The inconsistency predicate on De Morgan lattices

Abstract

We consider expansions of De Morgan lattices by an additional unary predicate interpreted in each De Morgan lattice by the ideal generated by all elements of the form a ∧ −a, and describe the finite lattice of strict universal Horn classes of such structures, thereby extending the description of the lattice of quasivarieties of De Morgan lattices due to Pynko. We also consider the same problem for expansions of De Morgan lattices by a constant interpreted as the maximal element of this ideal whenever it exists.

Keywords

• De Morgan lattice
• inconsistency
• Belnap–Dunn logic
• quasivariety
• universal Horn class