Chain polynomials of distributive lattices are 75% unimodal.
We give an affirmative answer to problem DJ from Fremlin’s list [8] which asks whether implies that every uncountable Boolean algebra has an uncountable set of pairwise incomparable elements.
Distributive ordered sets are characterized by so called generalized annihilators.
If is a class of partially ordered sets, let denote the system of all posets which are isomorphic to the system of all intervals of for some We give an algebraic characterization of elements of for being the class of all bounded posets and the class of all posets satisfying the condition that for each there exist a minimal element and a maximal element with respectively.
The concept of a 0-distributive poset is introduced. It is shown that a section semicomplemented poset is distributive if and only if it is 0-distributive. It is also proved that every pseudocomplemented poset is 0-distributive. Further, 0-distributive posets are characterized in terms of their ideal lattices.