Monadic pseudo BE-algebras

Abstract

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 BEalgebras 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.

Quantum B-algebras with involutions

“…− to introduce pairs of existential and universal quantifiers and study their properties with respect to each other ( [3], [4], [40], [11], [47], [51], [12]);…”

“…Recently, the theory of monadic MV-algebras has been developed in [1], [16], [24]. Monadic operators were defined and investigated on various algebras of fuzzy logic: Heyting algebras ( [2]), basic algebras ( [5]), GMV-algebras ( [41]), involutive pseudo BCK-algebras ( [35]), bounded commutative Rℓ-monoids ( [39]), bounded residuated lattices ( [40]), residuated lattices ( [36]), BE-algebras ( [51]), Wajsberg hoops ( [6]), BL-algebras ( [4]), bounded hoops ( [47]), pseudo equality algebras ( [29]), pseudo BCI-algebras ( [50]), NM-algebras ( [48]), pseudo BE-algebras ( [11]). The monadic operators on quantum B-algebras have been recently introduced in [12].…”