Algebraic geometry in first-order logic

In every variety of algebras Θ, we can consider its logic and its algebraic geometry. In previous papers, geometry in equational logic, i.e., equational geometry, has been studied. Here we describe an extension of this theory to first-order logic (FOL). The algebraic sets in this geometry are determined by arbitrary sets of FOL formulas. The principal motivation of such a generalization lies in the area of applications to knowledge science. In this paper, the FOL formulas are considered in the context of algebraic logic. For this purpose, we define special Halmos categories. These categories in algebraic geometry related to FOL play the same role as the category of free algebras Θ 0 play in equational algebraic geometry. This paper consists of three parts. Section 1 is of introductory character. The first part (Secs. 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 (Secs. 5-7). In the last part (Secs. 8-9), we consider applications of the previous material to knowledge science. © 2006 Springer Science+Business Media, Inc.