Collage of two-dimensional words

We consider a new operation on one-dimensional (resp. two-dimensional) word languages, obtained by piling up, one on top of the other, words of a given recognizable language (resp. two-dimensional recognizable language) on a previously empty one-dimensional (resp. two-dimensional) array. The resulting language is the set of words “seen from above”: a position in the array is labeled by the topmost letter. We show that in the one-dimensional case, the language is always recognizable. This is no longer true in the two-dimensional case which is shown by a counter-example, and we investigate in which particular cases the result may still hold.