The concept of variety with IBN (invariant basic number) propriety first appeared in ring theory. It is known that if we consider some field , the vector space over this field such that ring of all linear operators on this vector space ( ), then , i.e., the variety of all rights modules over the ring has not the IBN propriety.
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.
We will look at some examples where the IBN property of certain varieties can be proved directly. For example for every signature the variety defined by the empty set of identities has the IBN property.
We will discuss very simple but very useful
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.