SOME NUMERICAL CHARACTERIZATION OF FINITE DISTRIBUTIVE LATTICES
Joanna Grygiel · 2004
We regard a finite distributive lattice as the gluing of its maximal boolean intervals and describe some dependencies between dimensions of the boolean intervals and the skeleton of the lattice. In [7] we described the decomposition of finite lattices into blocks of its skeleton tolerance. In this paper we shall consider the other side of the phenomenon - the gluing of an atlas with overlapping neighbours. Both approaches are equivalent (see [2]). Let (Lx)x2K be a family of finite lattices and let the index set K be also a finite lattice. We call the family (Lx)x2K a K-atlas with overlapping neighbours if the following conditions hold for every x,y 2 K: 1. If Lx Ly then x = y. 2. If x y then Lx Ly 6 ;. 3. If x y and Lx Ly 6 ; then the orders of Lx and Ly coincide on the intersection Lx \Ly and the interval Lx \Ly is at the same time a filter of Lx and an ideal of Ly. 4. Lx Ly = Lx^y Lx_y. The structure L = h