Varieties of positive modal algebras and structural completeness

Abstrakt

Positive modal algebras are the〈∧,∨, 3, D, 0, 1〉-subreducts of modal algebras. We show that the variety of positive interior algebras is not locally finite. However, the free one-generated positive interior algebra has 37 elements. Moreover, we show that there are exactly 16 varieties of height at most 4 in the lattice of varieties of positive interior algebras. Building on this, we infer that there are only 3 non-trivial structurally complete varieties of positive K4-algebras. These are also the unique non-trivial hereditarily structurally complete such varieties. Moreover, we characterize passively structurally complete varieties of positive K4-algebras and show that there are infinitely many of them. These results are related to the study of structurally complete axiomatic extensions of an algebraizable Gentzen system for positive modal logic.