Some logical invariants of algebras and logical relations between algebras

Abstract

Let $\Theta$ be an arbitrary variety of algebras and $H$ an algebra in $\Theta$. Along with algebraic geometry in $\Theta$ over the distinguished algebra $H$, a logical geometry in $\Theta$ over $H$ is considered. This insight leads to a system of notions and stimulates a number of new problems. Some logical invariants of algebras $H\in \Theta$ are introduced and logical relations between different $H_1$ and $H_2$ in $\Theta$ are analyzed. The paper contains a brief review of ideas of logical geometry (§1), the necessary material from algebraic logic (§2), and a deeper introduction to the subject (§3). Also, a list of problems is given.