Category of fuzzy hyper BCK-algebras

0.15 MB·English

by  Joseph Dongho

CATEGORY OF FUZZY HYPER BCK-ALGEBRAS
J.DONGHO*
1
Abstract. In this paper we firstdefine the categoryof fuzzy hyper BCK-
1
0 algebras. After that we show that the category of hyper BCK-algebrashas
2 equalizers, coequalizers, products. It is a consequence that this category is
complete and hence has pullbacks.
n
a
J

3
1 1. Introduction

] The study of hyperstructure was initiated in 1934 by F. Marty at 8th congress
T
of Scandinavian Mathematiciens. Y.B. Jun et al. applied the hyperstructures
C
. to BCK-algebras, and introduces the notion of hyper BCK-algebra. Now we
h
follow [1,2,3,4] and introduce the category of fuzzy hyperBCK-algebra and
t
a
obtain some result, as mentioned in the abstarct.
m
[ 2. Preliminaries

1
v We now review some basic definitions that are very useful in the paper.
1
7 Definition 1. [3] Let H be an non empty set.
4
A hyperoperation ∗ on H is a mapping of H ×H family of non-empty subsets
2
. of H P∗(H)
1
0
Definition 2. Let ∗ be an hyperoperation on H and O a constant element of
1
1 H An hyperorder on H is subset < of P∗(H)×P∗(H) define by:
: for all x,y ∈ H,x < y iff O ∈ x ∗ y and for every A,B ⊆ H,A < B iff
v
i ∀a ∈ A,∃b ∈ B such that a < b.
X
r Definition 3. If ∗ is hyperoperation on H.
a
For all A,B ⊆ H,A∗B := a∗b
a∈AS,b∈B
Definition 4. [1] By hyper BCK-algebra we mean a non empty set H endowed
with a hyper-operation ∗ and a constant O satisfying the following axioms.
(HK1) (x∗z)∗(y ∗z) < (x∗y)
(HK2) (x∗y)∗z = (x∗z)∗y
(HK3) x∗H < {x}
Definition 5. AfuzzyhyperBCK-algebrais apair(H;µ ) whereH = (H;∗;O)
H
is hyper BCK-algebra and µ : H −→ [0,1] is a map satisfy the following prop-
H
erty:
inf(µ (x∗y)) ≥ min(µ (x),µ (y))
H H H
for all x,y ∈ H.
1
2 J.DONGHO*
Example 1. [5] Let n ∈ N∗. Define the hyperoperation ∗ on H = [n,+∞) as
follows:
[n,x] iff x < y

x∗y = (n,y] iff x > y 6= n

{x} iff y = n

for all x,y ∈ H. To show that (H,∗,n) is hyper BCK-algebra, it suffice to show
axiom HK3. For all x ∈ H,x∗H = x∗t. For all x ∈ H then x∗x ⊆ x∗H.
t∈SH
And then n ∈ [n,x]∗{x}
3. The category of fuzzyhyper BCK-algebras
Lemma 1. Let (H;µ ) be a fuzzy hyper BCK-algebra.
H
For all x ∈ H,µ (O) ≥ µ (x)
H H
Proof. For all x ∈ H,x < x; then O ∈ x∗x.
O ∈ x∗x imply µ (O) ≥ inf(µ (x∗y)) ≥ min(µ (x),µ (y))
H H H H
i.e µ (O) ≥ min(µ (x),µ (y)) = µ (x)
H H H H
i.e µ (O) ≥ µ (x).
H H
Definition 6. Let (H;µ ) be a fuzzy hyperBCK-algebra. µ is called a fuzzy
H H
map.
Lemma 2. Let (H;µ ) a fuzzy hyper BCK-algebra.The following properties
H
are trues:
i) If for all x,y ∈ H,x < y imply µ (x) ≤ µ (y)
H H
then for all x ∈ H,µ (x) = µ (O)
H H
ii) If µ (O) = 0 then µ (x) = 0
H H
Proof.
i) For all x ∈ H, x∗H < {x} then x∗O < x.
O < x ⇒ µ (O) ≤ µ (x).
H H
Then µ (x) ≤ µ (O) and µ (O) ≤ µ (x) for all x ∈ H.
H H H H
i.e µ (x) = µ (O) for all x ∈ H.
H H
ii) µ (O) = O ⇒ µ (O) ≤ µ (x), for all x ∈ H.
H H H
Then µ (x) = µ (O) for all x ∈ H.
H H
Definition 7. Let (H;µ ) and (F,µ ) two fuzzy hyperBCK-algebras. An
H F
homomorphism from (H,µ ) to (F,µ ) is an homomorphism f : H −→ F of
H F
hyper BCK-algebra such that for all x ∈ H, µ (f(x)) ≥ µ (x)
F H
Proposition1. Let (H,µ ) anhyperBCK-algebra. Let G,F ⊂ H two hyperBCK-
H
sualgebras of H. If there exist α ∈]0,1[ such that
µ (G∗) ⊂ [0,α[ and µ (F) ⊆]α,1]. Then any homorphism of hyper BCK-
H H
algebra f : G −→ F is homomorphism of fuzzy hyperBCK-algebra.
CATEGORY OF FUZZY HYPER BCK-ALGEBRAS 3
Proof. Suppose that there is α ∈]0,1] such that
µ (G∗) ⊂ [0,α[ and µ (F) ⊆]α,1[.
H H
Let f : G −→ F an homomorphism of hyper BCK-algebra.
For all x ∈ G∗,f(x) ∈ F. And µ (f(x)) > α > µ (x).
F H
Then µ (f(x)) > µ (x) for all x ∈ G∗
F H
f(O) = O then µ (f(O)) = µ (O) = µ (O)
F F H
i.e µ (f(O)) = µ (O). therefore, for all x ∈ x ∈ G,µ (x) ≥ µ (x)
F H F H
Example 2. [1] Define the hyper operation ”∗” on H = [1;+∞] as follow.
[1,x] if x ≤ y

x∗y = (1,y] if x > y 6= 1

{x} if y = 1

For all x,y ∈ H,(H,∗,1) is hyperBCK-algebra. Define the fuzzy structure µ
H
on H by:
µ : H −→ [0,1]
H
x 7→ 1
x
We show that (H,µ ) is a fuzzy hyper BCK-algebra.
H
Let x,y ∈ H.
(i) If x ≤ y, then x∗y = [1,x]; i.e for all t ∈ x∗y,1 ≤ t ≤ x ≤ y
and so 1 ≤ 1 ≤ 1. So, µ (t) ≥ 1 = min{1, 1} = min{µ (x),µ (y)}.
y x t H y y x H H
Then inf{x∗y} ≥ min{µ (x),µ (y)}
H H
(ii) If x > y 6= 1 then x∗y = (1,y]. For all t ∈ H ∩x∗y, 1 ≤ 1 ≤ 1 ≤ 1.
x y t
therefore, µ (t) = 1 ≥ 1 = min{µ (x),µ (y)} for all t ∈ x∗y. Then
H t x H H
∈ {µ (x∗y)} ≥ min{µ (x),µ (y)}.
H H H
(iii) If y = 1,x∗y = {x}, hence µ (x∗y) = {µ (x)} = {1}.
H H x
y = 1imply y ≤ x and 1 ≤ 1 for all x ∈ H; i.e;
x y
min{µ (x),µ (y)} = 1. Then µ (x∗y) = {1}.
H H x H x
Thus inf{µ (x∗y)} = 1 ≥ min{(µ (x),µ (y))}
H x H H
Proposition 2. The fuzzy hyperBCK-algebras and homomorphisms of fuzzy
hyperBCK-algebras form a category.
Proof. The proof is easy.
Notes 1. In the following we let H the category of hyperBCK-algebras; F the
H
category of fuzzy hyperBCK-algebras; H the fuzzy hyper BCK-algebra (H,µ )
H
For any fuzzy hyper BCK-algebra H, we associate for all α ∈ [0,1] the set
H := {x ∈ H,µ (x) ≥ α}
α H
Lemma 3. Let H a fuzzy hyper BCK-algebra. For all α ∈ [0,1],O ∈ H and
α
for all x,y ∈ H,x∗y ⊆ H
α
Proof. By lemma 1, for all x ∈ H,µ (x) ≤ µ (0).
H H
Then for all x ∈ H ,µ (O) ≥ µ (x) > α i.e O ∈ H .
α H H α
Let x,y ∈ H ;
α
4 J.DONGHO*
for all t ∈ x∗y, µ (t) ≥ inf{µ (x∗y) ≥ min{µ (x),µ (y)}} ≥ α
H H H H
then t ∈ H . therefore, x∗y ⊆ H
α α
Definition 8. Let(H,∗,O)be anhyperBCK-algebra. AnhyperBCK-subalgebra
of H is a non empty subset S of H such that O ∈ S and S is hyper BCK-algebra
with respect to the hyper operation ”∗” on H
Proposition 3. Let (H,∗,O) be an hyper BCK-algebra. A non empty subset
S of H is hyper BCK-subalgebra of H iff for all x,y ∈ S,x∗y ∈ S
Proof. The proof is easy.
Definition 9. A fuzzy hyper BCK-subalgebra of H is an hyper BCK-subalgebra
S of H with the restriction µ of µ on S.
S H
Proposition 4. For all α ∈ [0,1], (H ,µ ) is fuzzy hyper BCK-subalgebra of
α H
H
Proof. By lemma 3, H is hyper BCK-subalgebra of H
α
and inf{µ (x∗y)} ≥ min{µ (x),µ (y)}
H H H
Definition 10. Let H by an fuzzy hyper BCK-algebra. The fuzzy-hyperBCK-
subalgebra H := (H ;µ ) is calling hyper α-cut of H
α α H
Proposition 5. Let H be fuzzy hyper BCK-algebra. A hyper BCK-subalgebra
S of H is fuzzy hyper BCK-subalgebra iff S is hyper α-cut of H.
Proof. By prosition 4, any hyper α-cut is fuzzy hyper BCK-subalgebra.
Conversely, let S be fuzzy hyper BCK-subalgebra of H. Then µ (S) is subset
H
of [0,1].
If 0 ∈ µ (S), then S = H = H.
H 0
If 0 < inf(µ (S)), then S = H .
H inf(µH(S))
Proposition 6. Let H and F be two fuzzy hyper BCK algebras. An H-
morphism f : H −→ F is F -morphism iff for all α ∈ [0,1],f(H ) ⊆ F .
H α α
Proof. Suppose that f(H ) ⊆ F for all α ∈ [0,1] Let x ∈ [0,1] we need
α α
µ (x) ≤ µ (f(x)). Let α = µ (x); x ∈ H and f(x) ∈ f(H ) ⊆ F . Then
H F H α α α
µ (f(x)) > α = µ (x). whence for all x ∈ H,µ (f(x)) ≥ µ (x).
F H F H
Conversely, suppose that f : H −→ F is F -morphism.
H
For all x ∈ H for some α ∈ [0,1], µ (f(x)) ≥ µ (x) ≥ αi.e; f(x) ∈ [0,1].
α F H
Then f(H ) ⊆ F for all α ∈ [0,1].
α α
Proposition 7. A F -morphism f : H −→ F is F -iso iff it is both H-iso
H H
and µ = µ f.
H F
Proof. Suppose that f is H-iso and µ = µ f. there is g ∈ Hom (F,H);
H F H
g ◦f = Id and g ◦g = Id .
H F
Then, for all x ∈ F,µ (g(x)) = µ (f(g(x)))µ (x).
H F F
And then, g ∈ HomF (F,H).
H
CATEGORY OF FUZZY HYPER BCK-ALGEBRAS 5
Conversely, Suppose that f is F -iso.
H
There is g ∈ HomFH(F,H);g ◦f = IdF and f ◦g = IdH.
Since f ∈ HomFH(F,H),µH ≤ µFf.
Since g ∈ HomFH(H,H),µF ≤ µHg. x ∈ H imply f(x) ∈ F. Then µF(f(x)) ≤
µ (g(f(x))) = µ (x) i.e; µ f ≤ µ .
H H F H
therefore, µ f = µ
F H
Proposition 8. Let f ∈ HomF (F,H).
H
f is H -mono iff f is H-mono
H
Proof. Suppose that f is H -mono.
H
Forallh,g ∈ Hom (K,H),suchthatfh = fg,wedefineµ = min(µ (h(x);µ (g(x))))
H K H H
for all x ∈ K.
a) We show that (K;µ ) is fuzzy hyper BCK-algebra.
K
inf(µ (x∗y)) = inf{µ (h(x∗y);µ (g(x∗y)))}
K H H
= inf{µ (h(x)∗h(y);µ (g(x)∗g(y)))}
H H
= min{inf{µ (h(x)∗h(y)};inf{µ (g(x)∗g(y)))}}
H H
≥ min{min{µ (h(x);µ (h(y)};min{µ (g(x);g(y))}}
H H H
≥ min{min{µ (x);µ (y)}}
K K
≥ min{µ (x);µ (y)}
K K
Then, for all x,y ∈ K,inf(µ (x∗y)) ≥ min{µ (x),µ (y)}. therefore,
K K K
(K;µ ) is fuzzy hyper BCK-algebra.
K
b) We show that h and g are F -homomorphism.
H
For all x ∈ K, µ (x) = min{µ (h(x),µ (g(x)))}.
K H H
Then µ (x) ≤ µ (g(x)) and µ (x) ≤ µ (h(x)).
K H K H
therefore, h and g are FH-morphism.
Since f is FH-mono and h,g ∈ HomF (F,H), fh = fg imply f = g
H
Conversely, if f is F -mono, it is H-mono.
H
Lemma 4. The pair O = ({O},µ ) where
o
µ : {o} −→ [0,1]
o
o 7→ 0
is fuzzy hyper BCK-algebra
Proof. Easy
Lemma 5. O is final objet of F
H
Proposition 9. The category F has products.
H
Proof. Let (H ;µ ) a family of fuzzy hyper BCK-algebras.
i Hi i∈I
Denote H = H the H-product of (H ) with the projection morphisms
i i i∈I
iQ∈I
p : H −→ H . Consider the following map µ : H −→ [0,1] define by:
i i H
µ (x) = µ p (x)
H ^ Hi i
i∈I
6 J.DONGHO*
for all x ∈ H
a) We show that the pair (H;µ ) is fuzzy hyper BCK-algebra.
H
For all x,y ∈ H,p (x∗y) = p (x)∗p (y) for all iNI.
i i i
Then
inf(µ p (x∗y)) = inf(µ (p (x)∗p (y))
Hi i Hi i i
≥ min{µ (p (x));µ (p (y))}
Hi i Hi i
for all i ∈ I.
Then,
inf( µ p (x∗y)) ≥ inf{µ (p (x)∗p (y)}
iV∈I Hi i iV∈I Hi i i
≥ min{µ (p (x));µ (p (y))}
iV∈I Hi i Hi i .
≥ min{ µ p (x);µ p (y)}
iV∈I Hi i Hi i
≥ min{µ (x),µ (y)}
H H
b) For all i ∈ I,x ∈ H;µ p (x) ≥ ( µ p )(x).
Hi i iV∈I Hi i
Then each p is F -morphism.
i H
c) If q : F −→ H is family of F -morphism, there is unique H-morphism
j j H
ϕ : F −→ H such that the following diagram commute.
H pj // H
j
OO }>>
}
ϕ }}
}}} qj
F
i.e p ϕ = q for all j ∈ I
j j
For all x ∈ F, µ (x) ≤ µ q (x).
F Hi i
Then µ (x) ≤ µ p ϕ(x) for all x ∈ F,i ∈ I
F Hi i
i.e µ (x) ≤ µ p ϕ(x)
F iV∈I Hi i
≤ ( µ p )ϕ(x)
iV∈I Hi i
≤ µ (ϕ(x)) for all x ∈ F.
H
Then µ ≤ µ ϕ ≤
F H
Then ϕ is F -morphism.
H
Proposition 10. F have equalizers.
H
Proof. Let f,g ∈ HomF (H,F),K := {x ∈ H,f(x) = g(x)}.
H
It is prove in [1] that K is hyper BCK-subalgebra of H. It is clear that
(K,µ ) is fuzzy hyper BCK-algebra. Let i : K −→ H the inclusion map.
H
i ∈ HomF (K,F). For all x ∈ K,fi(x) = f(x) = g(x) = gi(x).
H
Let h ∈ HomF (L,F) such that fh = gh, for all x ∈ L,f(h(x)) = g(h(x)).
H
CATEGORY OF FUZZY HYPER BCK-ALGEBRAS 7
Then Imh ⊆ L. Define δ : L −→ K by δ(x) = h(x) for all x ∈ L. δ ∈
HomF (H,F) and iδ = h. So, the following diagram commute.
H
f
K i // H //// F
OO ~?? g
~
~
δ ~~
~~ h
L
Since i is monic, δ is unique F -morphism such that the above diagram com-
H
mute.
therefore, F have equalizers.
H
Proposition 11. F is complet.
H
Proof. By proposition 9, each family of objets of F has product.
H
By proposition 10, each pair of parallel arrows has an equalizer. Then F is
H
complet.
Corollary 1. F has pulbacks
H
Proof. By propositions 9 and 10, F has equalizers and products. therefore,
H
F has pulbacks.
H
Proposition 12. F have coequalizers
H
Proof. Let f,g ∈ HomF (H,K). Let
H
= {θ,θ regular congruence relation on K such that f(a)θg(a)∀a ∈ H}
X
fg
6= φ because K ×K ∈
fg fg
P P
Let ρ = θ. Then, ρ is regular congruence relation.
θ∈TP
fg
Define on K/ρ the following hyper operation
[x] ∗[y] = [x∗y] .
ρ ρ ρ
(K/ρ;∗;[0] ) is an objet of H (see [1]).
ρ
Define on K/ρ the following map
µ : K/ρ −→ [0,1]
K/ρ
µ ([x] ) 7−→ µ (a)
K/ρ ρ K
a∈W[x]ρ
a) We show that (K/ρ,µ ) is objet of F .
K/ρ H
If x,y ∈ K such that [x] = [y] .
ρ ρ
Then
µ (a) = µ (a)
_ K _ K
a∈[x]ρ a∈[y]θ
∀x ∈ K, µ (x) ≤ µ (a).
K K
a∈W[x]ρ
Then
µ ≤ µ ([x] ) = µ (π(x))
K K/ρ ρ K/ρ
8 J.DONGHO*
Then, the canonical projection π is an F −morphism
H
Since for all x ∈ H,f(x)ρg(x), then [f(x)] = [g(x)] .
ρ ρ
therefore, (π ◦f)(x) = (π ◦g)(x).
Then, π ◦f = π ◦g.
b) Universal property of coequalizer.
Let ϕ : K −→ L and F -morphism such that ϕ◦f = ϕ◦g.
H
Define the following mapping.
ψ : K/ρ −→ L
[x] 7−→ ϕ(x)
ρ
c) We prove that ψ is well define.
If [x] = [y] then, for all a ∈ H,ϕ(f(a)) = ϕ(g(a)) imply f(a)R g(a)
ρ ρ ϕ
because R is regular congruence on K. Then R ∈ . The mini-
ϕ ϕ f,g
P
mality of ρ on imply ρ ⊆ R .
f,g ϕ
P
therefore, [x] = [y] imply xρy.
ρ ρ
Then xR y. i.e ϕ(x) = ϕ(y)
ϕ
And then, ψ([x] ) = ψ([y] ) therefore, ψ is well define.
ρ ρ
For all x ∈ K,µ (ψ(π)(x)) = µ (ϕ(x)) ≥ µ (x),∀x ∈ K then for all
L L K
a ∈ [x] . µ (ϕ(a)) ≥ µ (a). By the minimality of ρ, [a] = [x] imply
ρ L K ρ ρ
aρx then aR x i.e ϕ(a) = ϕ(x) (because ρ ⊆ R ). Then
ϕ ϕ
L˜(ϕ(a)) = L˜(ϕ(x)) = L˜(ϕ(x))
_ _
a∈[x]ρ a∈[x]ρ
therefore
L˜(ϕ(x)) ≥ (a) = K˜/ρ([x] )
_ ρ
a∈[x]ρ
i.e L˜(ψ([x] )) ≥ k˜/ρ([x] )∀x ∈ H. It is clean that
ρ ρ
ψ(π(x)) = ψ([x] ) = ϕ(x),∀x ∈ H
ρ
i.e ψ ◦π = ϕ
This prove the commutativity of the following diagram:
f
H //// K π // K/ρ
g. CCϕCCCCCCC ψ
!! (cid:15)(cid:15)
L
The unicity of ψ is thus to the fat that π is epimorphism.
Then, F have coequalizer.
H
ACKNOWLEDGEMENTS
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****Department of Mathematics, University of Yaounde, BP 812, Cameroon
E-mail address: [email protected]