Monadic pseudo BE-algebras

TL;DR

This paper introduces monadic pseudo BE-algebras, explores their properties, and establishes connections with pseudo MV-algebras, focusing on quantifiers, deductive systems, and congruences.

Contribution

It defines monadic pseudo BE-algebras and studies their properties, including the behavior of quantifiers and the structure of deductive systems and congruences.

Findings

Existential and universal quantifiers form a residuated pair.

Quantifiers on bounded commutative pseudo BE-algebras extend to pseudo MV-algebras.

One-to-one correspondence between monadic congruences and deductive systems.

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 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.