Lattices of local two-dimensional languages

The aimof this paper is to study local two-dimensional languages froman algebraic point of view.We show that local two-dimensional languages over a finite alphabet, with the usual relation of set inclusion, form a lattice. The simplest case Loc1 of local languages defined over the alphabet consisting of one element yields a distributive lattice, which can be easily described. In the general case of the latticeLocn of local languages over an alphabet of n ≥ 2 symbols, we show that Locn is not semimodular, and we exhibit sublattices isomorphic to M5 and N5. We characterize the meet-irreducible elements, the coatoms, and the join-irreducible elements ofLocn.We point out someundecidable problemswhich arise in studying the lattices Locn, n ≥ 2. We study in some detail atoms and chains of Loc2. Finally we examine the latticeLoc2 of local string languages, i.e. the local languages over the binary alphabet consisting of objects of only one row.Loc2 is an ideal ofLoc2. As a lattice, it is not semimodular but satisfies the Jordan–Dedekind condition. © 2009 Published by Elsevier B.V.