Algebra and logic. Some problems

TL;DR

This paper discusses foundational concepts in algebra and logic, focusing on universal algebraic geometry, logical geometry, and types, influenced by model theory and classical algebraic ideas.

Contribution

It provides an overview of key notions, logical geometry, and types within the framework of universal algebraic geometry, integrating model theory perspectives.

Findings

Introduction of main notions in algebra and logic

Development of logical geometry within UAG

Analysis of types and isotypeness in algebraic structures

Abstract

The paper has a form of a talk on the given topic. It consists of three parts. The first part of the paper contains main notions, the second one is devoted to logical geometry, the third part describes types and isotypeness. The problems are distributed in the corresponding parts. The whole material oriented towards universal algebraic geometry (UAG), i.e., geometry in an arbitrary variety of algebras $\Theta$. We consider logical geometry (LG) as a part of UAG. This theory is strongly influenced by model theory and ideas of A.Tarski and A.I.Malcev.