On the representation and enumeration of fuzzy switching functions

This paper examines the properties of fuzzy switching functions of n variables with the help of combinatorial theory. A new concept of canonical representation is explained, and it is shown that any fuzzy switching function can be canonically represented as a sum of K ⩽ Nmax fundamental phrases, which form an antichain in the poset P of all fundamental phrases. Also, an algorithm to enumerate the distinct fuzzy switching functions and two formulae for the upper and lower bounds, which are better than those given by C. Clark, M. Ben-David, and A. Kandel [Fuzzy Sets Systems 5, 69–81 (1981)] and T. Kameda and E. Sadeh [Inform. Contr. 35, 139–145 (1977)], are given