De Morgan clones and four-valued logics

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