A syntactic congruence for languages of birooted trees

The study of languages of labelled birooted trees, that is, elements of the free inverse monoid enriched by a vertex labelling, has led to the notion of quasi-recognisability. It generalises the usual notion of recognisability by replacing homomorphisms by certain prehomomorphism into finite ordered monoids, called adequate, that only preserve some products: the so-called disjoint ones. In this paper we study the underlying partial algebra setting and we define a suitable notion of a syntactic congruence such that (i) having a syntactic congruence of finite index captures MSO-definability; (ii) a certain order-bisimulation refinement of the syntactic congruence captures quasi-recognisability in the same way.