“…The fragments of intuitionistic logic with double negation were also studied, e.g., in [2,10,14] as well as in [5], where ¬¬ was treated as a modal operator. Our construction works for some of these fragments, e.g., for (→, ¬¬) and (∧, →, ¬¬), where as the equivalent algebraic semantics one can take Hilbert algebras with regularization (HI r ) and Brouwerian semilattices with regularization (B r ), respectively, i.e., Hilbert algebras or Brouwerian semilattices endowed with a retraction r into the set of regular elements that is simultaneously a closure operator with respect to the natural order.…”