We study clones on a four-element set related to the clone \({\mathsf {DMA}}\) of all term functions of the subdirectly irreducible four-element De Morgan algebra \({\mathbf {DM}}_{{{\mathbf {4}}}}\). We find generating sets for the clones of all functions preserving the subalgebras of \({\mathbf {DM}}_{{{\mathbf {4}}}}\), the automorphisms of \({\mathbf {DM}}_{{{\mathbf {4}}}}\), the truth order and the information order on \({\mathbf {DM}}_{{{\mathbf {4}}}}\), 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.

我们就有关克隆了四元组研究克隆\（{\ mathsf {DMA}} \）的直不可约四元德摩根代数的所有项功能\（{\ mathbf {DM}} _ {{ {\ mathbf {4}}}} \）。我们发现发电机组的所有功能保留的子代数克隆\（{\ mathbf {DM}} _ {{{\ mathbf {4}}}} \），的自同构 \（{\ mathbf {DM}} _ {{{\ mathbf {4}}}} \），\（{\ mathbf {DM}} _ {{{\ mathbf {4}}}} \\）上的真相顺序和信息顺序，以及这些条件的结合定义的克隆。我们在四值克隆的格中标识\（{\ mathsf {DMA}} \）的封面，并描述\（{\ mathsf {DMA}} \）上方的克隆格。包含鉴别功能。最后，观察到\（{\ mathsf {DMA}} \）上方的每个克隆都定义了四值Belnap–Dunn逻辑的扩展，我们通过它们的元数据属性（特别是根据它们在Leibniz和Frege层次结构中的位置）对这些克隆进行分类。抽象代数逻辑。