Monadic NM-algebras

Abstract

In this paper, we introduce and investigate monadic NM-algebras: a variety of NM-algebras equipped with universal quantifiers. Also, we obtain some conditions under which monadic NM-algebras become monadic Boolean algebras. Besides, we show that the variety of monadic NM-algebras faithfully the axioms on quantifiers in monadic predicate NM logic. Furthermore, we discuss relations between monadic NM-algebras and some related structures, likeness modal NM-algebras and rough approximation spaces. In addition, we investigate monadic filters in monadic NMalgebras. In particular, we characterize simple and subdirectly irreducible monadic NM-algebras and obtain a representation theorem for monadic NM-algebras. Finally, we present monadic NM-logic and prove the (chain) completeness of monadic NM-logic based on monadic NM-algebras. These results constitute a crucial first step for providing a solid algebraic foundation for the monadic predicate NM logic.

Algebraic results on universal quantifiers in monoidal t-norm based logic

“…Therefore, ∀L = L ∀ . (13) ( 7) and (12) imply that → and ⊙ are preserved, respectively. ( 1) and ( 2) imply that 0, 1 ∈ ∀L.…”

“…The notion of a quantifier on a Boolean algebra was introduced by Halmos in [4] as an algebraic counterpart of the logical notion of an existential quantifier, and the algebras obtained in this way were called by Halmos monadic Boolean algebras. After then quantifiers have been considered by several authors in different algebras, for example, orthomodular lattices, Heyting algebras, distributive lattices, MV-algebras (Wajsberg algebras), BL-algebras, NM-algebras and BCI-algebras [5,6,7,8,9,10,11,12,13]. In the above-mentioned algebras, both MV-algebras and NM-algebras satisfy De Morgan and double negation laws, in the definition of the corresponding quantifiers, it is possible to use only one of the existential and universal quantifiers as primitive, the other being definable as the dual of the one defined.…”

On Subtractive Derivations of Rl‐Monoids

“…Residuated lattice ordered monoids (Rl-monoids, for short) were introduced by Swamy [1] as a common generalization of Abelian lattice ordered groups and Heyting algebras. Moreover, residuated lattice ordered monoids are in very close connections with algebras of t-norm-based fuzzy logics [2][3][4][5][6][7][8]. In particular, BL algebras and MV-algebras can be viewed as particular cases of such algebras.…”

Quantum B-algebras with involutions

“…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].…”