Relations between clones and full monoids

An endoprimal clone is defined via a set of unary operations. It was known before that the endoprimal clone for the set O₄⁽¹⁾ of all unary operations on, a k-element set is the least clone J_k and that the endoprimal clone for the symmetric group S_k strictly includes J_k. In this paper we consider monoids of unary operations and clones corresponding to such monoids. We define a descending sequence {N_i}_i=1 ^k=1 of monoids lying between O_k⁽¹⁾ and S_k, and show that the endoprimal clone for N_k-1 is distinct from J_k. Finally we present a characterization of the endoprimal clone for S_k