Van Douwen Families, Productivity and Ultrafilter Maximality

We study idealized maximal eventually different families through Van Douwen maximality, productivity, and ultrafilter maximality. We show that Van Douwen $\mathcal{J}$-MED families correspond to $(\mathcal{J}\times \mathrm{\varnothing })$-MAD families, and for coanalytic ideals containing the finite sets they are further reduced to infinite analytic $\mathcal{J}$-MAD families. This yields nonexistence results for analytic Van Douwen families for several standard ideals, together with a characterization of the finite case. We recall the definition of finite productivity and introduce finite-section and $\omega$-centered productivity. Finite-section productivity is equivalent to ${\mathcal{U}}^{\ast }$-maximality for some ultrafilter $\mathcal{U}$ extending ${\mathcal{J}}^{\ast }$, and these notions admit natural Stone-space characterizations. We construct Borel finitely productive families for uniformly weakly Ramsey ideals and Borel $\omega$-centered productive families for uniformly weakly $P^{+}$ ideals, while no analytic finite-section productive $\mathrm{F}\mathrm{i}\mathrm{n}$-MED family exists. Finally, for every Ramsey ultrafilter $\mathcal{U}$ and $1\le \alpha <{\omega }_1$, there is no analytic $({\mathcal{U}}^{\alpha }{)}^{\ast }$-MED family, while under $V=L$, there is a ${\mathbf{\Sigma }}_2^1$ Ramsey ultrafilter $\mathcal{U}$ where there is a coanalytic ${\mathcal{U}}^{\ast }$-MED family.