Gromov Hausdorff Distance For Quantum Metric Spaces Matrix Algebras Converge To The Sphere For Quantum Gromov Hausdorff Distance

Author : Joel Berman
Publisher : American Mathematical Soc.
ISBN 13 : 0821837079
Total Pages : 159 pages
Book Rating : 4.8/5 (218 download)

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Download or read book Generative Complexity in Algebra written by Joel Berman and published by American Mathematical Soc.. This book was released on 2005 with total page 159 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.