Partial metric monoids and semivaluation spaces

Stable partial metric spaces (or the equivalent invariant weightable quasi-metric spaces) form a fundamental concept in Quantitative Domain Theory. Indeed, all domains have been shown to be quantifiable via a stable partial metric (e.g. [22], [23], [26]). Monoid operations arise in Domain Theory in the context of power domains. These operations also arise naturally in a quantitative context and play a crucial role in several applications. This has motivated our study of partial metric monoids and quantitative monoids. The study of monoids in Domain Theory is gaining momentum. We remark that Jimmy Lawson has advocated the study of idempotent analysis (diods) in a domain theoretic context at the recent Domains Workshop in Birmingham. Here, we show that the structure of a stable partial metric monoid provides a suitable framework for a unified approach to some interesting examples of monoids, such as the interval domain, the domain of words and the dual complexity space. We also introduce the notion of a semivaluation monoid and show that there is a bijection between stable partial metric monoids and semivaluation monoids, extending the correspondence theorem of [22] to the context of monoids. Our results consist of a first exploration of the interesting topic of monoids on quantitative domains and relationships with Lawson’s program of idempotent analysis in Domain Theory will be explored in future work. AMS (2000) Subject classification: 22-04, 22A15, 22A26, 54-04, 54E35, 54H12, 68Q55. ∗The first-listed author acknowledges the support of the Spanish Ministry of Science and Technology, under grant BFM2000-1111