The variety of positive S4-algebras has a finite free algebra in one generator but is not locally finite, and exactly three nontrivial structurally complete varieties of positive K4-algebras exist.
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Varieties of positive modal algebras and structural completeness
The variety of positive S4-algebras has a finite free algebra in one generator but is not locally finite, and exactly three nontrivial structurally complete varieties of positive K4-algebras exist.