Maximal sublattices of finite distributive lattices (Q1272243)

The intersection of all maximal sublattices of a finite distributive lattice \(L\) is called the Frattini sublattice \(\Phi (L)\) of \(L.\) The authors use the well-known duality between the category \(D_{01}\) of all finite distributive lattices with \((0,1)\)-lattice homomorphisms and the category \(P\) of all finite posets with order-preserving maps to obtain all maximal (and hence Frattini) sublattices for a given lattice \(L\in D_{01}.\) The main tool is the theory of critical pairs in a poset \(P\) which is in 1-1 correspondence (using duality) with the lattice \(\Phi (L).\) They study also some arithmetic properties, e.g. the connection between the cardinals \(| L | \), \(\mu,\) the number of all maximal \((0,1)\)-sublattices of \(L\), and \(\mu ',\) the number of all maximal sublattices of \(L.\)