Generative Complexity in Algebra

Generative Complexity in Algebra

by Joel Berman, Pawel M. Idziak
     
 

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

See more details below

Overview

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.

Read More

Product Details

ISBN-13:
9780821837078
Publisher:
American Mathematical Society
Publication date:
03/30/2005
Series:
Memoirs of the American Mathematical Society Series, #175
Pages:
159
Product dimensions:
67.50(w) x 97.50(h) x 2.50(d)

Customer Reviews

Average Review:

Write a Review

and post it to your social network

     

Most Helpful Customer Reviews

See all customer reviews >