On Some Approaches to Solving the Dedekind Problem

This paper shows that the class of monotone functions coincides with the class of deadlock disjunctive normal forms. The paper introduces the concept of singletier deadlock disjunctive normal forms. A formula is obtained by calculating the number of these functions on all tiers of an n-dimensional cube. This paper proposes an algorithm for finding monotone functions that are single-tier deadlock disjunctive normal forms for n variables. Calculations are performed to calculate single-tier deadlock disjunctive normal forms for n=1,2,…13. Monotone functions of n variables can be defined on the vertices of an n-dimensional cube. A theorem is given that decomposes the monotone functions of n variables into two classes: the monotone functions of n-1 variable and the monotone functions of n variables that label only the variable x<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf>. A second method for computing single-tier deadlock disjunctive normal forms is proposed using this theorem.