Lattices of local two-dimensional languages

  • Authors:

  • F. De Carli;A. Frosini;S. Rinaldi;A. Sorbi

  • Affiliations:

  • Università di Siena, Dipartimento di Scienze Matematiche e Informatiche, Pian dei Mantellini, 44, 53100, Siena, Italy;Università di Siena, Dipartimento di Scienze Matematiche e Informatiche, Pian dei Mantellini, 44, 53100, Siena, Italy;Università di Siena, Dipartimento di Scienze Matematiche e Informatiche, Pian dei Mantellini, 44, 53100, Siena, Italy;Università di Siena, Dipartimento di Scienze Matematiche e Informatiche, Pian dei Mantellini, 44, 53100, Siena, Italy

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2009

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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 Loc"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 Loc"n of local languages over an alphabet of n=2 symbols, we show that Loc"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 Loc"n. We point out some undecidable problems which arise in studying the lattices Loc"n, n=2. We study in some detail atoms and chains of Loc"2. Finally we examine the lattice Loc"2^h of local string languages, i.e. the local languages over the binary alphabet consisting of objects of only one row. Loc"2^h is an ideal of Loc"2. As a lattice, it is not semimodular but satisfies the Jordan-Dedekind condition.