Memoirs of the American Mathematical Society 2005; 159 pp; softcover Volume: 175 ISBN10: 0821837079 ISBN13: 9780821837078 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/175/828
 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  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
