On the quasivariety of sub-Hilbert algebras

It is known that sub-Hilbert algebras are the implicative subreducts of subresiduated lattices. In this paper we give a new proof of this result by using some ideas employed to the study of the implicative-infimum subreducts of weak Heyting algebras. We also introduce open implicative filters in order to study the lattice of relative congruences of sub-Hilbert algebras. In particular, we introduce and study irreducible open implicative filters and we use this kind of open implicative filters in order to study the quasivariety of sub-Hilbert algebras generated by the class of its totally ordered members. Finally, we study a quasivariety which properly contains the variety of order Hilbert algebras.