We prove relations between regularity of two-dimensional and one-dimensional languages. Each two-dimensional language is corresponded to two sequences of one-dimensional languages corresponding to rows and columns of the two-dimensional language. For each of the following conditions the existence of both regular and nonregular two-dimensional languages is proved: all row and all column languages are regular; all row languages are regular, all column languages are nonregular; all column languages are regular, all row languages are nonregular; both all row and all column languages are nonregular.
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Korneeva et al. (2025) studied this question.
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