Uniformities on Free Semigroups

. It is known that the profinite completions of a free semigroup which are associated with a pseudovariety of semigroups or of ordered semigroups, can be defined by an 'ecart or a quasi-'ecart. We characterize those quasi-'ecarts and those quasi-uniformities which arise in this fashion. We also prove an Eilenberg-like one-to-one correspondence between pseudovarieties of ordered semigroups and so-called varieties of quasi-uniformities. 1 Introduction Let A + be a free semigroup. If V is a pseudovariety of finite semigroups, the profinite topology on A + associated with V (see [1, 2]) is defined by a metric d V , as soon as it is Hausdorff. Now the question arises to know which metrics are obtained in this way. We show that a metric on A + is uniformly equivalent with a metric of the form d V if and only if it satisfies the following properties (Theorem 5.10): (1) it is ultrametric, (2) A + is precompact, (3) concatenation is uniformly continuous, (4) morphisms from A + into A..