Decomposition of Intervals in the Space of Anti-Monotonic Functions

With the term ’anti-monotonic function’, we designate specific boolean functions on subsets of a finite set of positive integers which we call the universe. Through the well-known bijective relationship between the set of monotonic functions and the set of anti-monotonic functions, the study of the anti-monotonic functions is equivalent to the study of monotonic functions. The true-set of an anti-monotonic function is an antichain. If the universe is denoted by N , the set of anti-monotonic functions is denoted by AMF (N). This set can be partially ordered in a natural way. This paper studies enumeration in the resulting lattice of anti-monotonic functions. We define intervals of anti-monotonic functions according to this order and present four properties of such intervals, Finally we give a formula for the size of a general interval and a recursion formula for the n-th number of Dedekind.