FINITE CLONES CONTAINING ALL PERMUTATIONS

Let A be a finite set with Absolute value of A > 2. We describe all clones on A containing the set S(A) of all permutations of A among its unary operations. (A clone on A is a composition closed set of finitary operations on A containing all projections). With a few exceptions such a clone C is either essentially unary or cellular i.e. there exists a monoid M of self-maps of A containing S(A) such that either C = MBAR (= all essentially unary operations agreeing with some f is-an-element-of M) or C = M or GAMMA(h) where 1 < h less-than-or-equal-to Absolute value of A and GAMMA(h), consists of all finitary operations on A taking at most h values. The exceptions are subclones of Burle's clone or of its variant (provided Absolute value of A is even).