De Morgan clones and four-valued logics

TL;DR

This paper analyzes clones related to four-valued De Morgan algebras, identifying generating sets, lattice structures, and their logical properties, thereby advancing the algebraic understanding of four-valued logics.

Contribution

It provides a detailed classification of clones on a four-element set related to De Morgan algebras, including generating sets, lattice covers, and logical expansions.

Findings

Identified generating sets for clones preserving subalgebras, automorphisms, and orderings.

Described the lattice structure of four-valued clones and their covers.

Classified clones by their logical properties within algebraic logic hierarchies.

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 $\mathbf{DM_{4}}$. 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.