Valuation monoids, defining systems, and approximation theorems
type: Item
instance of: scholarly article
title: Valuation monoids, defining systems, and approximation theorems
zbMATH Open document ID: 0880.20047
DOI: 10.1007/PL00005910
published in: Semigroup Forum
publication date: 1997-12-08T00:00:00Z
review text: The author shows that ``essential parts of classical valuation theory can be developed in a purely multiplicative context''. All monoids considered are commutative and cancellative. Let \(Q\) be an abelian group and \(V\subset Q\) a submonoid. \(V\) is called a valuation monoid of \(Q\) if for all \(z\in Q\) we have \(z\in V\) or \(z^{-1}\in V\). A monoid \(V\) is called a valuation monoid if it is a valuation monoid of the quotient group \(Q(V)\) of \(V\). If \(H\) is any monoid, then \(V\) is called a valuation monoid for \(H\) if \(V\) is a valuation monoid satisfying \(H\subset V\subset Q(H)\). Different characterizations of valuation monoids are given and two existence theorems are proved. Then the classical notions of well-centered and essential valuations are extended to the general context. A defining system for a monoid \(H\) is a set \(\Omega\) of monoids \(V\) satisfying \(H\subset V\subset Q(H)\) such that \(H\) is the intersection of all monoids from \(\Omega\). A monoid \(H\) is a GSD-monoid if any finite set of elements of \(H\) possesses a greatest common divisor in \(H\). Every GSD-monoid possesses a defining system of essential valuation monoids. An arithmetical interpretation of the so-called Conrad's (F)-condition is given. The last chapter is devoted to a new approach towards approximation theorems and several applications. The paper is an extended version of an expository talk given at the Conference on Semigroups in Szeged, August 1994.
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 20M25
zbMATH DE Number: 1045578
zbMATH Keywords: valuation monoids
zbMATH Keywords: valuation theory
zbMATH Keywords: multiplicative arithmetic
zbMATH Keywords: defining systems
zbMATH Keywords: approximation theorems
zbMATH Keywords: commutative cancellative monoids
zbMATH Keywords: essential valuations
author: Franz Halter-Koch
reviewed by: Grigori I. Zhitomirski
MaRDI profile type: Publication
full work available at URL: pl00005910
OpenAlex ID: W2038342466
description: scientific article; zbMATH DE number 1045578