Fuzzy Boolean Filters Research Articles

Several types of hesitant fuzzy filters on residuated lattices

The present paper investigates the hesitant fuzzy filters on residuated lattices. A one-to-one correspondence between the set of all hesitant fuzzy filters and the set of all hesitant fuzzy congruences is established and a quotient residuated lattice with respect to a hesitant fuzzy filter is induced. Furthermore, several special types of hesitant fuzzy filters such as hesitant fuzzy implicative, regular and Boolean filters are introduced, and some alternative definitions of them are obtained, then some typical logical algebras are characterized by these identity forms.

On open problems based on fuzzy filters of pseudo BCK-algebras

We study the properties and relations of fuzzy pseudo-filters of pseudo-BCK algebras. After we discuss the equivalent conditions of fuzzy normal pseudo-filter of pseudo-BCK algebra (pP), we propose fuzzy implicative pseudo-filter and its relation with fuzzy Boolean filter of (bounded) pseudo-BCK algebras (pP). Then two open problems: In pseudo-BCK algebra or bounded pseudo-BCK algebra, Is the notion of implicative pseudo-filter equivalent to the notion of Boolean filter? and Prove or negate the following conclusion: A pseudo-BCK algebra is an implicative pseudo-BCK algebra if and only if every pseudo-filter of it is a Boolean filter(or an implicative pseudo-filter) are partly solved.

Fuzzy Weak Regular, Strong and Preassociative Filters in Residuated Lattices

In this paper, the notions of fuzzy weak regular, strong and preassociative filters are introduced with some properties of them investigated. In particular, under the context of Glivenko algebras, fuzzy weak regular filters and regular ones are equivalent and characterizations of Glivenko algebras are concluded by characterizations of fuzzy strong filters. At last the notion of fuzzy preassociative filters are defined, which are proved to coincide with fuzzy Boolean filters, and then some new alternative definitions of Boolean algebras are obtained by this type of filters.

Interval-valued intuitionistic (T, S)-fuzzy filters theory on residuated lattices

The aim of this paper is further to develop the filter theory on residuated lattices. Firstly, the notion of interval valued intuitionistic (T, S)-fuzzy filter(IVI (T, S)-fuzzy filter for short) on residuated lattices is introduced by linking the interval valued intuitionistic fuzzy set, t-norm, s-norm and filter theory of residuated lattices; the properties and equivalent characterizations of interval valued intuitionistic (T, S)-fuzzy filter are investigated; the relation between IVI (T, S)-fuzzy filter and filter is studied. Secondly, the notions of interval valued intuitionistic (T, S)-fuzzy implicative filter and interval valued intuitionistic (T, S)-fuzzy Boolean filter are introduced; the properties and equivalent characterizations of them are investigated; the intuitionistic (T, S)-fuzzy implicative filter is proved to be equivalent to the intuitionistic (T, S)-fuzzy Boolean filter in residuated lattices. Finally, the intuitionistic (T, S)-fuzzy positive implicative filter and intuitionistic (T, S)-fuzzy G (MV) filter are introduced; some equivalent characterizations of them are obtained and the relations among these fuzzy filters are investigated.

On filters of R0-algebras connected to intuitionistic fuzzy sets

Following Fuzzy implicative and Boolean filters of R0-algebras, Inf Sci 171 2005, 61--71, the notion of normal intuitionistic fuzzy filter of R0-algebras is introduced. Some methods are created to construct new intuitionistic fuzzy filters. Also some characteristic properties and equivalent conditions of this notion are investigated. Specially, it is shown that under certain conditions, a normal intuitionistic fuzzy filter is two-valued.

Intuitionistic fuzzy filter theory of BL-algebras

In this paper, the intuitionistic fuzzy filter theory of BL-algebras is researched. The basic knowledge of BL-algebras and intuitionistic fuzzy sets is firstly reviewed. The notions of intuitionistic fuzzy filters, lattice filters, prime filters, Boolean filters, implicative filters, positive implicative filters, ultra filters and obstinate filters are introduced, respectively. Their important properties are investigated. In intuitionistic fuzzy sets, intuitionistic fuzzy filters, Boolean filters, ultra filters are proved to be equivalent to lattice filters, implicative filters, obstinate filters, respectively. Each intuitionistic fuzzy Boolean filter is an intuitionistic fuzzy positive implicative filter, but the converse may not be true in BL-algebras. The conditions under an intuitionistic fuzzy positive implicative filter being an intuitionistic fuzzy Boolean filter are constructed. Finally, the concepts of the intuitionistic fuzzy ultra and obstinate filters are introduced, and the intuitionistic fuzzy ultra filter is proved to be equivalent to the intuitionistic fuzzy obstinate filter in BL-algebras.

On fuzzy filters of Heyting-algebras

The concept of fuzzy filter of Heyting-algebras was introduced and some important properties were discussed. Some special kinds of fuzzy filters were defined and we prove that fuzzy Boolean filter is equivelent to fuzzy implicative filter in Heyting-algebras. And the relation among the fuzzy filters were proposed.

On fuzzy filters of pseudo BL-algebras

Abstract We introduce the concept of fuzzy filters of pseudo-BL-algebras and obtain some of their important properties. We further give several kinds of fuzzy filters, such as fuzzy Boolean filter, fuzzy normal filter, fuzzy prime filter, fuzzy ultrafilter and fuzzy obstinate filter and discuss the relations among the above fuzzy filters. We establish the Prime Fuzzy Filter Theorem in pseudo BL-algebras. By studying the relation between fuzzy Boolean filter and fuzzy normal filter in pseudo-BL-algebras, we solve an open problem that whether or not every Boolean filter is normal in pseudo-BL-algebras.

Fuzzy Prime Boolean Filters and Their Operations in IMTL-algebras

Some characterizations of fuzzy prime Boolean filters of IMTL-algebras are given. The lattice operations and the order-reversing involution on the set PB(M) of all fuzzy prime Boolean filters of IMTL-algebras are defined. It is showed that the set PB(M) endowed with these operations is a complete quasi-Boolean algebra (a distributive complete lattice with an order-reversing involution). It is derived that the algebra M/F, which is the set of all cosets of F, is isomorphic to the Boolean algebra {0, 1} if F is a fuzzy prime Boolean filter. By introducing an adjoint pair on PB(M), it is proved that the set PB(M) is also a residuated lattice.

On (∈, ∈ ∨ q)‐fuzzy filters of R0‐algebras

AbstractIn this paper, we introduce the notions of (∈, ∈ ∨ q)‐fuzzy filters and (∈, ∈ ∨ q)‐fuzzy Boolean (implicative) filters in R0‐algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that a fuzzy set in R0‐algebras is an (∈, ∈ ∨ q)‐fuzzy Boolean filter if and only if it is an (∈, ∈ ∨ q)‐fuzzy implicative filter. Finally, we consider the concepts of implication‐based fuzzy Boolean (implicative) filters of R0‐algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Some types of generalized fuzzy filters of MTL-algebras

The aim of this paper is to introduce the concepts of (∈, ∈, ∨q)-fuzzy Boolean filters and (∈, ∈, ∨q)-fuzzy MV-filters in MTL-algebras. Some characterizations of these kinds of generalized fuzzy filters are derived. Finally, the concept of (∈, ∈, ∨q)-fuzzy G-filters in MTL-algebras is introduced. It is proved that an (∈, ∈, ∨q)-fuzzy filter of MTL-algebras is Boolean if and only if it is both an (∈, ∈, ∨q)-fuzzy G-filter and an (∈, ∈, ∨q)-fuzzy MV-filter.

Some types of generalized fuzzy filters of BL-algebras

The aim of this paper is to introduce the notions of interval valued ( ∈ , ∈ ∨ q ) -fuzzy Boolean (implicative, positive implicative and fantastic) filters in B L -algebras and to investigate some related properties. Some characterizations of these generalized fuzzy filters are derived. Finally, the relations among these generalized fuzzy filters are discussed. It is proved that an interval valued ( ∈ , ∈ ∨ q ) -fuzzy filter in B L -algebras is an interval valued ( ∈ , ∈ ∨ q ) -fuzzy Boolean (implicative) filter if and only if it is both an interval valued ( ∈ , ∈ ∨ q ) -fuzzy positive implicative filter and an interval valued ( ∈ , ∈ ∨ q ) -fuzzy fantastic filter.

On ( $$\in, \in \vee q$$ )-Fuzzy Filters of BL-Algebras

The authors introduce the notions of (\(\in, \in \vee q\))-fuzzy Boolean (implicative, positive implicative, and fantastic) filters in BL-algebras, present some characterizations on these generalized fuzzy filters, and describe the relations among these generalized fuzzy filters. It is proved that an (\(\in, \in \vee q\))-fuzzy filter in a BL-algebra is Boolean (implicative) if and only if it is both positive implicative and fantastic.

Generalized fuzzy filters of R 0-algebras

In this paper, we introduce the notions of interval valued $${(\in,\in\vee q)}$$-fuzzy filters and interval valued $${(\in,\in\vee q)}$$-fuzzy Boolean (implicative) filters in R 0-algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that an interval valued fuzzy set F in R 0-algebras is an interval valued $${(\in,\in\vee q)}$$-fuzzy Boolean filter if and only if it is an interval valued $${(\in,\in\vee q)}$$-fuzzy implicative filter.

Fuzzy Boolean and positive implicative filters of [formula omitted]-algebras

The aim of this paper is to introduce the notions of fuzzy Boolean filters and fuzzy positive implicative filters in BL-algebras and to investigate their properties. Several characterizations of fuzzy Boolean filters and fuzzy positive implicative filters are derived. The extension theorems of fuzzy Boolean filters and fuzzy positive implicative filters are obtained. The relation between fuzzy Boolean filters and fuzzy positive implicative filters is investigated and it is proved that every fuzzy Boolean filter is a fuzzy positive implicative filter, but the converse may not be true. Furthermore, the conditions under which a fuzzy positive implicative filter is a fuzzy Boolean filter are established.

Fuzzy implicative and Boolean filters of R0 algebras

The notions of fuzzy filters, fuzzy implicative filters and fuzzy Boolean filters are introduced in R 0 algebras, respectively. Some of their properties are investigated. The relations between fuzzy implicative filters and fuzzy Boolean filters are discussed. The characterizations of fuzzy implicative filters are studied and the following is proved to be equivalent (1) a fuzzy filter f is implicative, (2) f is a fuzzy Boolean filter, (3) ∀ x, y, z∈ L, f( x→ z)⩾ f( x→(¬ z→ y))∧ f( y→ z), (4) ∀ x, y∈ L, f( x→ y)= f( x→(¬ y→ y)), (5) ∀ x∈ L, n⩾1, f( nx)= f( x).