Journal of Mathematics (ISSN 1016-2526) Vol. 43 (2011) pp. 85-91
Anti Fuzzy Implicative Ideals in BCK-Algebras
Nora.O.Al-Shehri Department of mathematics Faculty of Sciences for Girls
King Abdulaziz University Jeddah, Saudi Arabia Email: n [email protected]
Abstract.The aim of this paper is to introduce the notion of anti fuzzy implicative ideals ofBCK-algebras and to investigate their properties.We give several characterizations of anti fuzzy implicative ideals. We also introduce the notion of anti Cartesian product of anti fuzzy implicative ideals, and then we study related properties.
AMS (MOS) Subject Classification Codes:06F35, 03G25
Key Words:Anti fuzzy ideal, Anti fuzzy implicative ideal,BCK-algebra.
1. INTRODUCTION
The main problem in fuzzy mathematics is how to carry out the ordinary concepts to the fuzzy case. The difficultylies in how to pick out the rational generalization from the large number of available approaches. It is worth noting that fuzzy ideals are different from ordinary ideals in the sense that one cannot say whichBCK-algebra element belongs to the fuzzy ideal under consideration and which one does not. The concept of fuzzy sets was introduced by Zadeh [13]. Since then these ideas have been applied to other algebraic structures such as semigroups, groups, rings, modules, vector spaces and topologies. In 1991, Xi [12] applied the concept of fuzzy sets toBCK -algebras which are introduced by Imai and Is´eki [5] . In [1], Biswas introduced the concept of anti fuzzy subgroups of groups. Modifying his idea, in [4], S. M. Hong and Y. B. Jun applied the idea toBCK -algebras. They introduced the notion of anti fuzzy ideals ofBCK-algebras. In this paper, we introduce the notion of anti fuzzy implicative ideal ofBCK-algebras, and investigate some related properties. We show that in an implicativeBCK-algebra, a fuzzy subset is an anti fuzzy ideal if and only if it is an anti fuzzy implicative ideal. We show that a fuzzy subset of aBCK-algebra is a fuzzy implicative ideal if and only if the complement of this fuzzy subset is an anti fuzzy implicative ideal. Moreover, we discuss the pre-image of anti fuzzy implicative ideals. Finally, we introduce the notion of anti Cartesian product of anti fuzzy implicative ideals, and then we characterize anti fuzzy implicative ideals by it .
2. PRELIMINARIES
In this section we cite the fundamental definitions that will be used in the sequel:
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Definition 1. [6] An algebra(X,∗,0)of type(2,0)is called aBCK- algebra if it satisfies the following axioms for allx,y,z ∈X :
(i) ((x∗y)∗(x∗y))∗(x∗y) = 0, (ii) (x∗(x∗y))∗y= 0,
(iii)x∗x= 0, (iv) 0∗x= 0,
(v)x∗y= 0and y∗x= 0imply x=y
We can define a partial ordering≤onXbyx≤yif and only ifx∗y= 0. Proposition 2. [6]In anyBCK-algebraX , the following hold for allx, y, z∈X:
(i) (x∗y)∗z= (x∗z)∗y, (ii)x∗y≤x,
(iii)x∗0 =x,
(iv) (x∗z)∗(y∗z)≤x∗y, (v)x∗(x∗(x∗y)) =x∗y,
(vi)x≤y implies x∗z≤y∗z and z∗y≤z∗x.
ABCK-algebra is said to be implicative if x∗(y∗x) =x f orall x, y∈X(see[6,9]).
Definition 3. [8]Anon-empty subset I of aBCK-algebraX is called an ideal ofX if it satisfies
(I1) 0∈I
(I2)x∗y∈Iand y∈I imply x∈I.
Definition 4. [8] A non-empty subsetIof aBCK-algebraXis called an implicative ideal ofXif it satisfies (I1)and(I3)x∈I whenever(x∗(y∗x))∗z∈I and z∈I f orall x, y, z∈X.
Definition 5. [13] LetS be a non-empty set. A fuzzy subsetAofS is a functionA:S
−→[0,1]. LetAbe a fuzzy subset ofS. Then fort∈[0,1], thet-level cut ofAis the set At = {x∈S |A(x)≥t}and the complement ofA, denoted byAcis the fuzzy subset ofS given byAc(x) = 1−A(x)for allx∈S.(see [2,3,7]).
Definition 6. [12] A fuzzy subsetAof aBCK-algebra is called a fuzzy subalgebra ofXif A(x∗y)≥min{A(x), A(y)}f orall x, y∈X.
Definition 7. [12] LetX be aBCK-algebra. A fuzzy subsetAofXis called a fuzzy ideal ofXif
(F1)A(0)≥A(x),
(F2)A(x)≥min {A(x∗y), A(y)}, f or all x, y∈X.
Definition 8. [10] A fuzzy subsetAof aBCK-algebraXis called a fuzzy implicative ideal ofXif it satisfies
(F1)and(F3) A(x)≥min {A((x∗(y∗x))∗z), A(z)}f orall x, y, z∈X.
Definition 9. [4] A fuzzy subsetAof aBCK-algebraX is called an anti fuzzy subalgebra ofXif
A(x∗y)≤ max {A(x), A(y)}f orall x, y∈X.
Definition 10. [4] A fuzzy subsetAof aBCK-algebraX is called an anti fuzzy ideal ofX if
(A1)A(0)≤A(x),
(A2)A(x)≤ max{A(x∗y), A(y)}, f orallx, y∈X.
Proposition 11. [4]Every anti fuzzy ideal of aBCK-algebra X is an anti fuzzy subalgebra ofX.
Definition 12. [4] LetAbe a fuzzy subset of aBCK-algebra. Then fort∈[0,1]the lower t-level cut ofAis the set
At={x∈X|A(x)≤t}
Definition 13. [4] LetAbe a fuzzy subset of aBCK-algebra. The fuzzification ofAt, t∈ [0,1]is the fuzzy subsetµAt ofXdefined by
Definition 14. [11] Let f:X →Y be a mapping ofBCK-algebras andAbe a fuzzy subset ofY . The mapAfis the inverse image ofAunderf ifAf(x) =A(f(x))∀x∈X.
3. ANTI FUZZY IMPLICATIVE IDEAL
Definition 15. A fuzzy subsetAof aBCK-algebraX is called an anti fuzzy implicative ideal ofXif it satisfies
(A1)and(A3)A(x)≤max{A((x∗(y∗x))∗z), A(z)}f orall x, y, z∈X.
Example 1. (1) Every constant functionA:X−→[0,1]is an anti fuzzy implicative ideal of .X(2) LetX ={0, a, b, c}be aBCK-algebra with Cayley table as follows:
∗ 0 a b c
0 0 0 0 0
a a 0 0 a b b a 0 b c c c c 0
Lett0,t1 be such that t0 ≺t1.DefineA: X −→[0,1]byA(0) = A(a) =A(b) = t0 andA(c) =t1Routine calculations give thatAis an anti fuzzy implicative ideal.
Proposition 16. Every anti fuzzy implicative ideal of aBCK-algebraXis order preserv- ing.
Proof. LetAbe an anti fuzzy implicative ideal of a BCK-algebraX and letx, y ∈Xbe such that x≤yThen
A(x)≤max{A((x∗(z∗x))∗y), A(y)},
=max{A((x∗y)∗(z∗x)), A(y)},
=max{A(0∗(z∗x)), A(y)},
=max{A(0), A(y)}=A(y).
Proposition 17. Every anti fuzzy implicative ideal of aBCK-algebraX is an anti fuzzy ideal.
Proof. LetAbe an anti fuzzy implicative ideal of aBCK-algebraX,so for allx, y, z ∈ X :
A(x)≤max{A((x∗(y∗x))∗z), A(z)}, Putting in y=x,andz=y:
A(x)≤max{A((x∗x)∗y), Ay}
=max{A(x∗y), A(y)}.
Combining Proposition 2.11 and 3.4 yields the following result.
Proposition 18. Every anti fuzzy implicative ideal of aBCK-algebraX is an anti fuzzy subalgebra ofX.
Remark 19. An anti fuzzy ideal (subalgebra) of aBCK-algebra X may not be an anti fuzzy implicative ideal ofX as shown in the following example:
Example 2. LetXbe theBCK-algebra in Example 3.2(2) and let t0,t1,t2∈[0,1]be such thatt0t1t2DefineA : X −→A(0) = t0, A(a) =A(b) = t1 andA(c) = t2Routine calculations give thatA is an anti fuzzy ideal (subalgebra) ofX. but not an anti fuzzy implicative ideal ofXbecause
A(a) =t1> max{A((a∗(b∗a))∗0), A(o)}
= max{A(0), A(0)}=t0.
Proposition 20. IfX is implicativeBCK-algebra, then every anti fuzzy ideal ofX is an anti fuzzy implicative ideal ofX.
Proof. LetAbe an anti fuzzy ideal of an implicativeBCK-al gebraX, so for allx, z∈X : A(x)≤max{A(x∗z), A(z)},
SinceXis an implicative, thenx∗(y∗x) =x f orall x, y∈X. Hence A(x)≤max{A((x∗(y∗x))∗z), A(z)},
This shows thatAis an anti fuzzy implicative ideal ofX. By applying Proposition 17 and 20 , we have
Theorem 21. IfX is an implicativeBCK-algebra, then a fuzzy subset A ofX is an anti fuzzy ideal ofXif and only if it is an anti fuzzy implicative ideal ofX.
Proposition 22. Afuzzy subsetAof aBCK-algebraXis a fuzzy implicative ideal ofXif and only if its complementAcis an anti fuzzy implicative ideal ofX.
Proof. LetAbe a fuzzy implicative ideal of aBCK-algebraX,and letx, y, z∈X. Then A(0) = 1−A(0)≤1−A(x) =Ac(x)and
Ac(x) = 1−A(x)≤1−min{A((x∗(y∗x))∗z), A(z)},
= 1−min{1−Ac((x∗(y∗x))∗z),1−Ac(z)},
=max{Ac((x∗(y∗x))∗z), Ac(z)}.
So,Acis an anti fuzzy implicative ideal ofX. Now letAcis an anti fuzzy implicative ideal ofX, and letx, y, z∈X. Then
A(0) = 1−Ac(0)≥1 =Ac(x) =A(x)and
A(x)−1Ac(x)≤1−max{Ac((x∗(y∗x))∗z), Ac(z)},
= 1−max{1−A((x∗(y∗x))∗z) 1−A(z)},
=min{A((x∗(y∗x))∗z), A(z)}.
Thus,Ais a fuzzy implicative ideal ofX.
Theorem 23. LetAbe an anti fuzzy implicative ideal of aBCK-algebraX. Then the set XA={x∈X |A(x) =A(0) },
is an implicative ideal ofX.
Proof. Clearly0∈X. Letx, y, z∈XAbe such that(x∗(x∗y))∗z∈X andz∈XA. Then
A((x∗(y∗x))∗z) =A(z) =A(0).
It follows that
A(x)≤max{A((x∗(y∗x))∗z), A(z)}
=max{A(0), A(0)}=A(0).
Combining Definition 15(A1), we getA(x) =A(0)and hencex∈X. Theorem 24. LetAbe a fuzzy subset A of aBCK-algebraX. ThenAis an anti fuzzy implicative ideal ofXif and only if for eacht∈[0,1], t≥A(0), the lowertt-level cutAt is an implicative ideal ofX.
Proof. Let Abe an anti fuzzy implicative ideal of X and lett ∈ [0,1]witht ≥ A(0).
Clearly0 ∈At. Letx, y, z ∈X be such that(x∗(y∗x))∗z ∈ Atand z ∈ At.Then A((x∗(y∗x))∗z)≤t , A(z)≤t ,
hence
A(x)≤max{A((x∗(y∗x))∗z)A(z)} ≤t. And sox∈At. HenceAtis an implicative ideal ofX.
Conversely, letAtis an implicative ideal ofX, we first showA(0)≤A(x)for allx∈X.
If not, then there existsx0∈X such thatA(0)¿A(x). Puttingt0= 12{A(0) +A(x0)} then0 ≤ A(x0)¡t0 ¡A(0) ≤ 1. Hence x0 ∈ At0, so thatAt0 ̸= ∅. ButAt0 is an implicative ideal ofX. Thus0 ∈ At0 , orA(0) ≤t0,a contradiction. HenceA(0) ≤ A(x)for allx∈X. Now we prove thatA(x)≤max{A((x∗(y∗))∗z), A(z)}for all x, y, z∈X. If not, then there existsx0, y0, z0∈X, such that
A(x)> max{A((x0∗(y0∗x0))∗z0), A(z0)}, Takings0= 12{A(x0) +max{((x0∗(y0∗x0))∗z0), A(z0)}};
then s0≤A(x0)and
0≤max{A((x0∗(y0∗x0))∗z0)}¡s0 ≤1.
Thus we haves0¿A((x0∗(y0∗x0))∗z0)ands0¿A(z0). Which imply that
(x0∗(y0∗x0))∗z0∈As0 and z0 ∈ As0 .But As0is an implicative ideal ofX . Thus x0∈As0 orA(x0)≤s0This is a contradiction, ending the proof.
Theorem 25. IfAis an anti fuzzy implicative ideal of aBCK-algebraX. ThenµAtis also an anti fuzzy implicative ideal ofX wheret∈[0,1], t,≥A(0).
Proof. From Theorem 24, it is sufficient to show that (µAt)s is an implicative ideal of X,wheres ∈ [0,1]ands≥ µAt(0). Clearly,0 = (µAt)Letx, y, z ∈ X be such that (x∗(y∗x))∗z∈(µAt)sandz∈(µAt)s,
hence
µAt((x∗(y∗x))∗z)≤s , and
µAt(z)≤s.We claim that x∈(µAt)or µAt(x)≤s. If(x∗(y∗x))∗z∈ Atandz∈At, thenx∈AtbecauseAtis an implicative ideal ofX. Hence.
µAt(x) =A(x)
≤ max{A((x∗(y∗x))∗z), A(z)},
= max{µAt((x∗(y∗x))∗z), µAt(z)} ≤s.
and sox∈(µAt)sIf(x∗(y∗x))∗z /∈Atorz ∈At, thenµAt((x∗(y∗x))∗z) = 0 orµAt(z) = 0then clearlyµAt(x) ≤ s, and sox ∈ (µAt)s. Therefore (µAt)sis an
implicative ideal ofX.
Theorem 26.
Theorem 3.13.
Let f :X−→Y be a homomorphism of BCK-algebras.IfAis an anti fuzzy implicative ideal ofY , thenAfis an anti fuzzy implicative ideal ofX.
Proof. SinceAis an anti fuzzy implicative ideal ofY, thenA(0′)≤A(f(x))f orany x∈X, and so,Af(0) =A(f(0)) =A(0′)≤A(f(x)) =Af(x). For anyx, y, z∈X, we have:
Af(x) =A(f(x))≤max{A((f(x)∗(f(y)∗f(x)))∗f(z)), A(f(z))},
=max{A((f(x)∗(f(y∗x)))∗f(z)), A(f(z))},
=max{A((f(x∗(y∗x)))∗f(z)), Af(z)},
=max{A(f((x∗(y∗x))∗z)), A(f(z))},
=max{
Af((x∗(y∗x))∗z), Af(z)} .
This completes the proof.
Theorem 27.
Theorem 3.14.
Let f :X −→Y be an epimorphism of BCK-algebras.IfAfis an anti fuzzy implicative ideal ofX, thenAis an anti fuzzy implicative ideal ofY. Proof. Lety∈Y , there existsx∈X such thatf(x) =y. Then
A(y) =A(f(x)) =Af(x)≥Af(0) =A(0′).
Letx′, y′, z′ ∈Y . Then there exist x, y, z ∈X . such thatf(x) =x′, f(y) =y′and f(z) =z′. It follows that
A(x′) =A(f(x)) =Af(x)≤max {
Af((x∗(y∗x))∗z), Af(z)} ,
=max{A(f((x∗(y∗x))∗z)), A(f(z))},
=max{A((f(x)∗(f(y∗x)))∗f(z)), A(f(z))},
=max{A((f(x)∗(f(y)∗f(x)))∗f(z)), A(f(z))},
=max{A((x′∗(y′∗x′))∗z′), A(z′)}.
HenceAis an anti fuzzy implicative ideal ofY.
4. ANTICARTESIAN PRODUCT OF ANTI FUZZY IMPLICATIVE IDEALS
Definition 28. Letλandµbe the fuzzy subsets in a setX. The anti Cartesian product λ×µ: X ×X −→[0, 1] is defined by(λ×µ) (x, y) = max {λ(x), µ(y)}f orall x, y∈X.
Theorem 29. Ifλandµare anti fuzzy implicative ideals of aBCK-algebraX, thenλ×µ is an anti fuzzy implicative ideal ofX×X.
Proof. Letx, x′∈X
(λ×µ) (0,0) =max{λ(0), µ(0)}
≤max{λ(x), µ(x)}= (λ×µ) (x, x′) For any(x, x′),(y, y′),(z, z′)∈X×X we have
(λ×µ) (x, x′) =max{λ(x), µ(x′)}
≤max{max{λ(x∗(y∗x))∗z), λ(z)}, max {µ(x′∗(y′∗x′))z′), µ(z′)}},
=max{max {λ(x∗(y∗x))∗z), µ(x′(y′∗x′))z′)}, max{λ(z), µ(z′)}},
=max{(λ×µ) ((x, x′)∗((y, y′)∗(x, x′)))∗(z, z′)),(λ×µ) (z, z′)}.
Henceλ×µis an anti fuzzy implicative ideal ofX×X
Theorem 30. Letλandµbe fuzzy subsets in aBCK-algebraXsuch thatλ×µis an anti fuzzy implicative ideal ofX×XThen:
(i) eitherλ(x)≥λ(0)orµ(x)≤µ(0)for allx∈X ;
(ii) ifλ(x)≥λ(0)for allx∈X ,then eitherλ(x)≥µ(0)orµ(x)≥µ(0) ; (iii) ifµ(x)≥µ(0)for allx∈X, then eitherλ(x)≥λ(0)orµ(x)≥λ(0). Proof. (i) Suppose thatλ(x)¡λ(0)andµ(y)¡µ(0)for somex, y∈X. Then
(λ×µ) (x, y) =max{λ(x, µ(y))}¡ max{λ(0), µ(0)}= (λ, µ) (0,0). This is a contradiction and we obtain (i).
(ii) Assume that there existx, y∈Xsuch thatλ(x)¡µ(0)andµ(x)¡µ(0). Then (λ×µ) (0,0) =max{λ(0), µ(0)}=µ(0).
It follows that
(λ×µ) (x, y) =max{λ(x), µ(y)}¡µ(0) = (λ×µ) (0,0), which is a contradiction. Hence (ii) holds.
(iii) Similar to (ii).
Theorem 31. Letλandµbe fuzzy subsets in aBCK-algebraXsuch thatλ×µis an anti fuzzy implicative ideal ofX×X. Then eitherµorλis an anti fuzzy implicative ideal of X.
Proof. By Theorem 30(i), without loss of generality we assume thatµ(x)≥µ(0)for all x∈X. From (iii) it follows that eitherλ(x)≥λ(0)orµ(x)≥λ(0).Ifµ(x)≥λ(0) for allx∈X, then(λ×µ) (0, x) =max{λ(0), µ(x)}=µ(x).Let(x, x′),(y, y′), (z, z′) ∈ X ×X, sinceλ×µis an anti fuzzy implicative ideal of X ×X.We have (λ×µ) (x, x′)≤max{(λ×µ) (((x, x′)∗(y, y′)∗(x, x′))∗(z, z′)),(λ×µ) (z, z′)},
=max(λ×µ) ((x∗(y∗x))∗z,(x′∗(y′∗x′))∗z′),(λ×µ) (z, z′)}. Puttingx=y=z= 0, then
µ(x′) = (λ×µ) (0, x′)≤max{(λ×µ) (0,(x′∗(y′∗x′))∗z′),(λ×µ) (0, z′)},
=max{max{λ(0), µ((x′(y′∗x′))∗z′)}, max {λ(0), µ(z′)}},
=max{µ((x′∗(y′∗x′))∗z′), µ(z′)}.
This proves thatµis an anti fuzzy implicative ideal ofX. The second part is similar. This
completes the proof.
5. CONCLUSION
We discussed the notion of anti fuzzy implicative ideals of BCK-algebras and gave several characterizations. Also, we introduced the notion of anti cartesian product of anti fuzzy implicative ideals. Our defintions probably can be applied in other kinds of anti ideals ofBCK-algebras.
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