Universal Algebra and Model Theory

本课题研究代数方程理论中不可约模型的分布和结构。主要目标是确定理论模型的结构含义，这些模型产生于理论有有限数量的不可约模型的假设。投影结果可应用于代数等价类的可决性、有限公理化性、范畴性和等价完备性。代数结构或代数是用作计算工具的数学对象。代数的一个典型例子是我们用于计数的数字系统，尽管更奇特的代数在物理、化学、逻辑和大多数数学分支中都有应用。代数中的基本问题，如一个数是否能除另一个数的问题，是通过将问题简化为有关不可约因子代数的相关问题来解决的。这项研究将促进对不可约代数的计算方法的理解，从而促进对一般代数的理解。

This project concerns the distribution and structure of irreducible models of equational theories of algebras. The chief goal is to determine the structural implications for the models of a theory which arise from the assumption that the theory has a bounded number of irreducible models. The projected results have applications to the decidability, finite axiomatizability, categoricity, and equational completeness of equational classes of algebras. Algebraic structures, or algebras, are mathematical objects which are used as devices for calculation. A typical example of an algebra is the number system we use for counting, although more exotic algebras find application in physics, chemistry, logic, and most branches of mathematics. Basic questions in algebra, such as the question of whether one number divides another, are solved by reducing the question to a related question about irreducible factor algebras. This research will advance the understanding of methods of calculation in irreducible algebras, and consequently in general algebras.