Several Classes Of Matchable And Non-matchable Distributive Lattices

Distributive lattices play a very important role in lattice theory.Day has studied a special sublattice(named cutting sublattice in the thesis)of a finite distributive lattice.Zhang et al.found that all perfect matchings of a finite plane weakly element bipartite graph equipped a special order could be treated as a finite distributive lattice.Recently,Zhang et al.introduced a concept of matchable distributive lattice,obtained some results on(non-)matchable distributive lattices.The thesis consists of four chapters.In the first chapter,we introduce the background,some preliminaries,and list the main results.In Chapter 2,a special sublattice of a finite distributive lattice studied by Day is first named cutting(sublattice).By the concept,a convex expansion for finite distributive lattices is considered.Thus,a more general method for drawing the Hasse diagram is given,the rank generating function of a finite distributive lattice is obtained.In addition,we have several enumerative properties on finite distributive lattices,and verify the generalized Euler formula for polyhedronsIn Chapter 3,we first introduce a special hexagon chain named lucasene which is similar to fibonaccene,a special poset named L-fence the Hasse diagram of which is isomorphic to the inner dual directed graph of the corresponding lucasene.A class of new cubes are named as matchable Lucas cubes according to the number of its vertices(elements),which are a series of directed or undirected Hasse diagrams of filter lattices of L-fences.The basic properties and several classes of polynomials,e.g.rank generating functions,cube polynomials and degree spectrum polynomials,of matchable Lucas cubes are obtained.Some special conclusions on the binomial coefficients,the Padovan sequence and the Lucas triangle are found.In Chapter 4,we introduce the meet-irreducible cell with respect to a perfect matching of a plane(weakly)elementary bipartite graph and give its equivalent characterizations.Using these,we extend a result on non-matchable distributive lattices with a cut element,obtain a class of new non-matchable distributive lattices.Moreover,we show that a result obtained by Yao and Zhang is could be extended,get a few of non-matchable distributive lattices with their structure.