  We investigate the lattice of clones that are generated by a set of functions
that are induced on a finite field $\mathbb{F}$ 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.
