A note on identities of two-dimensional languages

In this note we consider identical laws satisfied by two-dimensional (picture) languages. collections of rectangular arrays over a given alphabet. We prove that an identity alpha = beta holds for all picture languages if and only if alpha and beta represent the same bi-language (a subset of a free bi-monoid). As a consequence. we obtain decidability of the equational theory of picture languages, a description of free objects in the variety generated by picture language algebras, and prove that such a variety does not have a finite equational axiomatization. (C) 2004 Elsevier B.V. All rights reserved.