Enumerating fuzzy switching functions and free Kleene algebras (Q796494)

As Boolean algebras are related to two-valued logic and to switching functions, Kleene algebras are related to a three-valued logic and to fuzzy switching functions. The authors consider the free Kleene algebra FK(n) of n generators and determine the cardinality of FK(4) as 160,297,985,276. For general n, lower and upper bounds are given for the cardinality of FK(n), i.e. for the number of n-variable fuzzy switching functions. The methods are applied to determine also the numbers of regular (ternary) functions in 1, 2, 3, and 4 variables.