The lattice of monomial clones on finite fields

Abstract

AbstractWe investigate the lattice of clones that are generated by a set of functions that are induced on a finite field $${\mathbb {F}}$$ F by monomials. We study the atoms and coatoms of this lattice and investigate whether this lattice contains infinite ascending chains, or infinite descending chains, or infinite antichains.We give a connection between the lattice of these clones and semi-affine algebras. Furthermore, we show that the sublattice of idempotent clones of this lattice is finite and every idempotent monomial clone is principal.