Research Article The Relations among Fuzzy -Filters on ...of residuated lattices and B .Let be a...
Transcript of Research Article The Relations among Fuzzy -Filters on ...of residuated lattices and B .Let be a...
Research ArticleThe Relations among Fuzzy ๐ก-Filters on Residuated Lattices
Huarong Zhang1,2 and Qingguo Li1
1 College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China2Department of Mathematics, China Jiliang University, Hangzhou, Zhejiang 310018, China
Correspondence should be addressed to Qingguo Li; [email protected]
Received 8 July 2014; Accepted 21 September 2014; Published 20 October 2014
Academic Editor: Jianming Zhan
Copyright ยฉ 2014 H. Zhang and Q. Li. 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.
We give the simple general principle of studying the relations among fuzzy t-filters on residuated lattices. Using the general principle,we can easily determine the relations among fuzzy t-filters on different logical algebras.
1. Introduction
Residuated lattices, invented by Ward and Dilworth [1],constitute the semantics of Hohleโs Monoidal Logic [2].Residuated lattices are very useful and are basic algebraicstructures. Many logical algebras, such as Boolean algebras,MV-algebras, BL-algebras, MTL-algebras, Godel algebras,NM algebras, and R0-algebras, are particular residuatedlattices. Besides their logical interest, residuated lattices havelots of interesting properties. In [3], Idziak proved that thevarieties of residuated lattices are equational.
Filters play a vital role in investigating logical algebras andthe completeness of the corresponding nonclassical logics.From logical points of view, filters correspond to sets ofprovable formulae. At present, the filter theory on differentlogical algebras has been widely studied. Only on residuatedlattices, such literatures are as follows: [4โ11]. In [8, 9], Ma etal. found the common features of filters on residuated lattices.They, respectively, proposed the notion of ๐-filters and ๐ก-filters on residuated lattices. In [9], Vฤฑta studied some basicproperties of ๐ก-filters and gave the simple general frameworkof special types of filters.
After Zadeh [12] proposed the theory of fuzzy sets, it hasbeen applied to many branches in mathematics. The fuzzifi-cation of the filters was originated in 1995 [13]. Subsequently,a large amount of papers about special types of fuzzy filterswas published in many journals on different logical algebras[10, 11, 14โ24]. In [23], Vฤฑta found the common features offuzzy filters on residuated lattices. He proposed the notionof fuzzy ๐ก-filters and proved its basic properties. However,
the relations among fuzzy ๐ก-filters were not discussed. Usu-ally, when studying the relations among special types of fuzzyfilters, the equivalent characterizations of special types offuzzy filters were firstly discussed. Then, resorting to theproperties of the logical algebras, the relations among specialtypes of fuzzy filters were given. The proofs were tediousin many literatures. The motivation of this paper is to givethe simple general principle of studying the relations amongfuzzy ๐ก-filters on residuated lattices. In contrast to proofs ofparticular results for concrete special types of fuzzy filters,proofs of those general theorems in this paper are simple. Andthe general principle can be applied to all the subvarieties ofresiduated lattices.
2. Preliminary
Definition 1 (see [1, 25]). A residuated lattice is an algebra ๐ฟ =(๐ฟ, โง, โจ, โ, โ , 0, 1) such that for all ๐ฅ, ๐ฆ, ๐ง โ ๐ฟ,
(1) (๐ฟ, โง, โจ, 0, 1) is a bounded lattice;(2) (๐ฟ, โ, 1) is a commutative monoid;(3) (โ, โ ) forms an adjoint pair; that is, ๐ฅ โ ๐ฆ โค ๐ง if and
only if ๐ฅ โค ๐ฆ โ ๐ง.We denote ๐ฅ โ 0 = ๐ฅโ.
Definition 2 (see [11, 25โ30]). Let ๐ฟ be a residuated lattice.Then ๐ฟ is called
(i) an MTL-algebra if (๐ฅ โ ๐ฆ) โจ (๐ฆ โ ๐ฅ) = 1 for all ๐ฅ,๐ฆ โ ๐ฟ (prelinear axiom);
Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 894346, 5 pageshttp://dx.doi.org/10.1155/2014/894346
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(ii) an Rl-monoid if ๐ฅ โง ๐ฆ = ๐ฅ โ (๐ฅ โ ๐ฆ) for all ๐ฅ, ๐ฆ โ ๐ฟ(divisible axiom);
(iii) aHeyting algebra if๐ฅโ๐ฆ = ๐ฅโง๐ฆ for all๐ฅ,๐ฆ โ ๐ฟ, whichis equivalent to an idempotent residuated lattice; thatis, ๐ฅ = ๐ฅ โ ๐ฅ = ๐ฅ2 for ๐ฅ โ ๐ฟ;
(iv) a regular residuated lattice if it satisfies double nega-tion; that is, ๐ฅโโ = ๐ฅ for ๐ฅ โ ๐ฟ;
(v) a BL-algebra if it satisfies both prelinear and divisibleaxioms;
(vi) an MV-algebra if it is a regular Rl-monoid;(vii) a Godel algebra if it is an idempotent BL-algebra;(viii) a Boolean algebra if it is an idempotent MV-algebra;(ix) a R0-algebra (NM algebra) if it satisfies prelinear
axiom, double negation, and (๐ฅโ๐ฆ โ 0)โจ (๐ฅโง๐ฆ โ๐ฅ โ ๐ฆ) = 1.
Definition 3 (see [25, 31, 32]). Let ๐ฟ be a residuated lattice.Then, a nonempty subset ๐น of ๐ฟ is called a filter if
(1) for all ๐ฅ โ ๐น and ๐ฆ โ ๐ฟ, ๐ฅ โค ๐ฆ implies ๐ฆ โ ๐น,(2) for all ๐ฅ, ๐ฆ โ ๐น, ๐ฅ โ ๐ฆ โ ๐น.
Definition 4 (see [5โ11]). Let๐น be a filter of๐ฟ.Then,๐น is called
(i) an implicative filter if ๐ฅ โ ๐ฅ2 โ ๐น for all ๐ฅ โ ๐ฟ,(ii) a regular filter if ๐ฅโโ โ ๐ฅ โ ๐น for all ๐ฅ โ ๐ฟ,(iii) a divisible filter if (๐ฅ โง ๐ฆ) โ (๐ฅ โ (๐ฅ โ ๐ฆ)) โ ๐น for
all ๐ฅ, ๐ฆ โ ๐ฟ,(iv) a prelinear filter if (๐ฅ โ ๐ฆ) โจ (๐ฆ โ ๐ฅ) โ ๐น for all ๐ฅ,๐ฆ โ ๐ฟ,
(v) a Boolean filter if ๐ฅ โจ ๐ฅโ โ ๐น for all ๐ฅ โ ๐ฟ,(vi) a fantastic filter if (๐ฆ โ ๐ฅ) โ (((๐ฅ โ ๐ฆ) โ ๐ฆ) โ๐ฅ) โ ๐น for all ๐ฅ, ๐ฆ โ ๐ฟ,
(vii) an ๐-contractive filter if ๐ฅ๐ โ ๐ฅ๐+1 โ ๐น for all ๐ฅ โ ๐ฟ,where ๐ฅ๐+1 = ๐ฅ๐ โ ๐ฅ, ๐ โฅ 1.
Remark 5. On residuated lattices, ๐ฅ โ (๐ฆ โ ๐ง) = ๐ฆ โ(๐ฅ โ ๐ง) holds (see [31]). Using these properties, we havethat ๐น is a fantastic filter if ((๐ฅ โ ๐ฆ) โ ๐ฆ) โ ((๐ฆ โ๐ฅ) โ ๐ฅ) โ ๐น.
We now review some fuzzy concepts. A fuzzy set onresiduated lattice is a function ๐ : ๐ฟ โ [0, 1]. For any๐ผ โ [0, 1] and an arbitrary fuzzy set ๐, we denote the set{๐ฅ โ ๐ฟ | ๐(๐ฅ) โฅ ๐ผ} (i.e., the ๐ผ-cut) by the symbol ๐
๐ผ.
Definition 6 (see [10, 11]). A fuzzy set ๐ is a fuzzy filter on ๐ฟif and only if it satisfies the following two conditions for all๐ฅ, ๐ฆ โ ๐ฟ:
(1) ๐(๐ฅ โ ๐ฆ) โฅ min{๐(๐ฅ), ๐(๐ฆ)},(2) if ๐ฅ โค ๐ฆ, then ๐(๐ฅ) โค ๐(๐ฆ).
In the following, by the symbol ๐ฅ we denote the abbrevi-ation of ๐ฅ, ๐ฆ, . . .; that is, ๐ฅ is a formal listing of variables usedin a given content. By the term ๐ก, it is always meant as a termin the language of residuated lattices.
Definition 7 (see [9]). Let ๐ก be an arbitrary term on thelanguage of residuated lattices. A filter ๐น on ๐ฟ is a ๐ก-filter if๐ก(๐ฅ) โ ๐น for all ๐ฅ โ ๐ฟ.
Definition 8 (see [23]). A fuzzy filter ๐ on ๐ฟ is called a fuzzy๐ก-filter on ๐ฟ, if for all ๐ฅ โ ๐ฟ it satisfies ๐(๐ก(๐ฅ)) = ๐(1).
Example 9 (see [11]). Fuzzy Boolean filters are fuzzy ๐ก-filtersfor ๐ก equal to ๐ฅ โจ ๐ฅโ.
Example 10 (see [11]). Fuzzy regular filters are fuzzy ๐ก-filtersfor ๐ก equal to ๐ฅโโ โ ๐ฅ.
Example 11 (see [11]). Fuzzy fantastic filters are fuzzy ๐ก-filtersfor ๐ก equal to ((๐ฅ โ ๐ฆ) โ ๐ฆ) โ ((๐ฆ โ ๐ฅ) โ ๐ฅ).
Remark 12. Using the notion of fuzzy ๐ก-filter, fuzzy implica-tive filters are fuzzy ๐ก-filters for ๐ก equal to ๐ฅ โ ๐ฅ2. Fuzzydivisible filters are fuzzy ๐ก-filters for ๐ก equal to (๐ฅ โง ๐ฆ) โ(๐ฅ โ (๐ฅ โ ๐ฆ)) and so forth.
Let us assume that since now, ๐ก is an arbitrary term inthe language of residuated lattices.We also use another usefulconvention: given a varietyB of residuated lattices, we denoteits subvariety given by the equation ๐ก = 1 by the symbol B[๐ก];we call this algebra ๐ก-algebra.
The next part of this paper concerns fuzzy quotient con-structions. We recall some known results and constructionsconcerning fuzzy quotients residuated lattices.
Theorem 13 (see [11]). Let ๐ be a fuzzy filter on ๐ฟ and ๐ฅ, ๐ฆ โ ๐ฟ.For any ๐ง โ ๐ฟ, we define๐๐ฅ : ๐ฟ โ [0, 1],๐๐ฅ(๐ง) = min{๐(๐ฅ โ๐ง), ๐(๐ง โ ๐ฅ)}. Then, ๐๐ฅ = ๐๐ฆ if and only if ๐(๐ฅ โ ๐ฆ) =๐(๐ฆ โ ๐ฅ) = ๐(1).
Theorem 14 (see [11]). Let ๐ be a fuzzy filter on ๐ฟ and ๐ฟ/๐ :={๐๐ฅ
| ๐ฅ โ ๐ฟ}. For any ๐๐ฅ, ๐๐ฆ โ ๐ฟ/๐, we define๐๐ฅ
โง ๐๐ฆ
= ๐๐ฅโง๐ฆ,
๐๐ฅ
โจ ๐๐ฆ
= ๐๐ฅโจ๐ฆ,
๐๐ฅ
โ ๐๐ฆ
= ๐๐ฅโ๐ฆ,
๐๐ฅ
โ ๐๐ฆ
= ๐๐ฅโ๐ฆ.
Then, ๐ฟ/๐ = (๐ฟ/๐, โง, โจ, โ, โ , ๐0, ๐1) is a residuated latticecalled the fuzzy quotient residuated lattice.
Theorem 15 (quotient characteristics [23]). LetB be a varietyof residuated lattices and ๐ฟ โ B. Let ๐ be a fuzzy filter on ๐ฟ.Then, the fuzzy quotient ๐ฟ/๐ belongs to B[๐ก] if and only if ๐ isa fuzzy ๐ก-filter on ๐ฟ.
3. The General Principle of the Relationamong Fuzzy ๐ก-Filters and Its Application
In the following, letB be a variety of residuated lattices. ๐ฟ โ Band ๐ is a fuzzy filter on ๐ฟ.
Theorem 16. Suppose that there are fuzzy ๐ก1-filter and fuzzy
๐ก2-filter on ๐ฟ and B[๐ก
1] โ B[๐ก
2]. If ๐ is a fuzzy ๐ก
1-filter, then ๐
is a fuzzy ๐ก2-filter.
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Proof. ๐ is a fuzzy ๐ก1-filterโ ๐ฟ/๐ โ B[๐ก
1] โ ๐ฟ/๐ โ B[๐ก
2] โ
๐ is a fuzzy ๐ก2-filter.
Theorem 17. Suppose there are fuzzy ๐ก1-filter and fuzzy ๐ก
2-
filter on ๐ฟ. If B[๐ก1] = B[๐ก
2], then ๐ is a fuzzy ๐ก
1-filter if and
only if ๐ is a fuzzy ๐ก2-filter.
Proof. ๐ is a fuzzy ๐ก1-filterโ ๐ฟ/๐ โ B[๐ก
1] โ ๐ฟ/๐ โ B[๐ก
2] โ
๐ is a fuzzy ๐ก2-filter.
Remark 18. The above results give the general principle ofthe relations among fuzzy ๐ก-filters. If we want to judge therelations among fuzzy ๐ก-filter, we only resort to the relationsamong ๐ก-algebras. Since the relations among ๐ก-algebras areknown to us, we can easily obtain the relations among fuzzy๐ก-filters.
Theorem 19. Let ๐ฟ be a residuated lattice. If ๐ is a fuzzyimplicative filter, then ๐ is a fuzzy ๐-contractive filter.
Proof. It is obvious that B[๐ฅ โ ๐ฅ2] โ B[๐ฅ๐ โ ๐ฅ๐+1]. ByTheorem 16, the result is clear.
Lemma 20 (see [5, 27]). Let ๐ฟ be a residuated lattice. If ๐ฟ is aHeyting algebra, then ๐ฟ is an Rl-monoid.
Lemma 21 (see [11]). Let ๐ฟ be a residuated lattice. Then thefollowing are equivalent:
(1) ๐ฟ is an MV-algebra;(2) (๐ฅ โ ๐ฆ) โ ๐ฆ = (๐ฆ โ ๐ฅ) โ ๐ฅ, โ๐ฅ, ๐ฆ โ ๐ฟ.
Lemma 22 (see [25]). Let ๐ฟ be a residuated lattice. Then ๐ฟ isan MV-algebra if and only if ๐ฟ is a regular BL-algebra.
Lemma 23. Let ๐ฟ be a residuated lattice. Then the followingare equivalent:
(1) ๐ฟ is a Boolean algebra;(2) ๐ฅ โจ ๐ฅโ = 1, โ๐ฅ โ ๐ฟ;(3) ๐ฟ is regular and idempotent.
Proof. (1)โ(2) Reference [11], Proposition 2.10.(1)โ(3) If ๐ฟ is a Boolean algebra, then ๐ฟ is an idempotent
MV-algebra. Thus, ๐ฟ is regular and idempotent.(3)โ(1) If ๐ฟ is regular and idempotent, then ๐ฟ is a regular
Rl-monoid. Thus, ๐ฟ is an MV-algebra. Also, ๐ฟ is idempotent;therefore, ๐ฟ is a Boolean algebra.
Lemma 24. Let ๐ฟ be a residuated lattice. Then the followingare equivalent:
(1) ๐ฟ is a Boolean algebra;(2) ๐ฟ is an idempotent R0-algebra.
Proof. (1)โ(2) If๐ฟ is a Boolean algebra, then๐ฟ is a R0-algebra([30], Example 8.5.1) and ๐ฟ is idempotent.
(2)โ(1) Suppose ๐ฟ is an idempotent R0-algebra. Since ๐ฟis a R0-algebra, then ๐ฟ is regular. By Lemma 23, we have that๐ฟ is a Boolean algebra.
Lemma25. Let๐ฟ be a residuated lattice. ๐ก1and ๐ก2are arbitrary
terms on ๐ฟ. Then, B[๐ก1โ ๐ก2] = B[๐ก
1] โฉ B[๐ก
2].
Proof. ๐ฟ โ B[๐ก1โ ๐ก2] โ ๐ฟ โ B and ๐ก
1โ ๐ก2= 1 โ ๐ฟ โ B;
๐ก1= 1 and ๐ก
2= 1 โ ๐ฟ โ B[๐ก
1] โฉ B[๐ก
2].
Theorem 26. Let ๐ฟ be a residuated lattice. Then, ๐ is a fuzzyBoolean filter if and only if ๐ is both a fuzzy regular and a fuzzyimplicative filter.
Proof. ๐ is a fuzzy Boolean filterโ ๐ฟ/๐ โ B[๐ฅโจ๐ฅโ] โ ๐ฟ/๐ โB[(๐ฅโโ โ ๐ฅ)โ (๐ฅ โ ๐ฅ2)] โ ๐ฟ/๐ โ B[๐ฅโโ โ ๐ฅ]โฉB[๐ฅ โ
๐ฅ2
] โ ๐ฟ/๐ โ B[๐ฅโโ โ ๐ฅ] and ๐ฟ/๐ โ B[๐ฅ โ ๐ฅ2] โ ๐ isboth a fuzzy regular and a fuzzy implicative filter.
Remark 27. Using the same method, we can easily obtain thefollowing results.
Theorem 28. Let ๐ฟ be a residuated lattice. Then
(1) ๐ is a fuzzy Boolean filter if and only if ๐ is a fuzzyfantastic and fuzzy implicative filter;
(2) ๐ is a fuzzy fantastic filter if and only if ๐ is a fuzzyregular and fuzzy divisible filter;
(3) every fuzzy implicative filter is a fuzzy divisible one;(4) if ๐ is a fuzzy prelinear filter, then ๐ is a fuzzy fantastic
filter if and only if ๐ is both a fuzzy regular and a fuzzydivisible filter;
(5) if ๐ is a fuzzy Boolean filter, then ๐ is a fuzzy ๐-contractive filter.
Remark 29. The notion of fuzzy ๐ก-filter and the general prin-ciple are not only applicable on residuated lattices, but alsoeven transferable to all their subvarieties. Taking advantageof the relations among ๐ก-algebras, we can easily obtain thefollowing results.
Theorem 30. Let ๐ฟ be a Boolean-algebra. Then the fuzzyprelinear filter, fuzzy fantastic filter, fuzzy divisible filter, fuzzyregular filter, and fuzzy ๐-contractive and fuzzy Boolean filtercoincide.
Theorem 31. Let ๐ฟ be an MV-algebra. Then
(1) ๐ is a fuzzy Boolean filter if and only if ๐ is a fuzzyimplicative filter;
(2) the fuzzy prelinear filter, fuzzy fantastic filter, fuzzydivisible filter, and fuzzy regular filter coincide.
Theorem 32. Let ๐ฟ be a Godel-algebra. Then
(1) ๐ is a fuzzy Boolean filter if and only if ๐ is a fuzzyregular filter if and only if ๐ is a fuzzy fantastic filter;
(2) the fuzzy prelinear filter, fuzzy divisible filter, fuzzy ๐-contractive filter, and fuzzy implicative filter coincide.
Theorem 33. Let ๐ฟ be a BL-algebra; then
(1) ๐ is a fuzzy Boolean filter if and only if ๐ is both a fuzzyimplicative and a fuzzy regular filter;
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(2) ๐ is a fuzzy Boolean filter if and only if ๐ is both a fuzzyimplicative and a fuzzy fantastic filter;
(3) ๐ is a fuzzy fantastic filter if and only if ๐ is a fuzzyregular filter;
(4) the fuzzy prelinear filter and fuzzy divisible filtercoincide.
Theorem 34. Let ๐ฟ be an MTL-algebra. Then,
(1) ๐ is a fuzzy Boolean filter if and only if ๐ is both a fuzzyimplicative and a fuzzy regular filter;
(2) ๐ is a fuzzy Boolean filter if and only if ๐ is both a fuzzyimplicative and a fuzzy fantastic filter;
(3) ๐ is a fantastic filter if and only if ๐ is a regular anddivisible filter;
(4) if๐ is a fuzzy implicative filter, then๐ is a fuzzy divisiblefilter.
Theorem 35. Let ๐ฟ be a Heyting-algebra. Then
(1) ๐ is a fuzzy Boolean filter if and only if ๐ is a fuzzyregular filter if and only if ๐ is a fuzzy fantastic filter;
(2) the fuzzy implicative filter, fuzzy divisible filter, andfuzzy ๐-contractive filter coincide.
Theorem 36. Let ๐ฟ be a R0-algebra; then
(1) ๐ is a fuzzy Boolean filter if and only if ๐ is a fuzzyimplicative filter;
(2) every fuzzy Boolean filter is a fuzzy fantastic filter;(3) ๐ is a fuzzy fantastic filter if and only if ๐ is a fuzzy
divisible filter;(4) the fuzzy prelinear filter and fuzzy regular filter coin-
cide;(5) every fuzzy implicative filter is a fuzzy divisible one.
Theorem 37. Let ๐ฟ be a regular residuated lattice. Then
(1) ๐ is a fuzzy Boolean filter if and only if ๐ is a fuzzyimplicative filter;
(2) ๐ is a fuzzy fantastic filter if and only if ๐ is a fuzzydivisible filter;
(3) every fuzzy Boolean filter is a fuzzy fantastic filter;(4) every fuzzy implicative filter is a fuzzy divisible one.
Theorem 38. Let ๐ฟ be a Rl-monoid. Then
(1) ๐ is a fuzzy Boolean filter if and only if ๐ is a fuzzyimplicative and fuzzy fantastic filter;
(2) ๐ is a fuzzy Boolean filter if and only if ๐ is a fuzzyimplicative and fuzzy regular filter;
(3) ๐ is a fuzzy fantastic filter if and only if ๐ is a fuzzyregular filter;
(4) every fuzzy implicative filter is a fuzzy divisible one.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China, Grant nos. 11371130 and 61273018, andby the Research Fund for the Doctoral Program of HigherEducation of China, Grant no. 20120161110017.
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