

Remark. The affirmative answer to this question implies a positive answer to this problem. Namely, the strict inequality $\mathfrak b<\mathfrak b_a$ implies the exitence of a monotone cofinal map $\omega^\omega\to\mathfrak b^{\mathfrak b}$.

Answer 1

After thinking some time I realized that $\mathfrak b_a=\mathfrak b$, so this question has answer "No" and this "No" does not help to solve the original question.

To show that $\mathfrak b_a=\mathfrak b$, consider the binary tree $2^{<\omega}$ and take a family $\mathcal B$ of functions from $2^{<\omega}$ to $\omega$ of cardinality $|\mathcal F|=\mathfrak b$, which is not upper bounded in the set $\omega^{2^{<\omega}}$ endowed with the partial order $\le^*$ of almost dominance. Replacing each function $f\in\mathcal B$ by a larger function, we can assume that $f$ is constant on each level $2^n\subset 2^{<\omega}$, $n\in\omega$, of the tree $2^{<\omega}$. Let $\mathcal A$ be the (almost disjoint) set of branches of the binary tree $2^{<\omega}$. Choose any injective map $dom:\mathcal B\to\mathcal A$ and consider the DAD family $\mathcal F=\{f|dom(f)\}_{f\in\mathcal B}$ of partial functions from $2^{<\omega}$ to $\omega$.

We claim that the family $\mathcal F$ witnesses that $\mathfrak b_a=\mathfrak b$. Indeed, take any function $g:2^{<\omega}\to \omega$ and find a function $\hat g:2^{<\omega}\to\omega$ such that $\hat g\ge g$ and $\hat g$ is constant on each level $2^n$ of the tree $2^{<\omega}$. By the choice of the unbounded family $\mathcal B$, there exists a function $f\in\mathcal B$ such that $f\not\le^* \hat g$. Since $f$ and $\hat g$ are constant on each level and $dom(f)$ meets each level $2^n$ of the tree $2^{<\omega}$, the relation $f\not\le^* \hat g$ implies $f|dom(f)\not\le^* \hat g|dom(f)$ and hence $f|dom(f)\not\le^* g|dom(f)$.