  This paper studies partitions in the space of antimonotonic boolean functions
on sets of n elements. The antimonotonic functions are the antichains of the
partially ordered set of subsets. We analyse and characterise a natural partial
ordering on this set. We study the inter- vals according to this ordering. We
show how intervals of antimonotonic functions, and a fortiori the whole space
of antimonotonic functions can be partitioned as disjoint unions of certain
classes of intervals. These in- tervals are uniquely determined by
antimonotonic functions on smaller sets. This leads to recursive enumeration
algorithms and new recursion relations. Using various decompositions, we derive
new recursion formu- lae for the number of antimonotonic functions and hence
for the number of monotonic functions (i.e. the Dedekind number).
