Row-Column Combination of Dyck Words

We lift the notion of Dyck language from words to 2-dimensional arrays of symbols, i.e., pictures. We define the Dyck crossword language DCk as the row-column combination of Dyck word languages, which prescribes that each column and row is a Dyck word over an alphabet of size 4k. The standard relation between matching parentheses is represented in DCk by an edge of the matching graph situated on the picture array. Such edges form a circuit, of path length multiple of four, where row and column matches alternate. Length-four circuits are rectangular patterns, while longer ones exhibit a large variety of patterns. DCk languages are not recognizable by the Tiling Systems of Giammarresi and Restivo. DCk contains pictures where circuits of unbounded length occur, and where any Dyck word occurs in a row or in a column. We prove that the only Hamiltonian circuits of the matching graph of DCk have length four. A proper subset of DCk, called quaternate, includes only the rectangular patterns; we define a proper subset of quaternate pictures that (unlike the general ones) preserves a characteristic property of Dyck words: availability of a cancellation rule based on a geometrical partial order relation between rectangular circuits. Open problems are mentioned.