scholarly journals Fuzzy stabilizers in BL-algebras

Filomat ◽  
2018 ◽  
Vol 32 (5) ◽  
pp. 1783-1788 ◽  
Author(s):  
Xiaoyun Cheng ◽  
Xiaolong Xin ◽  
Juntao Wang ◽  
Pengfei He
Keyword(s):  
Fuzzy Logic ◽  
Residuated Lattice ◽  
Algebraic Foundation ◽  
Fuzzy Filters

The primary goal of this paper is to develop fuzzy stabilizer theory in BL-algebras. Two types of fuzzy stabilizers are introduced and their related properties are given. Also, the relationships between fuzzy stabilizers and several fuzzy filters are discussed. Finally, by means of fuzzy stabilizers, it is proven that the collection of all fuzzy filters in BL-algebras forms a residuated lattice. These results will provide a solid algebraic foundation for the consequence connectives in fuzzy logic.

scholarly journals Stabilizers in EQ-algebras

2019 ◽  
Vol 17 (1) ◽  
pp. 998-1013
Author(s):  
Xiao Yun Cheng ◽  
Mei Wang ◽  
Wei Wang ◽  
Jun Tao Wang
Keyword(s):  
Fuzzy Logic ◽  
Left And Right ◽  
Algebraic Foundation ◽  
Complemented Lattice

Abstract The main goal of this paper is to introduce the notion of stabilizers in EQ-algebras and develop stabilizer theory in EQ-algebras. In the paper, we introduce (fuzzy) left and right stabilizers and investigate some related properties of them. Then, we discuss the relations among (fuzzy) stabilizers, (fuzzy) prefilters (filters) and (fuzzy) co-annihilators. Also, we obtain that the set of all prefilters in a good EQ-algebra forms a relative pseudo-complemented lattice, where Str(F, G) is the relative pseudo-complemented of F with respect to G. These results will provide a solid algebraic foundation for the consequence connectives in higher fuzzy logic.

INTUITIONISTIC FUZZY FILTERS OF RESIDUATED LATTICES

2011 ◽  
Vol 07 (03) ◽  
pp. 499-513 ◽  
Author(s):  
SHOKOOFEH GHORBANI
Keyword(s):  
Fuzzy Sets ◽  
Complete Lattice ◽  
Residuated Lattice ◽  
Intuitionistic Fuzzy Sets ◽  
Residuated Lattices ◽  
Intuitionistic Fuzzy ◽  
Correspondence Theorem ◽  
Fuzzy Filters

In this paper, the concept of intuitionistic fuzzy sets is applied to residuated lattices. The notion of intuitionistic fuzzy filters of a residuated lattice is introduced and some related properties are investigated. The characterizations of intuitionistic fuzzy filters are obtained. We show that the set of all the intuitionistic fuzzy filters of a residuated lattice forms a complete lattice and we find the distributive sublattices of it. Finally, the correspondence theorem for intuitionistic fuzzy filters is established.

scholarly journals The Relations between Residuated Frames and Residuated Connections

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 295
Author(s):  
Yong Chan Kim ◽  
Ju-Mok Oh
Keyword(s):  
Fuzzy Logic ◽  
Complete Lattice ◽  
Rough Sets ◽  
Residuated Lattice ◽  
Macneille Completion ◽  
Relational Semantics ◽  
Fuzzy Rough Sets ◽  
Complete Residuated Lattice

We introduce the notion of (dual) residuated frames as a viewpoint of relational semantics for a fuzzy logic. We investigate the relations between (dual) residuated frames and (dual) residuated connections as a topological viewpoint of fuzzy rough sets in a complete residuated lattice. As a result, we show that the Alexandrov topology induced by fuzzy posets is a fuzzy complete lattice with residuated connections. From this result, we obtain fuzzy rough sets on the Alexandrov topology. Moreover, as a generalization of the Dedekind–MacNeille completion, we introduce R-R (resp. D R - D R ) embedding maps and R-R (resp. D R - D R ) frame embedding maps.

Several types of hesitant fuzzy filters on residuated lattices

2020 ◽  
Vol 39 (3) ◽  
pp. 3949-3956
Author(s):  
Zhi Qiang Liu ◽  
Zhen Ming Ma
Keyword(s):  
Residuated Lattice ◽  
Residuated Lattices ◽  
Fuzzy Filter ◽  
One To One ◽  
Boolean Filters ◽  
Fuzzy Filters

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.

Modal operators on bounded commutative residuated ℓ-monoids

2007 ◽  
Vol 57 (4) ◽  
Author(s):  
Jiří Rachůnek ◽  
Dana Šalounová
Keyword(s):  
Fuzzy Logic ◽  
Residuated Lattice ◽  
Heyting Algebras ◽  
Modal Operator ◽  
Modal Operators ◽  
Special Cases ◽  
Basic Fuzzy Logic ◽  
Commutative Residuated Lattice ◽  
Derived Algebras ◽  
Mv Algebras

AbstractBounded commutative residuated lattice ordered monoids (Rℓ-monoids) are a common generalization of, e.g., Heyting algebras and BL-algebras, i.e., algebras of intuitionistic logic and basic fuzzy logic, respectively. Modal operators (special cases of closure operators) on Heyting algebras were studied in [MacNAB, D. S.: Modal operators on Heyting algebras, Algebra Universalis 12 (1981), 5–29] and on MV-algebras in [HARLENDEROVÁ,M.—RACHŮNEK, J.: Modal operators on MV-algebras, Math. Bohem. 131 (2006), 39–48]. In the paper we generalize the notion of a modal operator for general bounded commutative Rℓ-monoids and investigate their properties also for certain derived algebras.

scholarly journals The Lattice of Intuitionistic Fuzzy Filters in Residuated Lattices

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zhen Ming Ma
Keyword(s):  
Distributive Lattice ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Bounded Distributive Lattice ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Filters

The notion of tip-extended pair of intuitionistic fuzzy filters is introduced by which it is proved that the set of all intuitionistic fuzzy filters in a residuated lattice forms a bounded distributive lattice.

Characterizations of hoops based on stabilizers

2020 ◽  
Vol 39 (3) ◽  
pp. 4341-4348
Author(s):  
Jun Tao Wang ◽  
R. A. Borzooei ◽  
M. Aaly Kologani
Keyword(s):  
Fuzzy Logic ◽  
Algebraic Structure ◽  
Open Problem ◽  
Fuzzy Systems ◽  
Wajsberg Hoops ◽  
Left And Right ◽  
Algebraic Foundation ◽  
The Relationship ◽  
Special Classes

In this paper, we characterize the algebraic structure of hoops via stabilizers. First, we further study left and right stabilizers in hoops and discuss the relationship between them. Then, we characterize some special classes of hoops, for example, Wajsberg hoops, local hoops, Gödel hoops and stabilizer hoops, in terms of stabilizers. Finally, we further determine the relationship between stabilizers and filters in hoops and obtain some improvement results. This results also give answer to open problem, which was proposed in [Stabilizers in MTL-algebras, Journal of Intelligent and Fuzzy Systems, 35 (2018) 717-727]. These results will provide a more general algebraic foundation for consequence connectives in fuzzy logic based on continuous t-norms.

scholarly journals Classes of fuzzy filters of residuated lattice ordered monoids

2010 ◽  
Vol 135 (1) ◽  
pp. 81-97
Author(s):  
Jiří Rachůnek ◽  
Dana Šalounová
Keyword(s):  
Residuated Lattice ◽  
Fuzzy Filters

scholarly journals A Theory of Approximate Reasoning with Type-2 Fuzzy Set

2021 ◽  
Vol 27 (1) ◽  
pp. 9-28
Author(s):  
Sudin Mandal ◽  
Injamam Ul Karim ◽  
Swapan Raha
Keyword(s):  
Fuzzy Logic ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Residuated Lattice ◽  
Approximate Reasoning ◽  
Truth Values ◽  
Logic Structure

In this paper, an attempt is made to study approximate reasoning based on a Type-2 fuzzy set theory. In the process, we have examined the underlying fuzzy logic structure on which the reasoning is formulated. We have seen that the partial/incomplete/imprecise truth-values of elements of a type-2 fuzzy set under consideration forms a lattice. We propose two new lattice operations which ultimately help us to define a residual and thereby making the structure of truth- values a residuated lattice. We have focused upon two typical rules of inference used mostly in ordinary approximate reasoning methodology based on Type-1 fuzzy set theory. Our proposal is illustrated with typical artificial examples.

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