De Morgan clones and four-valued logics

Abstract

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 $\mathbf{D}{\mathbf{M}}_4$. We find generating sets for the clones of all functions preserving the subalgebras of $\mathbf{D}{\mathbf{M}}_4$, the auto\-morphisms of~$\mathbf{D}{\mathbf{M}}_4$, the truth order and the information order on $\mathbf{D}{\mathbf{M}}_4$, 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.

arXiv comment42 pages, 10 figures