IBN-varieties of algebras

TL;DR

This paper extends the concept of IBN (invariant basic number) to arbitrary varieties of universal algebras, providing key theorems and applications, including for many-sorted algebras, to advance universal algebraic geometry.

Contribution

It introduces and proves fundamental theorems on IBN properties for various algebraic varieties, generalizing to many-sorted algebras and enhancing understanding in algebraic geometry.

Findings

Proved Theorems 2.1 and 3.2 on IBN properties

Extended IBN concepts to many-sorted algebras

Applied theorems to study algebraic geometry relations

Abstract

The concept of variety with IBN (invariant basic number) propriety first appeared in ring theory. But we can define this concept for arbitrary variety of universal algebras with arbitrary signature; see Definition 1.4. The proving of the IBN propriety of some variety is very important in universal algebraic geometry. This is a milestone in the study of the relation between geometric and automorphic equivalences of algebras of this variety. In this paper we prove very simple but very useful for studying of IBN proprieties of different varieties Theorems 2.1 and 3.2. We will consider applications of this theorem. We will consider many-sorted universal algebras as well as one-sorted. So all concepts and all results will by generalized for the many-sorted case.