The lattice of monomial clones on finite fields

We 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.

Ideal clones: Solution to a problem of Czédli and Heindorf

Given an infinite set X and an ideal I of subsets of X, the set of all finitary operations on X which map all (powers of) I-small sets to I-small sets is a clone. In [2], G. Czédli and L. Heindorf asked whether or not for two particular ideals I and J on a countably infinite set X, the corresponding ideal clones were a covering in the lattice of clones. We give an affirmative answer to this question.

ON DUALIZING CLONES AS LAWVERE THEORIES

In this paper we treat clones as Lawvere theories and then dualize them as categories, rather than as single objects of a category of algebras. The approach applies only to some primitive-positive clones, but in return, a structure with somewhat surprising properties is obtained. To illustrate the method, we thoroughly investigate the lattice of clones of operations over a finite boolean algebra.