Meet-reducible submaximal clones determined by nontrivial equivalence relations

The structure of the lattice of clones on a finite set has been proven to be very complex. To better understand the top of this lattice, it is important to provide a characterization of submaximal clones in the lattice of clones. It is known that the clones $Pol(\theta)$ and $Pol(\rho)$ (where $\theta$ is a nontrivial equivalence relation on $E_k = \{0,\dots, k-1\}$, and $\rho$ is among the six types of relations which characterize maximal clones) are maximal clones. In this paper, we provide a classification of relations (of Rosenberg's List) on $E_k$ such that the clone $Pol(\theta) \cap Pol(\rho)$ is maximal in $Pol(\theta)$.