Algebraic logic and logically-geometric types in varieties of algebras

TL;DR

This paper explores the deep connections between logical types, algebraic geometry, and model theory, revealing how types naturally emerge as logical kernels within algebraic logic and universal algebraic geometry.

Contribution

It establishes a novel link between types in logic and geometric properties of algebras using the framework of algebraic logic and Galois correspondence.

Findings

Types correspond to logical kernels in algebraic geometry.

A Galois correspondence connects filters in Halmos algebra with elementary sets.

The work bridges universal algebraic geometry and model theory.

Abstract

The main objective of this paper is to show that the notion of type which was developed within the frames of logic and model theory has deep ties with geometric properties of algebras. These ties go back and forth from universal algebraic geometry to the model theory through the machinery of algebraic logic. We show that types appear naturally as logical kernels in the Galois correspondence between filters in the Halmos algebra of first order formulas with equalities and elementary sets in the corresponding affine space.