Filters on Some Classes of Quantum B-Algebras

Abstract

In this paper, we continue the study of quantum B-algebras with emphasis on filters on integral quantum B-algebras. We then study filters in the setting of pseudo-hoops. First, we establish an embedding of a cartesion product of polars of a pseudo-hoop into itself. Second, we give sufficient conditions for a pseudohoop to be subdirectly reducible. We also extend the result of Kondo and Turunen to the setting of noncommutative residuated ∨-semilattices that, if prime filters and ∨-prime filters of a residuated ∨-semilattice A coincide, then A must be a pseudo MTL-algebra.

Q-Filters of Quantum B-Algebras and Basic Implication Algebras

“…In Reference [30], the notion of filter in quantum B-algebra is proposed. If X is a quantum B-algebra and F is a nonempty set of X, then F is called the filter of X if F∈U(X) and F•F ⊆ F. That is, F is a filter of X, if and only if: (1) F is a nonempty upper subset of X; (2) (z∈X, y∈F, y→z∈F) ⇒ z∈F.…”

“…Although the definition of a filter in a quantum B-algebra is given in Reference [30], quotient algebraic structures are not established by using filters. In fact, filters in special subclasses of quantum B-algebras are mainly discussed in Reference [30], and these subclasses require a unital element. In this paper, by introducing the concept of a q-filter in quantum B-algebras, we establish the quotient structures using q-filters in a natural way.…”

On Soft Quantum B-Algebras and Fuzzy Soft Quantum B-Algebras

“…Rump [11,12] investigated many implication algebras (for example, pseudo-BCK-algebras, po-groups, BL-algebras, MV-algebras, GPE-algebras, and resituated lattices). Botur and Paseka [13] studied filters on integral QB-As, and Zhang et al [14] established the quotient structures by using q-filters in QB-As and investigated the relation between basic implication algebras and QB-As. Han et al [15] constructed the unitality of QB-As and explained the injective hulls of QB-As in [16].…”