函数和关系可定义的语义条件
Semantical conditions for the definability of functions and relations
摘要：
设$\mathcal{L}\subseteq \mathcal{L}^{\prime }$为一阶语言，$R\in \mathcal{L}^{\prime }-\mathcal{L}$为关系符号，$% \mathcal{K}$为一类$\mathcal{L}^{\prime }$ -结构.在本文中，我们提出了语义条件等价于存在一个$\mathcal{L}$ -公式$φ\left( \vec{x}\right) $，使$\mathcal{K}\vDash φ(\vec{x})\leftrightarrow R(\vec{x})$，和$φ$有一个特定的语法形式（例如，量词自由、肯定和量词自由、存在角等）。对于这些关系的可定义性结果，我们也提出了一个类似的版本的功能的可定义性。在泛代数中自然可定义性问题的几个应用已经包括在内;最显著的是主同余的可定义性。本文的结论是看术语插值类的结构与相同的技术用于可定义性。在这里，我们得到了两个经典的项插值结果的推广：Pixley定理的拟原始代数，和Baker-Pixley定理的有限代数与多数项。
英文摘要：
Let $\mathcal{L}\subseteq \mathcal{L}^{\prime }$ be first order languages, let $R\in \mathcal{L}^{\prime }-\mathcal{L}$ be a relation symbol, and let $% \mathcal{K}$ be a class of $\mathcal{L}^{\prime }$-structures. In this paper we present semantical conditions equivalent to the existence of an $\mathcal{L}$-formula $φ\left( \vec{x}\right) $ such that $\mathcal{K}\vDash φ(\vec{x})\leftrightarrow R(\vec{x})$, and $φ$ has a specific syntactical form (e.g., quantifier free, positive and quantifier free, existential horn, etc.). For each of these definability results for relations we also present an analogous version for the definability of functions. Several applications to natural definability questions in universal algebra have been included; most notably definability of principal congruences. The paper concludes with a look at term-interpolation in classes of structures with the same techniques used for definability. Here we obtain generalizations of two classical term-interpolation results: Pixley's theorem for quasiprimal algebras, and the Baker-Pixley Theorem for finite algebras with a majority term.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
