  The aim of this paper is to define and study the involutive and weakly
involutive quantum B-algebras. We prove that any weakly involutive quantum
B-algebra is a quantum B-algebra with pseudo-product. As an application, we
introduce and investigate the notions of existential and universal quantifiers
on involutive quantum B-algebras. It is proved that there is a one-to-one
correspondence between the quantifiers on weakly involutive quantum B-algebras.
One of the main results consists of proving that any pair of quantifiers is a
monadic operator on weakly involutive quantum B-algebras. We investigate the
relationship between quantifiers on bounded sup-commutative pseudo BCK-algebras
and quantifiers on other related algebraic structures, such as pseudo
MV-algebras and bounded Wajsberg hoops.
