eBook ISBN:  9781470404291 
Product Code:  MEMO/175/828.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 
eBook ISBN:  9781470404291 
Product Code:  MEMO/175/828.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 175; 2005; 159 ppMSC: Primary 08; Secondary 03; 05
The Gspectrum or generative complexity of a class \(\mathcal{C}\) of algebraic structures is the function \(\mathrm{G}_\mathcal{C}(k)\) that counts the number of nonisomorphic 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 wellstudied 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


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The Gspectrum or generative complexity of a class \(\mathcal{C}\) of algebraic structures is the function \(\mathrm{G}_\mathcal{C}(k)\) that counts the number of nonisomorphic 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 wellstudied 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.

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