Varieties whose finitely generated members are free

TL;DR

This paper characterizes varieties of algebras where finitely generated members are free, showing they are essentially equivalent to well-known algebraic structures like sets, pointed sets, vector spaces, or affine vector spaces over division rings.

Contribution

It establishes a classification theorem for such varieties, identifying the specific algebraic structures that satisfy the property.

Findings

Finitely generated members being free implies the variety is one of the known types.

The varieties are either sets, pointed sets, vector spaces, or affine vector spaces over a division ring.

Provides a complete characterization of these algebraic varieties.

Abstract

We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine vector spaces over a division ring.