Valuation monoids, defining systems, and approximation theorems (Q1362885)

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.