The Lattice of  Intuitionistic Fuzzy Filters in Residuated Lattices

Ma, Zhen Ming

doi:https://doi.org/10.1155/2014/798670

Journal of Applied Mathematics

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Abstract Introduction Preliminaries Conclusions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 798670 | https://doi.org/10.1155/2014/798670

The Lattice of Intuitionistic Fuzzy Filters in Residuated Lattices

Zhen Ming Ma¹

Academic Editor: Ch. Tsitouras

Received06 Nov 2013

Accepted10 Apr 2014

Published29 Apr 2014

Abstract

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.

1. Introduction

Nowadays, it is generally accepted that in fuzzy logic the algebraic structure should be a residuated lattice which was introduced by Ward and Dilworth [1]. Some other logical algebras such as MTL-algebras [2], BL-algebras [3], MV-algebras [4], G-algebras, -algebras, and NM-algebras [2], which are also called -algebras [5], are all able to be considered particular classes of residuated lattices. (For details, see, e.g., [6].)

Filters are an important tool to study these logical algebras and the completeness of the corresponding nonclassical logics. On the one hand, filters are closely related to congruence relations with which one can associate quotient algebras [7]; on the other hand, various filters correspond to various sets of provable formula [3, 4]. A filter is also called a deductive system in BL-algebras [8]. It has been widely investigated in residuated lattices [7, 9–11] and particular residuated lattices [2, 3, 6, 8, 12–15].

Since Rosenfeld [16] applied the notion of fuzzy sets [17] to abstract algebra and introduced the notion of fuzzy subgroups, the literature of various fuzzy algebraic concepts has been growing very rapidly [18]. In particular, in [19–21], the notion of tip-extended pair of fuzzy sets was introduced to investigate the lattices of all fuzzy normal subgroups and -ideals.

The notion of fuzzy filters was introduced, and some properties of them were obtained [22–24]. Moreover, based on the notion of intuitionistic fuzzy sets (IFS) proposed by Atanassov [25], the concept of the intuitionistic fuzzy filter in BL-algebras was introduced in [26]. However, the study of residuated lattices from the point of lattice theory is less frequent.

In this paper, the intuitionistic fuzzy filter theory in residuated lattices is developed. This paper is organized as follows: in Section 2, some basic concepts and properties of intuitionistic fuzzy sets and intuitionistic fuzzy filters in residuated lattices are recalled. In Section 3, by introducing the notion of tip-extended pair of intuitionistic fuzzy filters, it is proved that the set of all intuitionistic fuzzy filters forms a bounded distributive lattice. The last section concludes this paper.

2. Preliminaries

The concepts of residuated lattices and intuitionistic fuzzy filters will be extensively used in the sequel. Therefore, we recall their definitions and summarize their main properties.

Let . A mapping is called a fuzzy set [17]. Let and be fuzzy sets on . Then tip-extended pair of and [19, 20] can be defined by

Let be two fuzzy sets satisfying for all . Then is called an intuitionistic fuzzy set [25] (or equivalently denoted by ). The family of all intuitionistic fuzzy sets on will be denoted by .

Basic operations on intuitionistic fuzzy sets are defined in the following way.

Let . One has

Definition 1 (see [3]). A residuated lattice is an algebra such that is a bounded lattice with the least element 0 and the greatest element 1, is a commutative monoid, and forms an adjoint pair; that is, if and only if for all .

A nonempty subset of is called a filter of if (i) ; (ii) implies or, equivalently, (iii) ; and (iv) implies .

The following alternative definitions of intuitionistic fuzzy filters were proved in [26], but they can be similarly verified in residuated lattices.

Definition 2. Let . Then is called an intuitionistic fuzzy filter if (1), for all ;(2) for all ;(3) for all .

The set of all intuitionistic fuzzy filters on a residuated lattice will be denoted by .

Theorem 3. Let . Then is an intuitionistic fuzzy filter if and only if implies and for all .

Theorem 4. Let . Then is an intuitionistic fuzzy filter if and only if the following assertions hold: (1) implies and for all ;(2) and for all .

3. Lattice of Intuitionistic Fuzzy Filters

In this section, we mainly investigate the lattice of all intuitionistic fuzzy filters by introducing the notion of tip-extended pair of intuitionistic fuzzy sets.

The following lemma is obvious but necessary.

Lemma 5. Let be intuitionistic fuzzy filters of . Then so is .

For , the intersection of all intuitionistic fuzzy filters containing is called the generated intuitionistic fuzzy filter by , denoted as .

Theorem 6. Let . Define a new intuitionistic fuzzy set by where for all . Then .

Proof. We complete the proof by two steps. Firstly, we verify that is an intuitionistic fuzzy filter. For all , such that , the definition of yields that and . For all , we have Thus is an intuitionistic fuzzy filter.
Secondly, let be an intuitionistic fuzzy filter such that . By the definition of intuitionistic fuzzy filter, it holds that and hence . Thus .

Example 7. Let with . The operations and are defined as Define as , and . Since is not a fuzzy filter. It is routine to verify that is an intuitionistic fuzzy filter, where , and from the above theorem.

Lemma 8. Let such that and . Then .

Proof. Not losing the generality, we assume that . Then . It is obvious that . Thus it holds that .

Theorem 9. Let be an intuitionistic fuzzy filter of and for all such that . Then is an intuitionistic fuzzy filter, where

Proof. It follows from Lemma 8 that . Now we prove that is an intuitionistic fuzzy filter.
If , we consider the following two cases:
Case 1 (). It is obvious that , .
Case 2 (). The definition of leads that , .
Thus , .
Let . We consider the following two cases:
Case 1 (). If , it is obvious that .
If or , it is a contradiction.
If , it holds that .
Case 2 (). If , it is a contradiction.
If or , it is obvious that .
If ,we have .
All in all, it yields that .
Thus is an intuitionistic fuzzy filter.

For given , the operation is defined by where

Furthermore, the tip-extended pair for intuitionistic fuzzy sets and are defined by where where

Theorem 10. Let . Then .

Proof. It is obvious that is order-preserving, and is order-reserving. For all , we have and hence .
Thus .

Theorem 11. Let . Then .

Proof. It is easy to prove that , and hence . Thus .
Assume that such that . If , we have . If , it holds that It follows from Theorem 10 that .

Associating with the above results, we prove the main theorem here. For , the operations and on are defined by

Theorem 12. is a bounded distributive lattice.

Proof. We only verify the distributivity. Obviously, it holds that , so we only prove that . Assume that for , we consider the following two cases.
Case 1 (). We have
Case 2 (). It holds that Let and ; it is easy to verify that , and then the above can be written as Let and ; it is easy to verify that , and then the above can be written as
Thus and ; that is, the distributivity holds.

4. Conclusions

In this paper, by the notion of tip-extended pair of intuitionistic fuzzy sets, it is verified that the set of all intuitionistic fuzzy filters forms a bounded distributive lattice.

Future research will focus on the lattice of fuzzy notions in other algebras by tip-extended pair.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by AMEP of Linyi University and the Natural Science Foundation of Shandong Province (Grant no. ZR2013FL006).

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Copyright

Copyright © 2014 Zhen Ming Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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