Details:
Title  The Diamond Lemma for Power Series Algebras  Author(s)  Lars Hellström  Type  PhD Theses  Abstract  The main result in this thesis is the generalisation of Bergman's diamond lemma for ring theory to power series rings. This generalisation makes it possible to treat problems in which there arise infinite descending chains. Several results in the literature are shown to be special cases of this diamond lemma and examples are given of interesting problems which could not previously be treated. One of these examples provides a general construction of a normed skew field in which a custom commutation relation holds.
There is also a general result on the structure of totally ordered semigroups, demonstrating that all semigroups with an archimedean element has a (up to a scaling factor) unique orderpreserving homomorphism to the real numbers. This helps analyse the concept of filtered structure. It is shown that whereas filtered structures can be used to induce pretty much any zerodimensional linear topology, a realvalued norm suffices for the definition of those topologies that have a reasonable relation to the multiplication operation.
The thesis also contains elementary results on degree (as of polynomials) functions, norms on algebras (in particular ultranorms), (Birkhoff) orthogonality in modules, and construction of semigroup partial orders from ditto quasiorders.  Keywords  diamond lemma, power series algebra, Gröbner basis, embedding into skew fields, archimedean element in semigroup, qdeformed Heisenberg–Weyl algebra, polynomial degree, ring norm, Birkhoff orthogonality, filtered structure  Length  246  ISBN  9173053279  ISSN  11028300  Copyright  (c) 2002 Lars Hellström 
File 
 URL 
http://abel.math.umu.se/~lars/diamond/ 
Language  English  Volume  23  Publisher  Umeå University, Department of Mathematics  Address  90187 Umeå, SWEDEN  Year  2002  Note  The file attached has hyperlinks in references and citations.  Edition  0  Translation 
No  Refereed 
No  Institution 
Department of Mathematics and Mathematical Statistics, Umeå University 
