Subalgebra lattices of totally reflexive sub-preprimal algebras

Abstract

For a set Q of relations on a finite set A, the set $$\mathrm{Pol}_{A}Q$$ , of all operations on A preserving all relations in Q, is a clone. The set of all clones on a given set forms a lattice under inclusion. Each of its maximal elements can be represented as $$\mathrm{Pol}_{A}\{\rho \}$$ where $$\rho $$ is one of the six types of relations determined by Ivo G. Rosenberg. Four types of them are totally reflexive. These relations are bounded orders, non-trivial equivalence relations, central relations, and universal relations. An algebra $$\underline{A}$$ is said to be totally reflexive sub-preprimal if its clone of term operations is where $$\rho _{1}$$ and $$\rho _{2}$$ are totally reflexive relations. We describe the subalgebra lattices of such algebras.