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 α=β holds for all picture languages if and only if α and β 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.