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TitleThe Diamond Lemma for Power Series Algebras
Author(s) Lars Hellström
TypePhD Theses
AbstractThe 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 order-preserving 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 zero-dimensional linear topology, a real-valued 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.
Keywordsdiamond lemma, power series algebra, Gröbner basis, embedding into skew fields, archimedean element in semigroup, q-deformed Heisenberg–Weyl algebra, polynomial degree, ring norm, Birkhoff orthogonality, filtered structure
Copyright(c) 2002 Lars Hellström
URL http://abel.math.umu.se/~lars/diamond/
PublisherUmeå University, Department of Mathematics
Address90187 Umeå, SWEDEN
NoteThe file attached has hyperlinks in references and citations.
Translation No
Refereed No
Institution Department of Mathematics and Mathematical Statistics, Umeå University