Lattices of local two-dimensional languages

Abstract

The aim of this paper is to study local two-dimensional languages from an 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 L oc 1 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 lattice L oc n of local languages over an alphabet of n ≥ 2 symbols, we show that L oc n is not semimodular, and we exhibit sublattices isomorphic to M 5 and N 5 . We characterize the meet-irreducible elements, the coatoms, and the join-irreducible elements of L oc n . We point out some undecidable problems which arise in studying the lattices L oc n , n ≥ 2 . We study in some detail atoms and chains of L oc 2 . Finally we examine the lattice L oc 2 h of local string languages, i.e. the local languages over the binary alphabet consisting of objects of only one row. L oc 2 h is an ideal of L oc 2 . As a lattice, it is not semimodular but satisfies the Jordan–Dedekind condition.