Geometrical aspect of databases and knowledge bases

Abstract

This work stands at intersection of two sciences: universal algebra on the one hand and a field we call knowledge science on the other. We view the latter as a science on languages of knowledge representation. It is strongly related to universal algebra and can be considered as an area of mathematics having motivation in computer science.¶Universal algebra deals with arbitrary algebraic operations of an arbitrary arity. They can be unary, binary, ternary, and of any finite arity. Traditionally, operations of arity over three are extremely rare in algebra. There is no such tradition in computer science and knowledge science, where operations and relations are free from restrictions. This circumstance may be one of the most significant reasons to consider operations of an arbitrary arity.¶The core point of the paper is elementary knowledge (First Order Knowledge). The main goal is to construct a model to represent some non-elementary knowledge about elementary knowledge using a universal algebraic approach. For the solution of this problem we join the methods of algebraic logic and universal algebraic geometry in logic, both defined over an arbitrary variety of algebras \( \Theta \). We use also the Galois theory in the logic over \( \Theta \) which generalizes the well-known theory by M. Krasner. (See also [P11]).