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Generative Complexity in Algebra
Joel Berman, University of Illinois, Chicago, IL, and Paweł M. Idziak, Jagiellonian University, Krakow, Poland

Memoirs of the American Mathematical Society
2005; 159 pp; softcover
Volume: 175
ISBN-10: 0-8218-3707-9
ISBN-13: 978-0-8218-3707-8
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/175/828
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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

  • Introduction
  • Background material
Part 1. Introducing Generative Complexity
  • Definitions and examples
  • Semilattices and lattices
  • Varieties with a large number of models
  • Upper bounds
  • Categorical invariants
Part 2. Varieties with Few Models
  • Types 4 or 5 need not apply
  • Semisimple may apply
  • Permutable may also apply
  • Forcing modular behavior
  • Restricting solvable behavior
  • Varieties with very few models
  • Restricting nilpotent behavior
  • Decomposing finite algebras
  • Restricting affine behavior
  • A characterization theorem
Part 3. Conclusions
  • Application to groups and rings
  • Open problems
  • Tables
  • Bibliography
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