Research ArticleThe Relations among Fuzzy π‘-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 Β© 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 Hohleβs 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: [4β11]. In [8, 9], Ma etal. found the common features of filters on residuated lattices.They, respectively, proposed the notion of π-filters and π‘-filters on residuated lattices. In [9], VΔ±ta studied some basicproperties of π‘-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, 14β24]. In [23], VΔ±ta found the common features offuzzy filters on residuated lattices. He proposed the notionof fuzzy π‘-filters and proved its basic properties. However,
the relations among fuzzy π‘-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 π‘-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 πΏ =(πΏ, β§, β¨, β, β , 0, 1) such that for all π₯, π¦, π§ β πΏ,
(1) (πΏ, β§, β¨, 0, 1) is a bounded lattice;(2) (πΏ, β, 1) is a commutative monoid;(3) (β, β ) forms an adjoint pair; that is, π₯ β π¦ β€ π§ if and
only if π₯ β€ π¦ β π§.We denote π₯ β 0 = π₯β.
Definition 2 (see [11, 25β30]). Let πΏ be a residuated lattice.Then πΏ is called
(i) an MTL-algebra if (π₯ β π¦) β¨ (π¦ β π₯) = 1 for all π₯,π¦ β πΏ (prelinear axiom);
Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 894346, 5 pageshttp://dx.doi.org/10.1155/2014/894346
2 The Scientific World Journal
(ii) an Rl-monoid if π₯ β§ π¦ = π₯ β (π₯ β π¦) for all π₯, π¦ β πΏ(divisible axiom);
(iii) aHeyting algebra ifπ₯βπ¦ = π₯β§π¦ for allπ₯,π¦ β πΏ, whichis equivalent to an idempotent residuated lattice; thatis, π₯ = π₯ β π₯ = π₯2 for π₯ β πΏ;
(iv) a regular residuated lattice if it satisfies double nega-tion; that is, π₯ββ = π₯ for π₯ β πΏ;
(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 (π₯βπ¦ β 0)β¨ (π₯β§π¦ βπ₯ β π¦) = 1.
Definition 3 (see [25, 31, 32]). Let πΏ be a residuated lattice.Then, a nonempty subset πΉ of πΏ is called a filter if
(1) for all π₯ β πΉ and π¦ β πΏ, π₯ β€ π¦ implies π¦ β πΉ,(2) for all π₯, π¦ β πΉ, π₯ β π¦ β πΉ.
Definition 4 (see [5β11]). LetπΉ be a filter ofπΏ.Then,πΉ is called
(i) an implicative filter if π₯ β π₯2 β πΉ for all π₯ β πΏ,(ii) a regular filter if π₯ββ β π₯ β πΉ for all π₯ β πΏ,(iii) a divisible filter if (π₯ β§ π¦) β (π₯ β (π₯ β π¦)) β πΉ for
all π₯, π¦ β πΏ,(iv) a prelinear filter if (π₯ β π¦) β¨ (π¦ β π₯) β πΉ for all π₯,π¦ β πΏ,
(v) a Boolean filter if π₯ β¨ π₯β β πΉ for all π₯ β πΏ,(vi) a fantastic filter if (π¦ β π₯) β (((π₯ β π¦) β π¦) βπ₯) β πΉ for all π₯, π¦ β πΏ,
(vii) an π-contractive filter if π₯π β π₯π+1 β πΉ for all π₯ β πΏ,where π₯π+1 = π₯π β π₯, π β₯ 1.
Remark 5. On residuated lattices, π₯ β (π¦ β π§) = π¦ β(π₯ β π§) holds (see [31]). Using these properties, we havethat πΉ is a fantastic filter if ((π₯ β π¦) β π¦) β ((π¦ βπ₯) β π₯) β πΉ.
We now review some fuzzy concepts. A fuzzy set onresiduated lattice is a function π : πΏ β [0, 1]. For anyπΌ β [0, 1] and an arbitrary fuzzy set π, we denote the set{π₯ β πΏ | π(π₯) β₯ πΌ} (i.e., the πΌ-cut) by the symbol π
πΌ.
Definition 6 (see [10, 11]). A fuzzy set π is a fuzzy filter on πΏif and only if it satisfies the following two conditions for allπ₯, π¦ β πΏ:
(1) π(π₯ β π¦) β₯ min{π(π₯), π(π¦)},(2) if π₯ β€ π¦, then π(π₯) β€ π(π¦).
In the following, by the symbol π₯ we denote the abbrevi-ation of π₯, π¦, . . .; that is, π₯ is a formal listing of variables usedin a given content. By the term π‘, it is always meant as a termin the language of residuated lattices.
Definition 7 (see [9]). Let π‘ be an arbitrary term on thelanguage of residuated lattices. A filter πΉ on πΏ is a π‘-filter ifπ‘(π₯) β πΉ for all π₯ β πΏ.
Definition 8 (see [23]). A fuzzy filter π on πΏ is called a fuzzyπ‘-filter on πΏ, if for all π₯ β πΏ it satisfies π(π‘(π₯)) = π(1).
Example 9 (see [11]). Fuzzy Boolean filters are fuzzy π‘-filtersfor π‘ equal to π₯ β¨ π₯β.
Example 10 (see [11]). Fuzzy regular filters are fuzzy π‘-filtersfor π‘ equal to π₯ββ β π₯.
Example 11 (see [11]). Fuzzy fantastic filters are fuzzy π‘-filtersfor π‘ equal to ((π₯ β π¦) β π¦) β ((π¦ β π₯) β π₯).
Remark 12. Using the notion of fuzzy π‘-filter, fuzzy implica-tive filters are fuzzy π‘-filters for π‘ equal to π₯ β π₯2. Fuzzydivisible filters are fuzzy π‘-filters for π‘ equal to (π₯ β§ π¦) β(π₯ β (π₯ β π¦)) and so forth.
Let us assume that since now, π‘ 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 π‘ = 1 by the symbol B[π‘];we call this algebra π‘-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 π be a fuzzy filter on πΏ and π₯, π¦ β πΏ.For any π§ β πΏ, we defineππ₯ : πΏ β [0, 1],ππ₯(π§) = min{π(π₯ βπ§), π(π§ β π₯)}. Then, ππ₯ = ππ¦ if and only if π(π₯ β π¦) =π(π¦ β π₯) = π(1).
Theorem 14 (see [11]). Let π be a fuzzy filter on πΏ and πΏ/π :={ππ₯
| π₯ β πΏ}. For any ππ₯, ππ¦ β πΏ/π, we defineππ₯
β§ ππ¦
= ππ₯β§π¦,
ππ₯
β¨ ππ¦
= ππ₯β¨π¦,
ππ₯
β ππ¦
= ππ₯βπ¦,
ππ₯
β ππ¦
= ππ₯βπ¦.
Then, πΏ/π = (πΏ/π, β§, β¨, β, β , π0, π1) is a residuated latticecalled the fuzzy quotient residuated lattice.
Theorem 15 (quotient characteristics [23]). LetB be a varietyof residuated lattices and πΏ β B. Let π be a fuzzy filter on πΏ.Then, the fuzzy quotient πΏ/π belongs to B[π‘] if and only if π isa fuzzy π‘-filter on πΏ.
3. The General Principle of the Relationamong Fuzzy π‘-Filters and Its Application
In the following, letB be a variety of residuated lattices. πΏ β Band π is a fuzzy filter on πΏ.
Theorem 16. Suppose that there are fuzzy π‘1-filter and fuzzy
π‘2-filter on πΏ and B[π‘
1] β B[π‘
2]. If π is a fuzzy π‘
1-filter, then π
is a fuzzy π‘2-filter.
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Proof. π is a fuzzy π‘1-filterβ πΏ/π β B[π‘
1] β πΏ/π β B[π‘
2] β
π is a fuzzy π‘2-filter.
Theorem 17. Suppose there are fuzzy π‘1-filter and fuzzy π‘
2-
filter on πΏ. If B[π‘1] = B[π‘
2], then π is a fuzzy π‘
1-filter if and
only if π is a fuzzy π‘2-filter.
Proof. π is a fuzzy π‘1-filterβ πΏ/π β B[π‘
1] β πΏ/π β B[π‘
2] β
π is a fuzzy π‘2-filter.
Remark 18. The above results give the general principle ofthe relations among fuzzy π‘-filters. If we want to judge therelations among fuzzy π‘-filter, we only resort to the relationsamong π‘-algebras. Since the relations among π‘-algebras areknown to us, we can easily obtain the relations among fuzzyπ‘-filters.
Theorem 19. Let πΏ be a residuated lattice. If π is a fuzzyimplicative filter, then π is a fuzzy π-contractive filter.
Proof. It is obvious that B[π₯ β π₯2] β B[π₯π β π₯π+1]. ByTheorem 16, the result is clear.
Lemma 20 (see [5, 27]). Let πΏ be a residuated lattice. If πΏ is aHeyting algebra, then πΏ is an Rl-monoid.
Lemma 21 (see [11]). Let πΏ be a residuated lattice. Then thefollowing are equivalent:
(1) πΏ is an MV-algebra;(2) (π₯ β π¦) β π¦ = (π¦ β π₯) β π₯, βπ₯, π¦ β πΏ.
Lemma 22 (see [25]). Let πΏ be a residuated lattice. Then πΏ isan MV-algebra if and only if πΏ is a regular BL-algebra.
Lemma 23. Let πΏ be a residuated lattice. Then the followingare equivalent:
(1) πΏ is a Boolean algebra;(2) π₯ β¨ π₯β = 1, βπ₯ β πΏ;(3) πΏ is regular and idempotent.
Proof. (1)β(2) Reference [11], Proposition 2.10.(1)β(3) If πΏ is a Boolean algebra, then πΏ is an idempotent
MV-algebra. Thus, πΏ is regular and idempotent.(3)β(1) If πΏ is regular and idempotent, then πΏ is a regular
Rl-monoid. Thus, πΏ is an MV-algebra. Also, πΏ is idempotent;therefore, πΏ is a Boolean algebra.
Lemma 24. Let πΏ be a residuated lattice. Then the followingare equivalent:
(1) πΏ is a Boolean algebra;(2) πΏ is an idempotent R0-algebra.
Proof. (1)β(2) IfπΏ is a Boolean algebra, thenπΏ is a R0-algebra([30], Example 8.5.1) and πΏ is idempotent.
(2)β(1) Suppose πΏ is an idempotent R0-algebra. Since πΏis a R0-algebra, then πΏ is regular. By Lemma 23, we have thatπΏ is a Boolean algebra.
Lemma25. LetπΏ be a residuated lattice. π‘1and π‘2are arbitrary
terms on πΏ. Then, B[π‘1β π‘2] = B[π‘
1] β© B[π‘
2].
Proof. πΏ β B[π‘1β π‘2] β πΏ β B and π‘
1β π‘2= 1 β πΏ β B;
π‘1= 1 and π‘
2= 1 β πΏ β B[π‘
1] β© B[π‘
2].
Theorem 26. Let πΏ be a residuated lattice. Then, π is a fuzzyBoolean filter if and only if π is both a fuzzy regular and a fuzzyimplicative filter.
Proof. π is a fuzzy Boolean filterβ πΏ/π β B[π₯β¨π₯β] β πΏ/π βB[(π₯ββ β π₯)β (π₯ β π₯2)] β πΏ/π β B[π₯ββ β π₯]β©B[π₯ β
π₯2
] β πΏ/π β B[π₯ββ β π₯] and πΏ/π β B[π₯ β π₯2] β π isboth a fuzzy regular and a fuzzy implicative filter.
Remark 27. Using the same method, we can easily obtain thefollowing results.
Theorem 28. Let πΏ be a residuated lattice. Then
(1) π is a fuzzy Boolean filter if and only if π is a fuzzyfantastic and fuzzy implicative filter;
(2) π is a fuzzy fantastic filter if and only if π is a fuzzyregular and fuzzy divisible filter;
(3) every fuzzy implicative filter is a fuzzy divisible one;(4) if π is a fuzzy prelinear filter, then π is a fuzzy fantastic
filter if and only if π is both a fuzzy regular and a fuzzydivisible filter;
(5) if π is a fuzzy Boolean filter, then π is a fuzzy π-contractive filter.
Remark 29. The notion of fuzzy π‘-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 π‘-algebras, we can easily obtain thefollowing results.
Theorem 30. Let πΏ be a Boolean-algebra. Then the fuzzyprelinear filter, fuzzy fantastic filter, fuzzy divisible filter, fuzzyregular filter, and fuzzy π-contractive and fuzzy Boolean filtercoincide.
Theorem 31. Let πΏ be an MV-algebra. Then
(1) π is a fuzzy Boolean filter if and only if π is a fuzzyimplicative filter;
(2) the fuzzy prelinear filter, fuzzy fantastic filter, fuzzydivisible filter, and fuzzy regular filter coincide.
Theorem 32. Let πΏ be a Godel-algebra. Then
(1) π is a fuzzy Boolean filter if and only if π is a fuzzyregular filter if and only if π is a fuzzy fantastic filter;
(2) the fuzzy prelinear filter, fuzzy divisible filter, fuzzy π-contractive filter, and fuzzy implicative filter coincide.
Theorem 33. Let πΏ be a BL-algebra; then
(1) π is a fuzzy Boolean filter if and only if π is both a fuzzyimplicative and a fuzzy regular filter;
4 The Scientific World Journal
(2) π is a fuzzy Boolean filter if and only if π is both a fuzzyimplicative and a fuzzy fantastic filter;
(3) π is a fuzzy fantastic filter if and only if π is a fuzzyregular filter;
(4) the fuzzy prelinear filter and fuzzy divisible filtercoincide.
Theorem 34. Let πΏ be an MTL-algebra. Then,
(1) π is a fuzzy Boolean filter if and only if π is both a fuzzyimplicative and a fuzzy regular filter;
(2) π is a fuzzy Boolean filter if and only if π is both a fuzzyimplicative and a fuzzy fantastic filter;
(3) π is a fantastic filter if and only if π is a regular anddivisible filter;
(4) ifπ is a fuzzy implicative filter, thenπ is a fuzzy divisiblefilter.
Theorem 35. Let πΏ be a Heyting-algebra. Then
(1) π is a fuzzy Boolean filter if and only if π is a fuzzyregular filter if and only if π is a fuzzy fantastic filter;
(2) the fuzzy implicative filter, fuzzy divisible filter, andfuzzy π-contractive filter coincide.
Theorem 36. Let πΏ be a R0-algebra; then
(1) π is a fuzzy Boolean filter if and only if π is a fuzzyimplicative filter;
(2) every fuzzy Boolean filter is a fuzzy fantastic filter;(3) π is a fuzzy fantastic filter if and only if π 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 πΏ be a regular residuated lattice. Then
(1) π is a fuzzy Boolean filter if and only if π is a fuzzyimplicative filter;
(2) π is a fuzzy fantastic filter if and only if π 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 πΏ be a Rl-monoid. Then
(1) π is a fuzzy Boolean filter if and only if π is a fuzzyimplicative and fuzzy fantastic filter;
(2) π is a fuzzy Boolean filter if and only if π is a fuzzyimplicative and fuzzy regular filter;
(3) π is a fuzzy fantastic filter if and only if π 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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