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Lattices of tilings and stability

Nathalie Caspard 1 Michel Morvan Éric Rémila Eric Thierry 2
2 TRIO - Real time and interoperability
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : Many tiling spaces such as domino tilings of fixed figures have an underlying lattice structure. This lattice structure corresponds to the dynamics induced by flips. In this paper, we further investigate the properties of these lattices of tilings. In particular, we point out a stability property: the set of all the shortest sequences of flips joining to fixed tilings also have a lattice structure close to the lattice of all tilings. We also show that some of these properties also apply to other discrete dynamical systems and more generally may be satisfied by some partially ordered sets. It gives a new perspective on the lattice structure of tiling spaces and enables to deduce some of their properties only by means of partial order theoretical tools.
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Nathalie Caspard, Michel Morvan, Éric Rémila, Eric Thierry. Lattices of tilings and stability. [Intern report] A03-R-424 || caspard03a, LIP- RR-2003-25, Laboratoire de l’Informatique du Parallélisme. 2003. ⟨inria-00099782⟩

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