Relations between clones and full monoids

Abstract

An endoprimal clone is defined via a set of unary operations. It was known before that the endoprimal clone for the set O/sub 4//sup (1)/ of all unary operations on, a k-element set is the least clone J/sub k/ and that the endoprimal clone for the symmetric group S/sub k/ strictly includes J/sub k/. In this paper we consider monoids of unary operations and clones corresponding to such monoids. We define a descending sequence {N/sub i/}/sub i=1//sup k=1/ of monoids lying between O/sub k//sup (1)/ and S/sub k/, and show that the endoprimal clone for N/sub k-1/ is distinct from J/sub k/. Finally we present a characterization of the endoprimal clone for S/sub k/.