We consider the family UREC of unambiguous recognizable
two-dimensional languages. We prove that there are recognizable
languages that are inherently ambiguous, that is UREC family is a
proper subclass of REC family. The result is obtained by showing a
necessary condition for unambiguous recognizable languages.
Further UREC family coincides with the class of picture languages
defined by unambiguous 2OTA and it strictly contains its
deterministic counterpart. Some closure and non-closure properties
of UREC are presented. Finally we show that it is undecidable
whether a given tiling system is unambiguous.