Classes of two-dimensional languages and recognizability conditions

The paper deals with some classes of two-dimensional recognizable languages of ``high complexity'', in a sense specified in the paper and motivated by some necessary conditions holding for recognizable and unambiguous languages. For such classes we can solve some open questions related to unambiguity, finite ambiguity and complementation. Then we reformulate a necessary condition for recognizability stated by O. Matz, introducing a new complexity function. We solve an open question proposed by O. Matz, showing that all the known necessary conditions for recognizability of a language and its complement are not sufficient. The proof relies on a family of languages defined by functions.