  Properties of intervals in the lattice of antichains of subsets of a universe
of finite size are investigated. New objects and quantities in this lattice are
defined. Expressions and numerical values are deduced for the number of
connected antichains and the number of fully distinguishing antichains. The
latter establish a connection with Stirling numbers of the second kind.
Decomposition properties of intervals in the lattice of antichains are proven.
A new operator allowing partitioning the full lattice in intervals derived from
lower dimensional sub-lattices is introduced. Special posets underlying an
interval of antichains are defined. The poset allows the derivation of a
powerful formula for the size of an interval. This formula allows computing
intervals in the six dimensional space. Combinatorial coefficients allowing
another decomposition of the full lattice are defined. In some specific cases,
related to connected components in graphs, these coefficients can be
efficiently computed. This formula allows computing the size of the lattice of
order 8 efficiently. This size is the number of Dedekind of order 8, the
largest one known so far.
