  We study clones on a four-element set related to the clone $\mathsf{DMA}$ of
all term functions of the sub\-directly irreducible four-element De~Morgan
algebra $\mathbf{DM_{4}}$. We find generating sets for the clones of all
functions preserving the subalgebras of $\mathbf{DM_{4}}$, the auto\-morphisms
of~$\mathbf{DM_{4}}$, the truth order and the information order on
$\mathbf{DM_{4}}$, as well as clones defined by conjunctions of these
conditions. We identify the covers of $\mathsf{DMA}$ in the lattice of
four-valued clones and describe the lattice of clones above $\mathsf{DMA}$
which contain the discriminator function. Finally, observing that each clone
above $\mathsf{DMA}$ defines an expansion of the four-valued Belnap--Dunn
logic, we classify these clones by their metalogical properties, specifically
by their position within the Leibniz and Frege hierarchies of abstract
algebraic logic.
