发表机构
Sichuan University; Université Claude Bernard Lyon 1; Southwest Jiaotong University（四川大学; 里昂第一大学; 西南交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究了理想化的极大最终不同族，建立了 Van Douwen 族与 MAD 族之间的联系，引入新的可乘性概念，并给出解析族的存在性与不存在性结果。
AI 中文摘要
我们通过 Van Douwen 极大性、可乘性和超滤极大性来研究理想化的极大最终不同族。我们证明了 Van Douwen $\mathcal{J}$-MED 族对应于 $(\mathcal{J}\times\emptyset)$-MAD 族，并且对于包含有限集的余解析理想，它们进一步归结为无限解析 $\mathcal{J}$-MAD 族。这为几个标准理想给出了解析 Van Douwen 族的不存在性结果，并给出了有限情形的刻画。我们回顾有限可乘性的定义，并引入有限截面可乘性和 $\omega$-中心可乘性。有限截面可乘性等价于某个超滤 $\mathcal{U}$（扩张 $\mathcal{J}^*$）的 $\mathcal{U}^*$-极大性，并且这些概念具有自然的 Stone 空间刻画。我们为一致弱 Ramsey 理想构造了 Borel 有限可乘族，为一致弱 $P^+$ 理想构造了 Borel $\omega$-中心可乘族，而不存在解析有限截面可乘的 $\mathrm{Fin}$-MED 族。最后，对于每个 Ramsey 超滤 $\mathcal{U}$ 和 $1\leq\alpha<\omega_1$，不存在解析的 $(\mathcal{U}^{\alpha})^*$-MED 族，而在 $V=L$ 下，存在一个 $\boldsymbol{\Sigma}^1_2$ Ramsey 超滤 $\mathcal{U}$，使得存在一个余解析的 $\mathcal{U}^*$-MED 族。
英文摘要
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 $(\mathcal{J}\times\emptyset)$-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 $ω$-centered productivity. Finite-section productivity is equivalent to $\mathcal{U}^*$-maximality for some ultrafilter $\mathcal{U}$ extending $\mathcal{J}^*$, and these notions admit natural Stone-space characterizations. We construct Borel finitely productive families for uniformly weakly Ramsey ideals and Borel $ω$-centered productive families for uniformly weakly $P^+$ ideals, while no analytic finite-section productive $\mathrm{Fin}$-MED family exists. Finally, for every Ramsey ultrafilter $\mathcal{U}$ and $1\leqα<ω_1$, there is no analytic $(\mathcal{U}^α)^*$-MED family, while under $V=L$, there is a $ \boldsymbolΣ^1_2 $ Ramsey ultrafilter $ \mathcal{U} $ where there is a coanalytic $ \mathcal{U}^* $-MED family.
