Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form ℒ∞λ. We prove that many naturally defined classes are anti-elementary, including the following: the class of all lattices of finitely generated convex ℓ-subgroups of members of any class of ℓ-groups containing all Archimedean ℓ-groups; the class of all semilattices of finitely generated ℓ-ideals of members of any nontrivial quasivariety of ℓ-groups; the class of all Stone duals of spectra of MV-algebras — this yields a negative solution to the MV-spectrum Problem; the class of all semilattices of finitely generated two-sided ideals of rings; the class of all semilattices of finitely generated submodules of modules; the class of all monoids encoding the nonstable K₀-theory of von Neumann regular rings, respectively, C*-algebras of real rank zero; (assuming arbitrarily large Erdős cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large 4-frame. The main underlying principle is that under quite general conditions, for a functor Φ:𝒜→ℬ, if there exists a noncommutative diagram D→ of 𝒜, indexed by a common sort of poset called an almost join-semilattice, such that ΦD→I is a commutative diagram for every set I, ΦD→≇ΦX→ for any commutative diagram X→ in 𝒜, then the range of Φ is anti-elementary.

中文翻译：

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从非交换图到反基本类

反元素性是确保给定一阶语言中的一类结构在关于以下形式的任何无限语言的基本等价下不封闭的一种强有力的方法ℒ∞λ. 我们证明了许多自然定义的类是反基本的，包括： 有限生成凸的所有格的类ℓ- 任何类别的成员的子组ℓ- 包含所有阿基米德的组ℓ-团体； 有限生成的所有半格的类ℓ- 任何非平凡准变量的成员的理想ℓ-团体； MV 代数谱的所有斯通对偶的类——这产生了MV谱问题; 有限生成的两侧环理想的所有半格的类； 模的有限生成子模的所有半格的类； 编码不稳定的所有幺半群的类ķ₀ - 冯诺依曼正则环理论，分别为 C* - 实秩零代数； （假设任意大的 Erdős 基数）所有可协调截面补模格的类4-框架。 主要的基本原理是，在非常一般的条件下，对于函子Φ：𝒜→ℬ, 如果存在非交换图D→的𝒜, 由一种称为 an 的常见 poset 索引几乎加入半格, 这样 ΦD→一世是每个集合的交换图一世, ΦD→≇ΦX→对于任何交换图X→在𝒜, 然后的范围Φ是反基本的。