Semigroups of valuations on local rings

TL;DR

This paper investigates which semigroups can be realized as valuation value semigroups on Noetherian local rings, providing examples that suggest minimal constraints beyond classical conditions.

Contribution

It demonstrates that many semigroups are realizable as valuation semigroups on local rings, challenging previous assumptions about necessary constraints.

Findings

Provided examples of semigroups with minimal constraints

Showed no additional restrictions beyond classical ones

Indicated broad realizability of semigroups in valuation theory

Abstract

In this paper the question of which semigroups are realizable as the semigroup of values attained on a Noetherian local ring which is dominated by a valuation is considered. We give some striking examples, indicating that there may be no constraints on the semigroup beyond those known classically.