Regular expressions for two-dimensional languages over one-letter alphabet

The aim of this paper is to give regular expressions for two-dimensional picture languages. The paper focuses on a one-letter alphabet case, that corresponds to the study of "shapes" of families of pictures. A new diagonal concatenation operation is defined. Languages denoted by regular expressions with union, diagonal concatenation and its closure are characterized both in terms of rational relations and in terms of two-dimensional automata moving only right and down. The class of languages denoted by regular expressions with union: column, row and diagonal concatenation, and their closures are included in REC and strictly contains languages defined by three-way automata, but they are not comparable with ones defined by four-way automata. In order to encompass a wider class of languages, we propose some new operations that define languages that still lie in REC.