We show that the variety of equivalential algebras with regularization gives the algebraic semantics for the -fragment of intuitionistic propositional logic. We also prove that this fragment is hereditarily structurally complete.

This paper provides a finer analysis of the well-known form of the Local Deduction Theorem in contraction-free logics . An infinite hierarchy of its natural strengthenings is introduced and studied. The main results are the separation of its initial four members and the subsequent collapse of the hierarchy.