Weightable quasi-metric semigroups and semilattices

In [Sch00] a bijection has been established, for the case of semilattices, between invariant partial metrics and semi valuations. Semivaluations are a natural generalization of valuations on lattices to the context of semilattices and arise in many different contexts in Quantitative Domain Theory ([Sch00]). Examples of well known spaces which are semivaluation spaces are the Baire quasi-metric spaces of [Mat95], the complexity spaces of [Sch95] and the interval domain ([EEP97]). In [Sch00a], we have shown that the totally bounded Scott domains of [Smy91] can also be represented as semivaluation spaces. In this extended abstract we explore the notion of a semivaluation space in the context of semigroups. This extension is a natural one, since for each of the above results, an invariant partial metric is involved. The notion of invariance has been well studied for semigroups as well (e.g. [Ko82]). As a further motivation, we discuss three Computer Science examples of semigroups, given by the domain of words ([Smy91]), the complexity spaces ([Sch95],[RS99]) and the interval domain ([EEP97]). An extension of the correspondence theorem of [Sch00] to the context of semigroups is obtained.