Overview:Generative Complexity in Algebra - Memoirs of the American Mathematical Society - 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.

This book is on page /a06b960ac9dc90216ed4e00b41bc8290/book/1603729738-9780821837078. It was written by the following authors: . Book Generative Complexity in Algebra - Memoirs of the American Mathematical Society, which can be read online, published by the company: American Mathematical Society. Other books on similar topics can be found in sections: Medicine, Science, Technology. The book was published on 2005-03-30. It has 159 pages and is published in Paperback format and weight 340 g. File for download Generative Complexity in Algebra - Memoirs of the American Mathematical Society has PDF format and is called generative-complexity-in-algebra-memoirs-of-the-american-mathematical-society.pdf. Other books you can download below. Our topcanal.eu site is not responsible for the content of PDF files.


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