We study the variety generated by the three-element equivalential algebra with conjunction on the dense elements. We prove the representation theorem which let us construct the free algebras in this variety.

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.

We effectively construct the finitely generated free equivalential algebras corresponding to the equivalential fragment of intuitionistic propositional logic.

We construct free algebras in the variety generated by the equivalential algebra with conjunction on dense elements and compute the formula for the free spectrum of this variety. Moreover, we describe the decomposition of free algebras into directly indecomposable factors.

We study the variety of equivalential algebras with zero and its subquasivariety that gives the equivalent algebraic semantics for the ‐fragment of intuitionistic propositional logic. We prove that this fragment is hereditarily structurally complete. Moreover, we effectively construct the finitely generated free equivalential algebras with zero.