Kruse, Eva (2004) Symmetrische Kettenzerlegungen von Verbänden und Intervallzerlegung des linearen Verbandes. Masters thesis.

Abstract

In the field of Combinatorics we are interested in the existence and structure of certain configurations. The solution of counting problems of a set often provides a survey of its structure and the composition of the elements. Thereby partially ordered sets (posets) and particularly lattices play an important role. If we know that a poset is decomposable into certain suborders like chains or intervals we will have a better survey of its structure. It is very interesting to answer the question whether there is a symmetric chain decomposition, i.e. a decomposition into disjoint, symmetric chains. SpernerÅœs theorem follows for example from the existence of such a decomposition of the Boolean lattice. The lattice of linear subspaces of a n-dimensional vector space- of which we have a rather bad survey- forms another class of into symmetric chains decomposable lattices. Frank Vogt and Bernd Voigt succeeded for the first time to give explicitely such a decomposition. Based on their construction and the recursion of the Galois Numbers there is the conjecture that it is possible to decompose the lattice of linear subspaces into disjoint intervals. This thesis gives the proof that- against oneÅœs expectations- the algorithm of Vogt and Voigt doesnÅœt affirm this conjecture.

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