# Generative complexity in algebra

### About this Title

**Joel Berman** and **PawełM. Idziak**

Publication: Memoirs of the American Mathematical Society

Publication Year
2005: Volume 175, Number 828

ISBNs: 978-0-8218-3707-8 (print); 978-1-4704-0429-1 (online)

DOI: http://dx.doi.org/10.1090/memo/0828

MathSciNet review: 2130585

MSC: Primary 08A05; Secondary 03C13, 03C45, 05A16, 08B20

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The G-*spectrum* or *generative complexity* of a
class $\mathcal{C}$ of algebraic structures is the function
$\mathrm{G}_\mathcal{C}(k)$ that counts the number of non-isomorphic
models in $\mathcal{C}$ that are generated by at most $k$
elements. We consider the behavior of $\mathrm{G}_\mathcal{C}(k)$ when
$\mathcal{C}$ is a locally finite equational class (variety) of
algebras and $k$ is finite. We are interested in ways that algebraic
properties of $\mathcal{C}$ lead to upper or lower bounds on generative
complexity. Some of our results give sharp upper and lower bounds so as to
place a particular variety or class of varieties at a precise level in an
exponential hierarchy. We say $\mathcal{C}$ has *many models* if
there exists $c>0$ such that $\mathrm{G}_\mathcal{C}(k) \ge
2^{2^{ck}}$ for all but finitely many $k$, $\mathcal{C}$
has *few models* if there is a polynomial $p(k)$ with
$\mathrm{G}_\mathcal{C}(k) \le 2^{p(k)}$, and $\mathcal{C}$
has *very few models* if $\mathrm{G}_\mathcal{C}(k)$ is bounded
above by a polynomial in $k$. Much of our work is motivated by a
desire to know which locally finite varieties have few or very few models, and
to discover conditions that force a variety to have many models. We present
characterization theorems for a very broad class of varieties including most
known and well-studied types of algebras, such as groups, rings, modules,
lattices. Two main results of our work are: a full characterization of locally
finite varieties omitting the tame congruence theory type **1** with very
few models as the affine varieties over a ring of finite representation type,
and a full characterization of finitely generated varieties omitting type
**1** with few models. In particular, we show that a finitely generated
variety of groups has few models if and only if it is nilpotent and has very
few models if and only if it is Abelian.

### Table of Contents

**Chapters**

- 1. Introduction
- 2. Background material
- Part 1. Introducing generative complexity
- 3. Definitions and examples
- 4. Semilattices and lattices
- 5. Varieties with a large number of models
- 6. Upper bounds
- 7. Categorical invariants
- Part 2. Varieties with few models
- 8. Types 4 or 5 need not apply
- 9. Semisimple may apply
- 10. Permutable may also apply
- 11. Forcing modular behavior
- 12. Restricting solvable behavior
- 13. Varieties with very few models
- 14. Restricting nilpotent behavior
- 15. Decomposing finite algebras
- 16. Restricting affine behavior
- 17. A characterization theorem
- Part 3. Conclusions
- 18. Application to groups and rings
- 19. Open problems
- 20. Tables