Monomial Clones: Local Results and Global Properties

Abstract

A monomial clone is a clone which is generated by a monomial. For a prime power k we consider monomial clones on the finite field GF(k). There are two monomial clones on a two-element set and six such clones on a three-element set. Ordered by set inclusion, they form lattices. In the general case, we put emphasis on monomial clones generated by unary and binary monomials. We determine unary minimal monomial clones and binary monomial clones of degree k. Furthermore, we list all unary and binary monomial clones on GF(5) and state several properties that hold on GF(7) and GF(11).