彩虹构造与Monk代数解决的整齐嵌入问题
Problems on neat embeddings solved by rainbow constructions and Monk algebras
摘要：
本文综述了（塔斯基）代数逻辑的最新结果和方法。我们主要研究柱形代数。固定2<n<ω。彩虹结构被用来解决问题的类组成的代数有一个整洁的嵌入性质基本上推广了开创性的结果霍金森以及赫希和霍金森原子正规性和完整的陈述，分别。为了证明非原子正规性的无限多品种逼近各种各样的表示代数的维数n，所谓的爆破和模糊的建设使用。彩虹建设相比，建设使用僧侣般的代数和情况下，这两个建设的工作。当分割方法失败时。彩虹结构被用来证明有限维n的可表示对角自由代数的对角自由簇不允许只包含100个变量的泛公理化.可表示性的概念，如完整的，弱的和强的，从原子结构提升到原子代数，并在整洁的嵌入性质方面进行了研究。利用Blow-up和Blur构造，重新证明了Monk和Maddux关于有限维n的可表示关系类和柱形代数类的非有限公理化性的经典结果.给出了一阶逻辑的n变量片段的应用。文末以表格形式总结了本文的主要结果。
英文摘要：
This paper is a survey of recent results and methods in (Tarskian) algebraic logic. We focus on cylindric algebras. Fix 2<n<ω. Rainbow constructions are used to solve problems on classes consisting of algebras having a neat embedding property substantially generalizing seminal results of Hodkinson as well as Hirsch and Hodkinson on atom-canonicity and complete representations, respectively. For proving non-atom-canonicity of infinitely many varieties approximating the variety of representable algebras of dimension n, so-called blow up and blur constructions are used. Rainbow constructions are compared to constructions using Monk-like algebras and cases where both constructions work are given. When splitting methods fail. rainbow constructions are used to show that diagonal free varieties of representable diagonal free algebras of finite dimension n, do no admit universal axiomatizations containing only finitely many variables. Notions of representability, like complete, weak and strong are lifted from atom structures to atomic algebras and investigated in terms of neat embedding properties. The classical results of Monk and Maddux on non-finite axiomatizability of the classes of representable relation and cylindric algebras of finite dimension n are reproved using also a blow up and blur construction. Applications to n-variable fragments of first order logic are given. The main results of the paper are summarized in tabular form at the end of the paper.
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