一阶逻辑中的Algebraic geometry
Algebraic geometry in First Order Logic
摘要：
在每一种代数$Θ$中，我们都可以考虑它的逻辑和代数几何。在以前的论文几何方程逻辑，即，研究了方程几何。在这里，我们描述了这个理论的一个扩展到一阶逻辑（FOL）。这种几何中的代数集合由任意的FOL公式集合确定。这种概括的主要动机在于知识科学的应用领域。在本文中，FOL公式被认为是在代数逻辑的背景下。为此，我们定义了特殊的Halmos类别。这些范畴在与FOL相关的代数几何中所起的作用与自由代数范畴$Θ^0$在方程代数几何中所起的作用相同。本文共分三个部分。第1节是介绍性的。第一部分（第2- 4节）包含给定代数种类中的代数逻辑背景知识$Θ$。第二部分是专门的代数几何有关的FOL（第5- 7节）。在最后一部分（第8- 9节）中，我们考虑前面的材料在知识科学中的应用。
英文摘要：
In every variety of algebras $Θ$ we can consider its logic and its algebraic geometry. In the previous papers geometry in equational logic, i.e., equational geometry has been studied. Here we describe an extension of this theory towards the First Order Logic (FOL). The algebraic sets in this geometry are determined by arbitrary sets of FOL formulas. The principal motivation of such generalization lies in the area of applications to knowledge science. In this paper the FOL formulae are considered in the context of algebraic logic. With this aim we define special Halmos categories. These categories in the algebraic geometry related to FOL play the same role as the category of free algebras $Θ^0$ play in the equational algebraic geometry. The paper consists of three parts. Section 1 is of introductory character. The first part (sections 2--4) contains background on algebraic logic in the given variety of algebras $Θ$. The second part is devoted to algebraic geometry related to FOL (sections 5--7). In the last part (sections 8--9) we consider applications of the previous material to knowledge science.
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