Order-sorted equational unification

Order-sorted equational unification is studied from an algebraic point of view. We show how order-sorted equational unification algorithms can be built when the order-sorted signature is regular (i.e. every term has a unique least sort) and the equational specification is sort-preserving (i.e. any A-equal terms have the same least sort). Under these conditions the transformations rules allowing to build unification algorithms in the unsorted framework can be extended to the order-sorted one. This allows us to generalize the known results to order-sorted equational unification, in particular when there exist overloaded symbols with different properties. An important application is order-sorted associative-commutative unification for which no direct algorithm was given until now