In this paper we develop the language theory of higher-dimensional automata
(HDAs). Regular languages of HDAs are sets of finite interval partially ordered
multisets (pomsets) with interfaces (iiPoms). We first show a pumping lemma
which allows us to expose a class of non-regular languages. We also give an
example of a regular language with unbounded ambiguity. Concerning decision and
closure properties, we show that inclusion of regular languages is decidable
(hence is emptiness), and that intersections of regular languages are again
regular. On the other hand, complements of regular languages are not regular.
We introduce a width-bounded complement and show that width-bounded complements
of regular languages are again regular