The Number of Maximal Closed Classes in the Set of Functions over a Finite Domain

Let k > 2 and EI, = (0, I,..., k I}. Let PI, be the set of all functions whose variables, finite in number, range over E, and whose values are in EI, . A _C PI, is complete iff any function from Pk can be expressed as a finite composition of functions from A. A is complete iff A is a subset of no maximal closed class in Pk . In 1965, all maximal closed classes were described by the author as classes of functions preserving some special relations on Ek (e.g., equivalence relations, partial orders, relations corresponding to permutations, Abelian groups, etc.) [4,6]. All repetitions in this list were established in 1969 in [6]. Also, in 1969 Zaharova, Kudrjavcev, and IablonskiI established an asymptotic estimate for the number n(k) of maximal classes in P, [8]. In the same paper, the numbers 41),..., 7r(7) are given. (However, the numbers 77(4) ~(7) and some statements seem incorrect.) In the present paper by a purely combinatorial argument, the number n(k) is found to be