  In this paper we define the monadic pseudo BE-algebras and investigate their
properties. We prove that the existential and universal quantifiers of a
monadic pseudo BE-algebra form a residuated pair. Special properties are
studied for the particular case of monadic bounded commutative pseudo
BE-algebras. Monadic classes of pseudo BE-algebras are investigated and it is
proved that the quantifiers on bounded commutative pseudo BE-algebras are also
quantifiers on the corresponding pseudo MV-algebras. The monadic deductive
systems and monadic congruences of monadic pseudo BE-algebras are defined and
their properties are studied. It is proved that, in the case of a monadic
distributive commutative pseudo BE-algebra there is a one-to-one correspondence
between monadic congruences and monadic deductive systems, and the monadic
quotient pseudo BE-algebra algebra is also defined.
