2 research outputs found
Quantum B-algebras with involutions
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
Monadic pseudo BE-algebras
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