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

• Definitions and examples
• Semilattices and lattices
• Varieties with a large number of models
• Upper bounds
• Categorical invariants

• 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

• Application to groups and rings
• Open problems
• Tables
• Bibliography