Partitioning in the space of antimonotonic functions

Abstract

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).