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De Morgan clones and four-valued logics

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Abstract

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.

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Acknowledgements

The author is grateful to the anonymous referee for their careful reading of the manuscript and useful comments, which helped to improve the paper.

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  1. Department of Mathematics, Vanderbilt University, Tennessee, 37235, USA

    Adam Přenosil

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  1. Adam Přenosil
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Correspondence to Adam Přenosil.

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Communicated by Presented by J. G. Raftery.

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Přenosil, A. De Morgan clones and four-valued logics. Algebra Univers. 82, 30 (2021). https://doi.org/10.1007/s00012-021-00726-5

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