Iteratoal Joural of Fuzzy Mathematcs ad Systems. ISSN 2248-9940 Volume 3, Number 5 (2013), pp. 375-379 Research Ida Publcatos http://www.rpublcato.com O L- Fuzzy Sets T. Rama Rao, Ch. Prabhakara Rao, Dawt Solomo Ad Derso Abeje. Prof. of Mathematcs, Vkas College of Egeerg, Vjayawada. (ramaraothota99@yahoo.com). Assoc. Prof. of Mathematcs, Dad Isttute of Egeerg ad Techology, Aakapalle. (raoprabhakar_ch@redffmal.com). Dept. of Mathematcs, Uversty of Godar, Ethopa. (devatasol@gmal.com. (deab02@yahoo.com). Abstract It s foud that the terval [0, 1] of real umbers s suffcet to have the truth values of geeral fuzzy statemets. I ths paper t s argued that a complete lattce L satsfyg the fte meet dstrbutve law s a best caddate to assume the truth values of fuzzy statemets. Such a lattce s called a frame. A thorough dscusso s made of fuzzy subsets of a set havg truth values a abstract frame. Keywords: Lattce, complete lattce, Ifte meet dstrbutve law, Frame, Fuzzy statemets. Itroducto Ever sce Zadeh [10] troduced the oto of a fuzzy subset of a set X as fucto X to the ut terval [0, 1] of real umbers, several mathematcas took terest the study of fuzzy subsets, partcular, o fuzzy sub algebras of several algebrac structures. Fuzzy statemets usually take truth values the terval [0, 1] of real umbers, whle the ordary (or covetoal or crsp) statemets take truth vales the two-elemet set {F, T } or {0, 1}where F ad O stad for false ad T ad 1 stad for true. However, the terval [0, 1] s foud to be suffcet to have the truth values of geeral fuzzy statemets [2ad9]. For example, cosder the statemet Ida s a good coutry. The truth value of ths statemet may ot a real umber [0, 1]. Beg good coutry may have several compoets: good educatoal facltes, good publc trasport system, good poltcal awareess amog the ctzes, good medcal facltes, good for toursm, etc. The truth value
376 T. Rama Rao et al correspodg to each compoet may be a real umber [0, 1]. I s the umber of such compoets uder cosderato, the the truth value of the above fuzzy statemet s a -tuple of real umbers [0, 1]; that s, a elemet of [0, 1]. If C s the collecto of all coutres o ths earth ad G s the collecto of good coutres, the G s ot a subset of C, but t s a fuzzy subset of C, sce beg good s fuzzy. That s, G ca be cosdered as a fucto of C to a set lke [0, 1], for a postve teger. Such a G s called a L-fuzzy subset of C where L= [0, 1]. The usual orderg of real umbers makes [0, 1] as a totally ordered set But [0, 1] s ot a totally ordered set whe > 1, uder the usual coordate-wse orderg. However, [0, 1] satsfes certa rch lattce theoretc propertes such as the fte meet dstrbutvty, amely a (supx) = sup{a x x X} for ay elemet a ad ay subset X. Ths dstrbutvty satsfes almost all the major requremets to develop the theory of geeral fuzzy subsets. PRELIMINARIES We brefly recall certa elemetary cocepts ad otatos from the theory of partally ordered sets ad lattces [1]. A bary relato o a set X s called a partal order o X f t s reflexve, trastve ad at-symmetrc. A par (X, ) s called a partally ordered set or, smply, poset f X s a oempty set ad s a partal order o X. A poset (X, ) s called a lattce (complete lattce) f every oempty fte subset (respectvely, every arbtrary subset) of X has greatest lower boud ad least upper boud X whch are respectvely called fmum ad supremum also; for ay subset A of X, we wrte f A or glb A or Aor a for the greatest lower boud (or fmum ) of A ad supa or luba or Aor a for the least upper boud (or a A supremum) of A. If A= {a 1,., a }, the we wrte for the f A ad a or a a... a for the supa. = 1 1 2 If (L, ) s a lattce, the a b = f {a, b} ad a b = sup{a, b} gve two bary operatos ad o L whch are both assocatve, commutatve ad dempotet ad satsfy the absorpto laws a (a b) = a = a coversely f ad are bary operatos o a oempty set L satsfyg all the above propertes ad f the partal order o L s defed by a b a = a b ( a b = b), the (L, ) s a lattce whch a b ad a b are respectvely the fmum ad supremum of {a, b}. A elemet a a poset (L, ) s called the smallest (greatest) elemet f a x (respectvely x a) for all x L. The smallest ad greatest elemets, f they exst, are usually deoted by 0 ad 1 respectvely. A poset s called bouded f t has both smallest ad greatest elemets. A complete lattce s ecessarly bouded. Logcally, the fmum ad supremum of the empty subset of a poset, f they exst, are respectvely the greatest elemet ad smallest elemet. A complete lattce (L, ) s called a frame, f t satsfes the fte meet dstrbutve law; that s, a (supx) = sup{a x x X} for all a L ad X L. It s kow that a complete lattce (L, ) s a frame f ad oly f, for ay a ad b L, there exst a largest elemet, deoted by a A
O L- Fuzzy Sets 377 a b, L such that x a b x a b for all x L[9]. A poset (P, ) s called a totally ordered set f, for ay a ad b P, ether a b or b a. A subset C of a poset (P, ) s called a cha P f (C, ) s totally ordered. L-FUZZY SUBSETS OF A SET It s well kow that f A s a algebrac structure ad X s ay oempty set X, the the set A X of all mappgs of X to A ca be made as a algebrac structure of the same type as A by defg the fudametal as a algebrac structure of the same type as A by defg the fudametal type as A by defg the fudametal operatoally defable algebra (lke a group or a rg or a module or a lattce), the A X s a algebra belogg to the varety geerated by A. I partcular, f 2 s the two elemet lattce {0, 1} wth 0 < 1, the 2 X s a Boolea algebra for ay oempty set X, sce 2 s a Boolea algebra. Also, recall that 2 X s somorphc to the Boolea algebra P(X) of all subsets of X, uder the mappg whch maps ay A X wth ts characterstc map χ A defed by χ A (x) = 1 or 0 accordg as x s A or ot A. Therefore, the usual (or crsp) subsets of X ca be detfed wth mappgs of X to 2. Wth ths backgroud, we defe the followg. Defto 3.1. Let X be a oempty set ad L = (L, ) be a frame. Ay mappg of X to L s called a L-fuzzy subset of X. The set of all L-fuzzy subsets of X s deoted by FS L (X). Sce L s a complete lattce, t has smallest elemet 0 ad greatest elemet 1 ad hece 2 ca be treated as a subset of L. Ths facltates us to treat the usual subsets of X as L- fuzzy subsets of X. For the sake of dstgushg the subsets of X from the L- fuzzy subsets of X, the subsets of X are usually called the crsp subsets of X. The lattce structure o L ca be exteded to FS L (X)(= L X ) as gve below. Defto 3.2. For ay L-fuzzy subsets A ad B of X, defe A B f ad oly f A(x) B(x) for all x X. Clearly s a partal order o FS L (X). Also, for ay crsp subsets S ad T of X, we have χs χt S T. The followg s a easy verfcato. Theorem 3.3. (FS L (X), ) s a frame whch, for ay {A }, the fmum ad supremum of {A } are respectvely gve by ( f{ }( x)) = f ( x) ad ( sup{ A })( x) = sup A ( x ), for ay x X. A I A Also, for ay A ad B FS L (X), (A B)(x) = A(x) B (x) for all x X. Defto 3.4. For ay L- fuzzy subset A of X ad for ay α L, defe A α = A 1 ([α, 1]) = {x X : α A(x)}.The Aα s called the α - cut of A. The followg s a straght forward verfcato.
378 T. Rama Rao et al Theorem 3.5. (1) For ay A, B FS L (X), A B A α B α for all α L (2) For ay {A } FS L (X) ad A = A, Aα = I A α for all α L. Eve though the α-cut of the fmum of A s s smply the set tersecto of the α-cuts of A s, the α-cut of the supremum of A s may ot be the set uo of the α- cuts of A s. However the α-cuts of A ca be ca be expressed terms of the α- cuts of A s. Frst, let us recall that, for ay α L ad M L, M s sad to be a cover of α (or α s sad to be covered by M) f α supm. The followg ca be easly proved. Theorem 3.6. Let {A } be a oempty class of L-fuzzy subsets of a set X ad A = A, The the α-cut of A s gve by Aα = U{ I( U A β ) / M s a cov er of α } For ay L-fuzzy subset A of X, {Aα / α L} s a class of crsp subsets of X such that Aα = A supm for all M L. α M The coverse of ths s also true, the followg sese. Theorem 3.7. For ay class {Sα / α L} of crsp subsets of X such that α M Sα = SsupM for all M L, there exsts a uque L-fuzzy subset A of X whose α-cut s precsely Sα for all α L. The followg s a useful tool workg wth L-fuzzy subsets or crsp subsets of a set X. A class {A } of L-fuzzy subsets of X s sad to be drected above f, for ay ad j I, there exsts k I such that A A k ad A j A k. Theorem 3.8. Let {A} be a drected above class of L- fuzzy subsets of a set X ad x 1, x 2,..., x X, the ( A ( x )) = ( A ( x )) = 1 r r= 1 r Proof. Let α ad β deote respectvely the left sde ad rght sde of the above requred equato. It s clear that β α. Also, by the fte meet dstrbutvty L, we have α = A ( x ) A ( x )... A ( x ))..(*) 1, 2,.. I ( 1 1, 2 2 Now, for ay 1, 2,..., I, there exsts j I such that Ar Aj for all 1 r ad hece A ( 1 x 1 ) A ( 2 x 2 )... A (x ) A j (x 1 ) A j (x 2 )... A j (x ) β. From ths ad ( ) above, we get that α β. Thus α = β. For ay α L, the L-fuzzy subset A of X defed by A(x) = α for all x X s called a costat L-fuzzy subset of X ad s deoted by α. Note that 0(= χφ) ad T (= χ X ) are respectvely the smallest ad greatest elemets FS L (X). For ay α L ad a crsp subset Y of X, we defe α Y : X L by 1, f x Y α Y (x) = { α, f x Y. β M
O L- Fuzzy Sets 379 The followg s a easy verfcato. Theorem 3.9. Let X be a oempty set ad L a frame. For ay 1 α L, Y α Y s a embeddg of the lattce P (X) of all crsp subsets of X to the lattce FS L (X) of L- fuzzy subsets of X. Also, for ay proper crsp subset Y of X, α α Y s a embeddg of L to the lattce FS L (X) of L-fuzzy subsets of X. Also, for ay proper crsp subset Y of X, α α Y s a embeddg of L to the lattce FS L (X). Note that FS L (X) s precsely the set L X of all mappgs of X to Y. Sce L s a frame, L X s also a frame. L beg a complete lattce satsfyg the fte meet dstrbutvty, we have, for ay α ad β L, the largest elemet α β wth the property that α γ β γ α β for ay γ L. Ifact, α β = sup{γ L α γ β}. Therefore, for ay A ad B FS L (X), we have A B FS L (X) gve by (A B)(x) = A(x) B(x) for all x X ad hece we have the followg. Theorem 3.10. For ay oempty set X ad for ay frame L, The set FS L (X) of all L fuzzy subsets of X s a frame uder the pot wse orderg. The Authors thak Prof. U.M.Swamy for hs help preparg ths paper. Refereces [1] Brkhoff, G., Lattce Theory, Amer. Math.Soc. colloq publ.,.3rd edto, 1967 [2] Gogue, J., L-fuzzy sets, Jour. Math. Aal. Appl., 18(1967), 145-174 [3] Swamy, U. M. ad Raju, D.V., Algebrac Fuzzy systems, Fuzzy sets ad Systems, 41(1991), 187-194 [4] Swamy, U. M ad Raju, D.V., Irreducblty algebrac Fuzzy systems, Fuzzy sets ad Systems, 41(1991), 233-241 [5] Swamy, U. M., Rama Rao.T. ad Rao, Ch. P., O the truth values of fuzzy statemets, Iter. J. of Appl. comp. Sc. ad Math., 3(2013), 1-6 [6] Zadeh, L., Fuzzy sets, Iformato ad cotrol, 8(1965), 338-353.