模的无穷正存在范式
Infinitary positive existential normal forms for modules
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中文总结 AI 辅助
该数学论文针对左R-模，证明了任意秩不超过α的无参数L_{∞,θ}公式等价于特定陪集的无穷布尔组合，核心是利用组合引理证明相关完全布尔代数在投影下封闭。
中文摘要 AI 辅助
设θ为正则基数，R为环，M为左R-模。我们证明，对任意序数α，存在大小至多为beth_α(|R|^{<θ})的集合I_α⊆M^{<θ}，使得每个秩不超过α的无参数L_{∞,θ}公式，在M中等价于陪集的无穷布尔组合ā+φ(M)，其中ā∈I_α，φ为秩不超过α的无穷正存在公式。证明的核心是一个组合引理：给定阿贝尔群的κ个子群，存在大小至多为2^κ的点集，可检测每个成员要么为空集要么为对应子群陪集的族是否覆盖整个群。证明通过将该引理逐纤维应用，说明正存在可定义陪集的相关完全布尔代数在投影下是封闭的。
英文摘要
Let $θ$ be a regular cardinal, $R$ a ring, and $M$ a left $R$-module. We prove that for every ordinal $α$ there is a set $I_α\subseteq M^{<θ}$ of size at most $\beth_α(|R| + θ)$ such that every parameter-free $L_{\infty,θ}$ formula of rank at most $α$ is equivalent in $M$ to an infinitary Boolean combination of cosets $\overline{a} + ϕ(M)$, where $\overline{a} \in I_α$ and $ϕ$ is an infinitary positive existential formula of rank at most $α$. The main ingredient in the proof is a combinatorial lemma which says that given $κ$ subgroups of an abelian group, there is a set of at most $2^κ$ points which tests whether any family in which each member is either empty or a coset of the corresponding subgroup covers the whole group. The proof proceeds by applying this lemma fiberwise to show that the relevant complete Boolean algebras of positive-existentially definable cosets are closed under projections.
