Category of fuzzy hyper BCK-algebras

CA TEGOR Y OF FUZZY HYP ER BCK-ALGEB RAS J.DONGHO* Abstract. In this pap er w e first define the category of fuzzy hyper BCK- algebras . After that we show that the ca tegory of hyper BCK -algebras has equalizers, co equa lizers, pro ducts. It is a co nsequence that this categor y is complete and hence has pullbacks. 1. Intro duction The study of h yp erstructure w as initiated in 1934 by F. Marty at 8th congress of Scandin a vian Mathematiciens . Y.B. Jun et al. applied the h yperstructures to BCK-algebras, and in tro duces the notio n of hy p er BCK-a lgebra. No w w e follo w [1 ,2,3,4] and in tro duce the category of fuzzy h yperBCK-a lgebra and obtain some result, as men tioned in the abstarct. 2. Preliminaries W e now review some basic definitions that are v ery useful in the pap er. Definition 1. [ 3 ] L et H b e an non empty se t. A hyp er op er ation ∗ on H is a mapping of H × H family of non-em p ty subsets of H P ∗ ( H ) Definition 2. L et ∗ b e an hyp e r op er ation on H and O a c onstant element of H An hyp er or der on H is subset < of P ∗ ( H ) × P ∗ ( H ) define by: for al l x, y ∈ H , x < y iff O ∈ x ∗ y and for every A, B ⊆ H , A < B iff ∀ a ∈ A, ∃ b ∈ B such that a < b. Definition 3. If ∗ is hyp er op er ation on H . F o r al l A, B ⊆ H , A ∗ B := S a ∈ A,b ∈ B a ∗ b Definition 4. [1] By hyp er BCK-algebr a we me an a non em pty set H endowe d with a h yp er-op er ation ∗ and a c onstant O satisfying the fol lowing axi o ms. (HK1) ( x ∗ z ) ∗ ( y ∗ z ) < ( x ∗ y ) (HK2) ( x ∗ y ) ∗ z = ( x ∗ z ) ∗ y (HK3) x ∗ H < { x } Definition 5. A fuzzy hyp er BCK-algeb r a i s a p air ( H ; µ H ) whe r e H = ( H ; ∗ ; O ) is hyp er BCK-alg e br a and µ H : H − → [0 , 1] is a map satisfy the f ol lowing pr op- erty: inf ( µ H ( x ∗ y )) ≥ min( µ H ( x ) , µ H ( y )) for al l x, y ∈ H . 1 2 J.DONGHO* Example 1. [ 5 ] L e t n ∈ N ∗ . Define the hyp er op er ation ∗ on H = [ n, + ∞ ) as fol lows: x ∗ y =    [ n, x ] iff x < y ( n, y ] iff x > y 6 = n { x } iff y = n for al l x, y ∈ H . T o show that ( H , ∗ , n ) is hyp er BCK-algebr a, it suffic e to sh ow axiom H K 3 . F or al l x ∈ H , x ∗ H = S t ∈ H x ∗ t. F o r al l x ∈ H then x ∗ x ⊆ x ∗ H . A nd then n ∈ [ n, x ] ∗ { x } 3. The c a tegor y of fuzzyhyper BCK-algebras Lemma 1. L et ( H ; µ H ) b e a fuzzy hyp er BCK-algebr a. F o r al l x ∈ H , µ H ( O ) ≥ µ H ( x ) Pro of. F or all x ∈ H , x < x ; then O ∈ x ∗ x . O ∈ x ∗ x imply µ H ( O ) ≥ inf ( µ H ( x ∗ y )) ≥ min( µ H ( x ) , µ H ( y )) i.e µ H ( O ) ≥ min( µ H ( x ) , µ H ( y )) = µ H ( x ) i.e µ H ( O ) ≥ µ H ( x ) . Definition 6. L et ( H ; µ H ) b e a fuzzy hyp erBCK-algebr a . µ H is c al le d a fuzzy map. Lemma 2. L et ( H ; µ H ) a fuzzy hyp er BCK-algebr a.The fol lowing pr op erties ar e trues: i) If f o r al l x, y ∈ H , x < y imply µ H ( x ) ≤ µ H ( y ) then for al l x ∈ H , µ H ( x ) = µ H ( O ) ii) If µ H ( O ) = 0 then µ H ( x ) = 0 Pro of. i) F o r all x ∈ H , x ∗ H < { x } then x ∗ O < x . O < x ⇒ µ H ( O ) ≤ µ H ( x ) . Then µ H ( x ) ≤ µ H ( O ) a nd µ H ( O ) ≤ µ H ( x ) fo r all x ∈ H . i.e µ H ( x ) = µ H ( O ) f or all x ∈ H . ii) µ H ( O ) = O ⇒ µ H ( O ) ≤ µ H ( x ) , fo r all x ∈ H. Then µ H ( x ) = µ H ( O ) for all x ∈ H . Definition 7. L et ( H ; µ H ) and ( F , µ F ) two fuzzy hyp erBCK-algebr as . A n homomorphism fr om ( H , µ H ) to ( F , µ F ) is an homomorph i s m f : H − → F of hyp er BCK-algebr a such that for al l x ∈ H , µ F ( f ( x )) ≥ µ H ( x ) Prop osition 1. L et ( H , µ H ) an hyp erBC K - a lgebr a. L et G , F ⊂ H two hyp erB CK- sualgebr as of H . If ther e e xist α ∈ ]0 , 1[ such that µ H ( G ∗ ) ⊂ [0 , α [ and µ H ( F ) ⊆ ] α , 1] . Then a ny homorphism of hyp er BCK- algebr a f : G − → F is homom o rphism of fuzzy hyp erBCK-algebr a. CA TEGOR Y OF FUZZY HYPER BCK-ALGEBRAS 3 Pro of. Suppose that there is α ∈ ]0 , 1] suc h that µ H ( G ∗ ) ⊂ [0 , α [ and µ H ( F ) ⊆ ] α , 1[ . Let f : G − → F an homomorphism of h yp er BCK-algebra. F or a ll x ∈ G ∗ , f ( x ) ∈ F . And µ F ( f ( x )) > α > µ H ( x ) . Then µ F ( f ( x )) > µ H ( x ) fo r all x ∈ G ∗ f ( O ) = O then µ F ( f ( O )) = µ F ( O ) = µ H ( O ) i.e µ F ( f ( O )) = µ H ( O ) . t herefore, for a ll x ∈ x ∈ G, µ F ( x ) ≥ µ H ( x ) Example 2. [ 1 ] Define the hyp er op er ation ” ∗ ” on H = [1; + ∞ ] as fol low. x ∗ y =    [1 , x ] if x ≤ y (1 , y ] if x > y 6 = 1 { x } if y = 1 F o r al l x, y ∈ H , ( H , ∗ , 1 ) is hyp erB CK-algebr a. Defin e the fuzzy structur e µ H on H by: µ H : H − → [0 , 1] x 7→ 1 x W e show that ( H , µ H ) is a fuzzy h yp er BCK-algebr a. L et x, y ∈ H . (i) If x ≤ y , then x ∗ y = [1 , x ] ; i.e for al l t ∈ x ∗ y , 1 ≤ t ≤ x ≤ y and so 1 y ≤ 1 x ≤ 1 t . So, µ H ( t ) ≥ 1 y = min { 1 y , 1 x } = min { µ H ( x ) , µ H ( y ) } . Then inf { x ∗ y } ≥ min { µ H ( x ) , µ H ( y ) } (ii) If x > y 6 = 1 then x ∗ y = (1 , y ] . F or al l t ∈ H ∩ x ∗ y , 1 x ≤ 1 y ≤ 1 t ≤ 1 . ther efor e, µ H ( t ) = 1 t ≥ 1 x = min { µ H ( x ) , µ H ( y ) } for al l t ∈ x ∗ y . Then ∈ { µ H ( x ∗ y ) } ≥ min { µ H ( x ) , µ H ( y ) } . (iii) If y = 1 , x ∗ y = { x } , henc e µ H ( x ∗ y ) = { µ H ( x ) } = { 1 x } . y = 1 i mply y ≤ x and 1 x ≤ 1 y for al l x ∈ H ; i.e; min { µ H ( x ) , µ H ( y ) } = 1 x . Then µ H ( x ∗ y ) = { 1 x } . Thus inf { µ H ( x ∗ y ) } = 1 x ≥ min { ( µ H ( x ) , µ H ( y )) } Prop osition 2. Th e fuzzy hyp erBCK-algebr as and h o momorphisms of fuzzy hyp erBCK - a lgebr as form a c ate gory. Pro of. The pro o f is easy . Notes 1. In the fol lowing we let H the c ate gory of hyp erBCK-algebr as; F H the c ate gory of fuzzy h yp erBCK-algebr as; H the fuzzy hyp er BCK-alg e b r a ( H , µ H ) F or an y fuzzy h yp er BCK-algebra H , w e associate for all α ∈ [0 , 1] the set H α := { x ∈ H , µ H ( x ) ≥ α } Lemma 3. L et H a fuzzy hyp er B CK-algebr a. F or al l α ∈ [0 , 1] , O ∈ H α and for al l x, y ∈ H , x ∗ y ⊆ H α Pro of. By lemma 1, for all x ∈ H , µ H ( x ) ≤ µ H (0) . Then for all x ∈ H α , µ H ( O ) ≥ µ H ( x ) > α i.e O ∈ H α . Let x, y ∈ H α ; 4 J.DONGHO* for all t ∈ x ∗ y , µ H ( t ) ≥ inf { µ H ( x ∗ y ) ≥ min { µ H ( x ) , µ H ( y ) }} ≥ α then t ∈ H α . therefore, x ∗ y ⊆ H α Definition 8. L et ( H , ∗ , O ) b e an hyp er BCK-alg e br a. An hyp er BCK-sub algebr a of H is a non empty subset S of H such that O ∈ S an d S is hyp er BCK - algebr a with r esp e c t to the hyp er op er ation ” ∗ ” on H Prop osition 3. L e t ( H , ∗ , O ) b e an hyp er BC K-algebr a . A non empty subset S of H is hyp er BCK-sub algebr a of H iff for al l x, y ∈ S, x ∗ y ∈ S Pro of. The pro o f is easy . Definition 9. A fuzzy hyp e r BC K-sub algebr a of H is an hyp er BCK-sub algebr a S of H with the r estriction µ S of µ H on S. Prop osition 4. F or al l α ∈ [0 , 1] , ( H α , µ H ) is f uzzy hyp e r BCK-sub algebr a o f H Pro of. By lemma 3, H α is hy p er BCK- subalgebra of H and inf { µ H ( x ∗ y ) } ≥ min { µ H ( x ) , µ H ( y ) } Definition 10. L et H by an fuzzy hyp er BCK-algebr a. The fuzzy-hyp erBCK- sub algebr a H α := ( H α ; µ H ) is c al ling hyp er α -cut o f H Prop osition 5. L et H b e fuzzy hyp er B CK-algebr a. A hyp er BCK-sub algebr a S of H is fuzzy hyp er BCK-sub algebr a iff S is hyp er α -cut of H . Pro of. By prosition 4 , an y h yper α - cut is fuzzy h yper BCK- subalgebra. Con v ersely , let S b e fuzzy h yper BCK-subalgebra of H . Then µ H ( S ) is subset of [0 , 1] . If 0 ∈ µ H ( S ) , then S = H 0 = H . If 0 < inf ( µ H ( S )) , then S = H inf ( µ H ( S )) . Prop osition 6. L et H and F b e two fuzzy hyp e r B CK algebr a s. A n H - morphism f : H − → F i s F H -morphism iff for al l α ∈ [0 , 1] , f ( H α ) ⊆ F α . Pro of. Supp ose that f ( H α ) ⊆ F α for all α ∈ [0 , 1] Let x ∈ [0 , 1] w e need µ H ( x ) ≤ µ F ( f ( x )) . Let α = µ H ( x ); x ∈ H α and f ( x ) ∈ f ( H α ) ⊆ F α . Then µ F ( f ( x )) > α = µ H ( x ) . whence for all x ∈ H , µ F ( f ( x )) ≥ µ H ( x ) . Con v ersely , supp ose that f : H − → F is F H -morphism. F or all x ∈ H α for some α ∈ [0 , 1] , µ F ( f ( x )) ≥ µ H ( x ) ≥ α i.e; f ( x ) ∈ [0 , 1] . Then f ( H α ) ⊆ F α for all α ∈ [0 , 1] . Prop osition 7. A F H -morphism f : H − → F is F H -iso iff it is b oth H -iso and µ H = µ F f . Pro of. Suppose that f is H -iso and µ H = µ F f . the re is g ∈ H om H ( F , H ) ; g ◦ f = I d H and g ◦ g = I d F . Then, for all x ∈ F , µ H ( g ( x )) = µ F ( f ( g ( x ))) µ F ( x ) . And then, g ∈ H om F H ( F , H ) . CA TEGOR Y OF FUZZY HYPER BCK-ALGEBRAS 5 Con v ersely , Supp ose that f is F H -iso. There is g ∈ H om F H ( F , H ); g ◦ f = I d F and f ◦ g = I d H . Since f ∈ H om F H ( F , H ) , µ H ≤ µ F f . Since g ∈ H o m F H ( H , H ) , µ F ≤ µ H g . x ∈ H imply f ( x ) ∈ F . Then µ F ( f ( x )) ≤ µ H ( g ( f ( x ))) = µ H ( x ) i.e; µ F f ≤ µ H . therefore, µ F f = µ H Prop osition 8. L et f ∈ H om F H ( F , H ) . f is H H -mono iff f is H -mono Pro of. Suppose that f is H H -mono. F or all h, g ∈ H om H ( K , H ) , such that f h = f g , we define µ K = min( µ H ( h ( x ); µ H ( g ( x )))) for all x ∈ K . a) W e sho w that ( K ; µ K ) is fuzzy hy p er BCK-algebra. inf ( µ K ( x ∗ y )) = inf { µ H ( h ( x ∗ y ); µ H ( g ( x ∗ y ))) } = inf { µ H ( h ( x ) ∗ h ( y ); µ H ( g ( x ) ∗ g ( y ))) } = min { inf { µ H ( h ( x ) ∗ h ( y ) } ; inf { µ H ( g ( x ) ∗ g ( y ))) }} ≥ min { min { µ H ( h ( x ); µ H ( h ( y ) } ; min { µ H ( g ( x ); g ( y )) }} ≥ min { min { µ K ( x ); µ K ( y ) }} ≥ min { µ K ( x ); µ K ( y ) } Then, for all x, y ∈ K , inf ( µ K ( x ∗ y )) ≥ min { µ K ( x ) , µ K ( y ) } . therefore, ( K ; µ K ) is fuzzy h yper BCK-algebra. b) W e show that h and g are F H -homomorphism. F or all x ∈ K , µ K ( x ) = min { µ H ( h ( x ) , µ H ( g ( x ))) } . Then µ K ( x ) ≤ µ H ( g ( x )) and µ K ( x ) ≤ µ H ( h ( x )) . therefore, h and g are F H -morphism. Since f is F H -mono and h, g ∈ H om F H ( F , H ) , f h = f g imply f = g Con v ersely , if f is F H -mono, it is H -mono. Lemma 4. T he p ai r O = ( { O } , µ o ) wher e µ o : { o } − → [0 , 1] o 7→ 0 is fuzzy hyp er B C K-algebr a Pro of. Easy Lemma 5. O is fin al objet of F H Prop osition 9. The c ate gory F H has pr o ducts. Pro of. Let ( H i ; µ H i ) i ∈ I a family of fuzzy h yper BCK-algebras. Denote H = Q i ∈ I H i the H -pro duct of ( H i ) i ∈ I with the pro jection morphisms p i : H − → H i . Conside r the following map µ H : H − → [0 , 1] define b y: µ H ( x ) = ^ i ∈ I µ H i p i ( x ) 6 J.DONGHO* for all x ∈ H a) W e sho w that the pair ( H ; µ H ) is fuzzy hy p er BCK-algebra. F or all x, y ∈ H , p i ( x ∗ y ) = p i ( x ) ∗ p i ( y ) for all i N I . Then inf ( µ H i p i ( x ∗ y )) = inf ( µ H i ( p i ( x ) ∗ p i ( y )) ≥ min { µ H i ( p i ( x )); µ H i ( p i ( y )) } for all i ∈ I . Then, inf ( V i ∈ I µ H i p i ( x ∗ y )) ≥ V i ∈ I inf { µ H i ( p i ( x ) ∗ p i ( y ) } ≥ V i ∈ I min { µ H i ( p i ( x )); µ H i ( p i ( y )) } ≥ min { V i ∈ I µ H i p i ( x ); µ H i p i ( y ) } ≥ min { µ H ( x ) , µ H ( y ) } . b) F o r all i ∈ I , x ∈ H ; µ H i p i ( x ) ≥ ( V i ∈ I µ H i p i )( x ) . Then eac h p i is F H -morphism. c) If q j : F − → H j is family of F H -morphism, there is unique H -morphism ϕ : F − → H suc h that the followin g diagram comm ute. H p j / / H j F ϕ O O q j > > } } } } } } } i.e p j ϕ = q j for all j ∈ I F or all x ∈ F , µ F ( x ) ≤ µ H i q i ( x ) . Then µ F ( x ) ≤ µ H i p i ϕ ( x ) for all x ∈ F , i ∈ I i.e µ F ( x ) ≤ V i ∈ I µ H i p i ϕ ( x ) ≤ ( V i ∈ I µ H i p i ) ϕ ( x ) ≤ µ H ( ϕ ( x )) for all x ∈ F . Then µ F ≤ µ H ϕ ≤ Then ϕ is F H -morphism. Prop osition 10. F H have e qualizers. Pro of. Let f , g ∈ H om F H ( H , F ) , K := { x ∈ H , f ( x ) = g ( x ) } . It is pro v e in [1] that K is h yp er BCK-subalgebra of H . It is clear that ( K , µ H ) is fuzzy h yper BCK-algebra. Let i : K − → H the inclusion map. i ∈ H om F H ( K , F ) . F or all x ∈ K , f i ( x ) = f ( x ) = g ( x ) = g i ( x ) . Let h ∈ H om F H ( L , F ) suc h that f h = g h, f or all x ∈ L, f ( h ( x )) = g ( h ( x )) . CA TEGOR Y OF FUZZY HYPER BCK-ALGEBRAS 7 Then I mh ⊆ L . Define δ : L − → K by δ ( x ) = h ( x ) for all x ∈ L. δ ∈ H om F H ( H , F ) and iδ = h. So, the follo wing diagra m comm ute. K i / / H f / / g / / F L δ O O h ? ? ~ ~ ~ ~ ~ ~ ~ Since i is monic, δ is unique F H -morphism suc h that the ab o v e diagram com- m ute. therefore, F H ha v e equalizers. Prop osition 11. F H is c omplet. Pro of. By prop osition 9, eac h family of ob jets of F H has pro duct. By prop osition 10, each pair of parallel arrows has an equalizer. Then F H is complet. Corollary 1. F H has pulb acks Pro of. By prop ositions 9 and 10, F H has equalizers and pro ducts . therefore, F H has pulbac ks. Prop osition 12. F H have c o e qualize rs Pro of. Let f , g ∈ H om F H ( H , K ) . Let X f g = { θ , θ regular congruence relation on K such that f ( a ) θ g ( a ) ∀ a ∈ H } P f g 6 = φ b ecause K × K ∈ P f g Let ρ = T θ ∈ P f g θ . Th en, ρ is regular congruence relation. Define on K /ρ the follo wing h yp er op eration [ x ] ρ ∗ [ y ] ρ = [ x ∗ y ] ρ . ( K /ρ ; ∗ ; [0] ρ ) is an ob jet of H (see [ 1 ] ). Define on K /ρ the follo wing map µ K/ρ : K /ρ − → [0 , 1] µ K/ρ ([ x ] ρ ) 7− → W a ∈ [ x ] ρ µ K ( a ) a) W e sho w that ( K/ ρ, µ K/ρ ) is ob jet of F H . If x, y ∈ K suc h that [ x ] ρ = [ y ] ρ . Then _ a ∈ [ x ] ρ µ K ( a ) = _ a ∈ [ y ] θ µ K ( a ) ∀ x ∈ K , µ K ( x ) ≤ W a ∈ [ x ] ρ µ K ( a ) . Then µ K ≤ µ K/ρ ([ x ] ρ ) = µ K/ρ ( π ( x )) 8 J.DONGHO* Then, the canonical pro jection π is an F H − mor phism Since for a ll x ∈ H, f ( x ) ρg ( x ) , then [ f ( x )] ρ = [ g ( x )] ρ . therefore, ( π ◦ f )( x ) = ( π ◦ g )( x ) . Then, π ◦ f = π ◦ g . b) Univ ersal prop ert y of co equ alizer. Let ϕ : K − → L and F H -morphism suc h that ϕ ◦ f = ϕ ◦ g . Define the follo wing mapping. ψ : K /ρ − → L [ x ] ρ 7− → ϕ ( x ) c) W e prov e that ψ is w ell define. If [ x ] ρ = [ y ] ρ then, for all a ∈ H , ϕ ( f ( a )) = ϕ ( g ( a )) imply f ( a ) R ϕ g ( a ) b ecause R ϕ is regular congruence on K . Then R ϕ ∈ P f ,g . The mini- malit y of ρ on P f ,g imply ρ ⊆ R ϕ . therefore, [ x ] ρ = [ y ] ρ imply xρy . Then xR ϕ y . i.e ϕ ( x ) = ϕ ( y ) And then, ψ ([ x ] ρ ) = ψ ([ y ] ρ ) therefore, ψ is w ell define. F or all x ∈ K , µ L ( ψ ( π )( x )) = µ L ( ϕ ( x )) ≥ µ K ( x ) , ∀ x ∈ K then for all a ∈ [ x ] ρ . µ L ( ϕ ( a )) ≥ µ K ( a ) . By the minimalit y of ρ , [ a ] ρ = [ x ] ρ imply aρx then aR ϕ x i.e ϕ ( a ) = ϕ ( x ) (b ecause ρ ⊆ R ϕ ). Th en _ a ∈ [ x ] ρ ˜ L ( ϕ ( a )) = _ a ∈ [ x ] ρ ˜ L ( ϕ ( x )) = ˜ L ( ϕ ( x )) therefore ˜ L ( ϕ ( x )) ≥ _ a ∈ [ x ] ρ ( a ) = ˜ K /ρ ([ x ] ρ ) i.e ˜ L ( ψ ([ x ] ρ )) ≥ ˜ k /ρ ([ x ] ρ ) ∀ x ∈ H . It is clean that ψ ( π ( x )) = ψ ([ x ] ρ ) = ϕ ( x ) , ∀ x ∈ H i.e ψ ◦ π = ϕ This pro v e the comm utativit y o f the f ollo wing diagram: H f / / g . / / K π / / ϕ ! ! C C C C C C C C C K/ ρ ψ   L The unicit y of ψ is thus to the fat that π is epimorphism. Then, F H ha v e co equalizer. A CKNO WLEDGEMENTS References [1] H. Harizavi,J. Mac do nald AND A. Bor zo o ei Cate gory of b ck-algebr as :Scientiae Math- ematicae J ap onicae Online, e-200 6, 5 29-53 7. CA TEGOR Y OF FUZZY HYPER BCK-ALGEBRAS 9 [2] R. A. Bo rzo o ei H.Harizavi. R e gular Congruenc e R elations on hyp er BC K-algebr as : vol.61, No .1(2005),8 3 -97 [3] Eun Hwan ROH, B. Davv a z And Ky ung Ho Kim T-fuzzy subhyp erne a-rings of hyp erne ar-ring , Scien tiae Mathema tica e Jap onica e Online, e-2 005,19 - 29 . [4] Caro l L . W alker, Cate gory of fuzz y sets . [5] R.A. Borzo e i a nd M.M. Zahedi, P ositive implic ative hyp erK-ide als , Scientiae Ma the- maticae J a po nicae Online, V o l. 4,(200 1), 381-3 89. [6] C. LELE and M. Salissou, Discussiones Mathematic ae, gener al Al gebr a and Applic a- tion 26 : 11 1-135 (2 006). ****Dep ar tment of Ma thema tics, University of Y aounde, BP 812, Cameroon E-mail addr ess : jo sephdo ngho@y ahoo.fr