 Structures: A many-sorted structure has several non-empty domains, one for each sort, and variables of a given sort take values over the corresponding domain. An n-ary relation could be freely established between elements of different domains or only between certain ones. These two options are called liberal and strict. A liberal relation may relate objects of arbitrary domains and only the arity (a natural number) has to be stated. For a strict relation,  the sorts involved have to be specified as well as the arity.