Generalizations of Unification

We define a general notion of (syntactic) unification, the ( -unification, whose special cases are matching, unification, semi-unification, and weak-unification. We settle all the implications between the various cases of ( -unification leading to a classification and a decidability result. We show that some distinctions holding for finite terms collapse for infinite terms. We give some positive results on the existence of most general weak-unifiers