Completeness in finite algebras with a single operation

Abstract

An algebra (A, (wi)icT) is said to be complete if for each n all functions f: An-*A are definable in terms of the operations wi (iI the latter theorem in turn generalizes results of Post [1] (| A =2) and Jablonskil [4] (IA| =3).1 As a corollary of the main result we show that a finite algebra (A, c) with more than two elements is complete iff co generates a doubly transitive subgroup of the symmetric group on A. This improves results of Salomaa [5] and Schofield [7]. In order to state Rosenberg's theorem we need to introduce certain definitions. For k>O we denote by Ek the set {0, 1, . . , k-1}. An h-ary relation is a subset R of Ek, and is called universal if R =-E we write R(xi, * * *, Xh) when (xi, * * , Xh)C-R. Iff is a function of n arguments and R is an h-ary relation we say that f preserves R if R(f(xl, * , xl), , f(x^, , x()) whenever R(xl, ,x ** *,R(xl, (el An h-ary relation is totally reflexive if R(xi, Xh) whenever Xl, * * *, Xh are not all distinct, and totally symmetric if for each permutation a of the integers 1, . . , h, R(xi, * * *, Xh)4R(x,(1), * . ., X?(h)). The centre of a totally symmetric h-ary relation is the set of elements c such that R(xi, , Xh-l, c) holds for all xi, * * *, XhCleEk. A relation is said to be central if it is totally reflexive and totally symmetric and has centre C, 0 C CCEk. Thus the singularly central relations are just the proper nonnull subsets of Ek. For aEh-m we denote by [a], the Ith digit (1=0, *, m-1) in the expansion a= Z:_%1[a]1h1 of a in the scale of h. We may now state the theorem of Rosenberg as follows: