ON n-FOLD FUZZY POSITIVE IMPLICATIVE IDEALS OF BCK-ALGEBRAS

PII. S0161171201006135 http://ijmms.hindawi.com

© Hindawi Publishing Corp.

ON n-FOLD FUZZY POSITIVE IMPLICATIVE IDEALS OF BCK-ALGEBRAS

YOUNG BAE JUN and KYUNG HO KIM (Received 11 March 2000)

Abstract.We consider the fuzzification of the notion of ann-fold positive implicative ideal. We give characterizations of ann-fold fuzzy positive implicative ideal. We establish the extension property forn-fold fuzzy positive implicative ideals, and state a characteri- zation of PIⁿ-Noetherian BCK-algebras. Finally we study the normalization ofn-fold fuzzy positive implicative ideals.

2000 Mathematics Subject Classification. 06F35, 03G25, 03E72.

1. Introduction. For the general development of BCK-algebras, the ideal theory plays an important role. In 1999, Huang and Chen [1] introduced the notion ofn-fold positive implicative ideals in BCK-algebras. In this paper, we consider the fuzzifica- tion ofn-fold positive implicative ideals in BCK-algebras. We first define the notion of n-fold fuzzy positive implicative ideals of BCK-algebras, and then discuss the related properties. We give the relation between a fuzzy ideal and ann-fold fuzzy positive implicative ideal. We state a condition for a fuzzy ideal to be ann-fold fuzzy positive implicative ideal. Using level sets, we give a characterization of ann-fold fuzzy posi- tive implicative ideal. We establish the extension property for ann-fold fuzzy positive implicative ideal. Using a family ofn-fold fuzzy positive implicative ideals, we make a newn-fold fuzzy positive implicative ideal. We define the notion of PIⁿ-Noetherian BCK-algebras, and give its characterization. Furthermore, we study the normalization of ann-fold fuzzy positive implicative ideal.

2. Preliminaries. By aBCK-algebrawe mean an algebra(X;∗,0)of type (2, 0) sat- isfying the axioms

(I) ((x∗y)∗(x∗z))∗(z∗y)=0, (II) (x∗(x∗y))∗y=0,

(III) x∗x=0, (IV) 0∗x=0,

(V) x∗y=0 andy∗x=0 implyx=y,

for all x, y, z∈X. We can define a partial ordering ≤ on X by x ≤y if and only ifx∗y=0.A BCK-algebraXis said to ben-fold positive implicative(see Huang and Chen [1]) if there exists a natural numbernsuch thatx∗yⁿ⁺¹=x∗yⁿfor allx, y∈X.

In any BCK-algebraX, the following hold:

(P1) x∗0=x, (P2) x∗y≤x,

(P3) (x∗y)∗z=(x∗z)∗y,

(P4) (x∗z)∗(y∗z)≤x∗y,

(P5) x≤yimpliesx∗z≤y∗zandz∗y≤z∗x.

Throughout this paperXwill always mean a BCK-algebra unless otherwise specified.

A nonempty subsetIofXis called anidealofXif it satisfies (I1) 0∈I,

(I2) x∗y∈Iandy∈Iimplyx∈I.

A nonempty subsetIofXis said to be apositive implicative idealif it satisfies (I1) 0∈I,

(I3) (x∗y)∗z∈Iandy∗z∈Iimplyx∗z∈I.

Theorem2.1(see [3, Theorem 3]). A nonempty subsetIofXis a positive implicative ideal ofXif and only if it satisfies

(I1) 0∈I,

(I4) ((x∗y)∗y)∗z∈Iandz∈Iimplyx∗y∈I.

We now review some fuzzy logic concepts. A fuzzy set in a setX is a function µ:X→[0,1]. For a fuzzy setµinXandt∈[0,1]defineU (µ;t)to be the setU (µ;t):= {x∈X|µ(x)≥t}.

A fuzzy setµinXis said to be afuzzy idealofXif (F1) µ(0)≥µ(x)for allx∈X,

(F2) µ(x)≥min{µ(x∗y), µ(y)}for allx, y∈X.

Note that every fuzzy idealµofXis order reversing, that is, ifx≤ythenµ(x)≥ µ(y).

A fuzzy setµinXis called afuzzy positive implicative idealofXif it satisfies (F1) µ(0)≥µ(x)for allx∈X,

(F3) µ(x∗z)≥min{µ((x∗y)∗z), µ(y∗z)}for allx, y, z∈X.

Theorem2.2(see [2, Proposition 1]). For any fuzzy idealµofX, we have µ

x∗y

≥µ x∗y

∗y

⇐⇒µ

(x∗z)∗ y∗z

≥µ x∗y

∗z

∀x, y, z∈X.

(2.1) 3. n-fold fuzzy positive implicative ideals. For any elementsx andyof a BCK- algebra,x∗yⁿdenotes

···

x∗y

∗y

∗···

∗y (3.1)

in whichy occursntimes. UsingTheorem 2.1, Huang and Chen [1] introduced the concept of ann-fold positive implicative ideal as follows.

Definition3.1. A subsetAofXis called ann-fold positive implicative idealofXif (I1) 0∈A,

(I5) x∗yⁿ∈Awhenever(x∗yⁿ⁺¹)∗z∈Aandz∈Afor everyx, y, z∈X.

We try to fuzzify the concept ofn-fold positive implicative ideal.

Definition3.2. Letnbe a positive integer. A fuzzy setµinXis called ann-fold fuzzy positive implicative idealofXif

(F1) µ(0)≥µ(x)for allx∈X,

(F4) µ(x∗yⁿ)≥min{µ((x∗yⁿ⁺¹)∗z), µ(z)}for allx, y, z∈X.

Notice that the 1-fold fuzzy positive implicative ideal is a fuzzy positive implicative ideal.

Example3.3. LetX= {0, a, b}be a BCK-algebra with the following Cayley table:

∗ 0 a b

0 0 0 0

a a 0 0

b b b 0

Define a fuzzy set µ : X→[0,1] by µ(0)=t0, µ(a)=t1, and µ(b)= t2 where t0> t1> t2in[0,1]. Thenµis ann-fold fuzzy positive implicative ideal ofXfor every natural numbern.

Proposition3.4. Everyn-fold fuzzy positive implicative ideal is a fuzzy ideal for every natural numbern.

Proof. Letµbe ann-fold fuzzy positive implicative ideal ofX. Then µ(x)=µ

x∗0ⁿ

≥min µ

x∗0ⁿ⁺¹

∗z , µ(z)

=min

µ(x∗z), µ(z)

∀x, z∈X. (3.2) Henceµis a fuzzy ideal ofX.

The following example shows that the converse ofProposition 3.4may not be true.

Example3.5. LetX=N∪ {0}, whereNis the set of natural numbers, in which the operation ∗is defined by x∗y=max{0, x−y}for all x, y∈X. ThenX is a BCK-algebra [1, Example 1.3]. Letµbe a fuzzy set inXgiven byµ(0)=t0> t1=µ(x) for allx(=0)∈X. Thenµis a fuzzy ideal ofX. Butµis not a 2-fold fuzzy positive implicative ideal ofX becauseµ(5∗2²)=µ(1)=t1and µ((5∗2³)∗0)=µ(0)=t0, and so

µ 5∗2²

≥min µ

5∗2³

∗0 , µ(0)

. (3.3)

LetXbe ann-fold positive implicative BCK-algebra and letµbe a fuzzy ideal ofX.

For anyx, y, z∈Xwe have µ

x∗yⁿ

=µ

x∗yⁿ⁺¹

≥min µ

x∗yⁿ⁺¹

∗z , µ(z)

. (3.4)

Hence µ is an n-fold fuzzy positive implicative ideal of X. Combining this and Proposition 3.4, we have the following theorem.

Theorem 3.6. In ann-fold positive implicative BCK-algebra, the notion ofn-fold fuzzy positive implicative ideals and fuzzy ideals coincide.

Proposition3.7. Letµ be a fuzzy ideal of X. Thenµ is an n-fold fuzzy positive implicative ideal ofXif and only if it satisfies the inequalityµ(x∗yⁿ)≥µ(x∗yⁿ⁺¹) for allx, y∈X.

Proof. Suppose thatµ is ann-fold fuzzy positive implicative ideal ofXand let x, y∈X. Then

µ x∗yⁿ

≥min µ

x∗yⁿ⁺¹

∗0 , µ(0)

=min µ

x∗yⁿ⁺¹ , µ(0)

=µ

x∗yⁿ⁺¹ .

(3.5)

Conversely, letµbe a fuzzy ideal ofXsatisfying the inequality µ

x∗yⁿ

≥µ

x∗yⁿ⁺¹

∀x, y∈X. (3.6) Then

µ x∗yⁿ

≥µ

x∗yⁿ⁺¹

≥min µ

x∗yⁿ⁺¹

∗z , µ(z)

∀x, y, z∈X. (3.7) Henceµis ann-fold fuzzy positive implicative ideal ofX.

Corollary 3.8. Everyn-fold fuzzy positive implicative idealµ ofX satisfies the inequalityµ(x∗yⁿ)≥µ(x∗y^n+k)for allx, y∈Xandk∈N.

Proof. UsingProposition 3.7, the proof is straightforward by induction.

Lemma3.9. LetAbe a nonempty subset ofXand letµbe a fuzzy set inXdefined by

µ(x):=





t1 ifx∈A,

t2 otherwise, (3.8)

wheret1> t2in[0,1]. Thenµis a fuzzy ideal ofXif and only ifAis an ideal ofX.

Proof. LetAbe an ideal ofX. Since 0∈A, thereforeµ(0)=t1≥µ(x)for allx∈X.

Suppose that (F2) does not hold. Then there exista, b∈X such thatµ(a)=t2and min{µ(a∗b), µ(b)} =t1. Thus µ(a∗b)=t1=µ(b), and so a∗b∈Aand b∈A.

It follows from (I2) that a∈Aso that µ(a)=t1. This is a contradiction. Suppose thatµ is a fuzzy ideal ofX. Sinceµ(0)≥µ(x)for allx∈X, we haveµ(0)=t1and hence 0∈A. Let x, y∈X be such that x∗y ∈A and y∈A. Using (F2), we get µ(x)≥min{µ(x∗y), µ(y)} =t1and soµ(x)=t1, that is,x∈A. Consequently,Ais an ideal ofX.

Proposition3.10. LetAbe a nonempty subset ofX, na positive integer, andµa fuzzy set inXdefined as follows:

µ(x):=





t1 ifx∈A,

t2 otherwise, (3.9)

wheret1> t2in[0,1]. Thenµis ann-fold fuzzy positive implicative ideal ofXif and only ifAis ann-fold positive implicative ideal ofX.

Proof. Assume thatµis ann-fold fuzzy positive implicative ideal ofX. Thenµis a fuzzy ideal ofX. It follows fromLemma 3.9thatAis an ideal ofX. Letx, y∈Xbe such thatx∗yⁿ⁺¹∈A. UsingProposition 3.7, we getµ(x∗yⁿ)≥µ(x∗yⁿ⁺¹)=t1and so

µ(x∗yⁿ)=t1, that is,x∗yⁿ∈A. Hence by [1, Theorem 1.5], we conclude thatAis an n-fold positive implicative ideal ofX. Conversely, suppose thatAis ann-fold positive implicative ideal ofX. ThenA is an ideal ofX (see [1, Proposition 1.2]). It follows fromLemma 3.9thatµ is a fuzzy ideal ofX. For anyx, y∈X, eitherx∗yⁿ∈Aor x∗yⁿ∉A. The former inducesµ(x∗yⁿ)=t1≥µ(x∗yⁿ⁺¹). In the latter, we know that x∗yⁿ⁺¹∉A by [1, Theorem 1.5]. Hence µ(x∗yⁿ)=t2=µ(x∗yⁿ⁺¹). From Proposition 3.7it follows thatµis ann-fold fuzzy positive implicative ideal ofX.

Proposition3.11. A fuzzy setµinXis ann-fold fuzzy positive implicative ideal ofXif and only if it satisfies

(F1) µ(0)≥µ(x),

(F5) µ(x∗zⁿ)≥min{µ((x∗y)∗zⁿ), µ(y∗zⁿ)},for allx, y, z∈X.

Proof. Suppose thatµ is ann-fold fuzzy positive implicative ideal ofXand let x, y, z∈X. Thenµis a fuzzy ideal ofX(seeProposition 3.4), and soµis order revers- ing. It follows from (P3), (P4), and (P5) that

µ

x∗z²ⁿ

∗

y∗zⁿ

=µ x∗zⁿ

∗

y∗zⁿ

∗zⁿ

≥µ x∗y

∗zⁿ

. (3.10) Using (F2) andCorollary 3.8, we get

µ x∗zⁿ

≥µ

x∗z²ⁿ

≥min µ

x∗z²ⁿ

∗

y∗zⁿ , µ

y∗zⁿ

≥min µ

x∗y

∗zⁿ , µ

y∗zⁿ

, (3.11)

which proves (F5). Conversely, assume thatµsatisfies conditions (F1) and (F5). Taking z=0 in (F5) and using (P1), we conclude that

µ(x)=µ(x∗0)≥min µ

x∗y

∗0ⁿ , µ

y∗0ⁿ

=min µ

x∗y , µ

y

. (3.12)

Henceµis a fuzzy ideal ofX. Puttingz=yin (F5) and applying (III), (IV), and (F1), we have

µ x∗yⁿ

≥min µ

x∗y

∗yⁿ , µ

y∗yⁿ

=min µ

x∗yⁿ⁺¹ , µ(0)

=µ

x∗yⁿ⁺¹

. (3.13)

ByProposition 3.7, we know thatµis ann-fold fuzzy positive implicative ideal ofX.

Now we give a condition for a fuzzy ideal to be ann-fold fuzzy positive implicative ideal.

Theorem3.12. A fuzzy setµinXis ann-fold fuzzy positive implicative ideal ofX if and only ifµis a fuzzy ideal ofXin which the following inequality holds:

(F6) µ((x∗zⁿ)∗(y∗zⁿ))≥µ((x∗y)∗zⁿ)for allx, y, z∈X.

Proof. Assume that µ is an n-fold fuzzy positive implicative ideal of X. By Proposition 3.4, it follows thatµis a fuzzy ideal ofX. Leta=x∗(y∗zⁿ)andb=x∗y.

Then

µ

(a∗b)∗zⁿ

=µ x∗

y∗zⁿ

∗ x∗y

∗zⁿ

≥µ y∗

y∗zⁿ

∗zⁿ

=µ(0), (3.14)

and soµ((a∗b)∗zⁿ)=µ(0). Using (F5) we obtain µ

x∗zⁿ

∗

y∗zⁿ

=µ x∗

y∗zⁿ

∗zⁿ

=µ a∗zⁿ

≥min µ

(a∗b)∗zⁿ , µ

b∗zⁿ

=min µ(0), µ

b∗zⁿ

=µ b∗zⁿ

=µ x∗y

∗zⁿ ,

(3.15)

which is condition (F6). Conversely, letµ be a fuzzy ideal ofX satisfying condition (F6). It is sufficient to show thatµsatisfies condition (F5). For anyx, y, z∈Xwe have

µ x∗zⁿ

≥min µ

x∗zⁿ

∗

y∗zⁿ , µ

y∗zⁿ

≥min µ

x∗y

∗zⁿ , µ

y∗zⁿ

, (3.16)

which is precisely (F5). Henceµis ann-fold fuzzy positive implicative ideal ofX.

Theorem3.13. Letµbe a fuzzy set inXand letnbe a positive integer. Thenµis an n-fold fuzzy positive implicative ideal ofXif and only if the nonempty level setU (µ;t) ofµis ann-fold positive implicative ideal ofXfor everyt∈[0,1].

Proof. Assume thatµis ann-fold fuzzy positive implicative ideal ofXandU (µ;t)

= ∅for everyt∈[0,1]. Then there existsx∈U (µ;t). It follows from (F1) thatµ(0)≥ µ(x)≥tso that 0∈U (µ;t). Letx, y, z∈Xbe such that(x∗yⁿ⁺¹)∗z∈U (µ;t)and z∈U (µ;t). Thenµ((x∗yⁿ⁺¹)∗z)≥tandµ(z)≥t, which imply from (F4) that

µ x∗yⁿ

≥min µ

x∗yⁿ⁺¹

∗z , µ(z)

≥t, (3.17)

so thatx∗yⁿ∈U (µ;t). ThereforeU (µ;t)is ann-fold positive implicative ideal of X. Conversely, suppose thatU (µ;t)(= ∅)is ann-fold positive implicative ideal ofX for everyt∈[0,1]. For anyx∈X, letµ(x)=t. Thenx∈U (µ;t). Since 0∈U (µ;t), we getµ(0)≥t=µ(x)and soµ(0)≥µ(x)for all x∈X. Now assume that there exista, b, c∈X such that µ(a∗bⁿ) <min{µ((a∗bⁿ⁺¹)∗c), µ(c)}. Selectings0= (1/2)(µ(a∗bⁿ)+min{µ((a∗bⁿ⁺¹)∗c), µ(c)}), then

µ a∗bⁿ

< s0<min µ

a∗bⁿ⁺¹

∗c , µ(c)

. (3.18)

It follows that(a∗bⁿ⁺¹)∗c∈U (µ;s0),c∈U (µ;s0), anda∗bⁿ∉U (µ;s0). This is a contradiction. Henceµis ann-fold fuzzy positive implicative ideal ofX.

Theorem3.14. Ifµis ann-fold fuzzy positive implicative ideal ofX, then the set

Xµ:=

x∈X|µ(x)=µ(0)

(3.19) is ann-fold positive implicative ideal ofX.

Proof. Letµbe ann-fold fuzzy positive implicative ideal ofX. Clearly 0∈Xµ. Let x, y, z∈Xbe such that(x∗yⁿ⁺¹)∗z∈Xµandz∈Xµ. Then

µ x∗yⁿ

≥min µ

x∗yⁿ⁺¹

∗z , µ(z)

=µ(0). (3.20) It follows from (F1) thatµ(x∗yⁿ)=µ(0)so thatx∗yⁿ∈Xµ. HenceXµis ann-fold positive implicative ideal ofX.

Theorem3.15(extension property forn-fold fuzzy positive implicative ideals). Let µandνbe fuzzy ideals ofXsuch thatµ(0)=ν(0)andµ⊆ν, that is,µ(x)≤ν(x)for allx∈X. Ifµis ann-fold fuzzy positive implicative ideal ofX, then so isν.

Proof. UsingProposition 3.7, it is sufficient to show thatνsatisfies the inequality ν(x∗yⁿ)≥ν(x∗yⁿ⁺¹)for allx, y∈X. Letx, y∈X. Then

ν(0)=µ(0)=µ x∗

x∗yⁿ⁺¹

∗yⁿ⁺¹

≤µ x∗

x∗yⁿ⁺¹

∗yⁿ

=µ x∗yⁿ

∗

x∗yⁿ⁺¹

≤ν x∗yⁿ

∗

x∗yⁿ⁺¹ .

(3.21)

Sinceνis a fuzzy ideal, it follows from (F1) and (F2) that ν

x∗yⁿ

≥min ν

x∗yⁿ

∗

x∗yⁿ⁺¹ , ν

x∗yⁿ⁺¹

≥min ν(0), ν

x∗yⁿ⁺¹

=ν

x∗yⁿ⁺¹

. (3.22)

This completes the proof.

4. PIⁿ-Noetherian BCK-algebras

Definition4.1. A BCK-algebraXis said to satisfy the PIⁿ-ascending (resp., PIⁿ- descending)chain condition(briefly, PIⁿ-ACC (resp., PIⁿ-DCC)) if for every ascending (resp., descending) sequenceA1⊆A2⊆ ··· (resp.,A1⊇A2⊇ ···) ofn-fold positive implicative ideals ofXthere exists a natural numberrsuch thatAr=Akfor allr≥k.

IfXsatisfies the PIⁿ-ACC, we say thatXis a PIⁿ-Noetherian BCK-algebra.

Theorem4.2. Let{Ak|k∈N}be a family ofn-fold positive implicative ideals ofX which is nested, that is,A1⊋A2⊋···.Letµbe a fuzzy set inXdefined by

µ(x)=







 k

k+1 ifx∈Ak\A_k+1, k=0,1,2, . . . , 1 ifx∈ ∩^∞_k=0Ak,

(4.1)

for allx∈X, whereA0stands forX. Thenµ is ann-fold fuzzy positive implicative ideal ofX.

Proof. Clearlyµ(0)≥µ(x)for allx∈X. Letx, y, z∈X. Suppose that x∗yⁿ⁺¹

∗z∈Ak\Ak+1, z∈Ar\Ar+1 (4.2) fork=0,1,2, . . .;r=0,1,2, . . . .Without loss of generality, we may assume thatk≤r.

Then obviouslyz∈Ak. SinceAkis ann-fold positive implicative ideal, it follows that x∗yⁿ∈Akso that

µ x∗yⁿ

≥ k

k+1=min µ

x∗yⁿ⁺¹

∗z , µ(z)

. (4.3)

If(x∗yⁿ⁺¹)∗z∈ ∩^∞k=0Akandz∈ ∩^∞k=0Ak,thenx∗yⁿ∈ ∩^∞k=0Ak. Hence µ

x∗yⁿ

=1=min µ

x∗yⁿ⁺¹

∗z , µ(z)

. (4.4)

If (x∗yⁿ⁺¹)∗z∉∩^∞_k=0Ak and z∈ ∩^∞_k=0Ak,then there existsi∈Nsuch that(x∗ yⁿ⁺¹)∗z∈Ai\Ai+1. It follows thatx∗yⁿ∈Aiso that

µ x∗yⁿ

≥ i

i+1=min µ

x∗yⁿ⁺¹

∗z , µ(z)

. (4.5)

Finally, assume that(x∗yⁿ⁺¹)∗z∈ ∩^∞_k=0Akandz∉∩^∞_k=0Ak.Thenz∈Aj\A_j+1for somej∈N. Hencex∗yⁿ∈Aj, and thus

µ x∗yⁿ

≥ j

j+1=min µ

x∗yⁿ⁺¹

∗z , µ(z)

. (4.6)

Consequently,µis ann-fold fuzzy positive implicative ideal ofX.

Theorem 4.2tells that if everyn-fold fuzzy positive implicative ideal ofX has a finite number of values, thenXsatisfies the PIⁿ-DCC.

Now we consider the converse ofTheorem 4.2.

Theorem4.3. LetX be a BCK-algebra satisfyingPIⁿ-DCC and letµ be an n-fold fuzzy positive implicative ideal ofX. If a sequence of elements ofIm(µ)is strictly in- creasing, thenµhas a finite number of values.

Proof. Let{tk}be a strictly increasing sequence of elements of Im(µ). Hence 0≤ t1< t2<··· ≤1.ThenU (µ;r ):= {x∈X|µ(x)≥tr}is ann-fold positive implicative ideal ofX for all r =2,3, . . . . Let x∈U (µ;r ). Then µ(x)≥tr ≥t_r−1, and sox∈ U (µ;r−1). Hence U (µ;r )⊆U (µ;r−1). Sincetr−1∈Im(µ), there existsxr−1∈X such thatµ(x_r−1)=t_r−1. It follows thatx_r−1∈U (µ;r−1), butx_r−1∉U (µ;r ).Thus U (µ;r )⊊U (µ;r−1),and so we obtain a strictly descending sequence

U µ; 1

⊋U µ; 2

⊋U µ; 3

⊋··· (4.7)

ofn-fold positive implicative ideals ofXwhich is not terminating. This contradicts the assumption thatXsatisfies the PIⁿ-DCC. Consequently,µ has a finite number of values.

Theorem4.4. The following are equivalent.

(i) Xis aPIⁿ-Noetherian BCK-algebra.

(ii) The set of values of anyn-fold fuzzy positive implicative ideal ofX is a well- ordered subset of[0,1].

Proof. (i)⇒(ii). Letµ be ann-fold fuzzy positive implicative ideal ofX. Assume that the set of values ofµ is not a well-ordered subset of[0,1]. Then there exists a strictly decreasing sequence{tk}such thatµ(xk)=tk. It follows that

U µ; 1

⊊U µ; 2

⊊U µ; 3

⊊··· (4.8)

is a strictly ascending chain ofn-fold positive implicative ideals ofX, whereU (µ;r )= {x∈X|µ(x)≥tr}for everyr=1,2, . . . .This contradicts the assumption thatXis PIⁿ-Noetherian.

(ii)⇒(i). Assume that condition (i) is satisfied andXis not PIⁿ-Noetherian. Then there exists a strictly ascending chain

A1⊊A2⊊A3⊊··· (4.9)

ofn-fold positive implicative ideals ofX. LetA= ∪k∈NAk. ThenAis ann-fold positive implicative ideal ofX. Define a fuzzy setνinXby

ν(x):=









0 ifx∉Ak, 1

r wherer=min

k∈Nx∈Ak

.

(4.10)

We claim thatνis ann-fold fuzzy positive implicative ideal ofX. Since 0∈Akfor all k=1,2, . . . ,we haveν(0)=1≥ν(x)for allx∈X. Letx, y, z∈X. If(x∗yⁿ⁺¹)∗z∈ Ak\A_k−1andz∈Ak\A_k−1fork=2,3, . . . ,thenx∗yⁿ∈Ak. It follows that

ν x∗yⁿ

≥1

k=min ν

x∗yⁿ⁺¹

∗z , ν(z)

. (4.11)

Suppose that(x∗yⁿ⁺¹)∗z∈Akandz∈Ak\Ar for allr < k. SinceAkis ann-fold positive implicative ideal, it follows thatx∗yⁿ∈Ak. Hence

ν x∗yⁿ

≥1 k≥ 1

r+1≥ν(z), ν x∗yⁿ

≥min ν

x∗yⁿ⁺¹

∗z , ν(z)

. (4.12) Similarly for the case(x∗yⁿ⁺¹)∗z∈Ak\Arandz∈Ak, we have

ν x∗yⁿ

≥min ν

x∗yⁿ⁺¹

∗z , ν(z)

. (4.13)

Thusνis ann-fold fuzzy positive implicative ideal ofX. Since the chain (4.9) is not ter- minating,νhas a strictly descending sequence of values. This contradicts the assump- tion that the value set of anyn-fold fuzzy positive implicative ideal is well ordered.

ThereforeXis PIⁿ-Noetherian. This completes the proof.

We note that a set is well ordered if and only if it does not contain any infinite descending sequence.

Theorem 4.5. LetS= {tk|k=1,2, . . .} ∪ {0}where{tk}is a strictly descending sequence in(0,1). Then a BCK-algebraXisPIⁿ-Noetherian if and only if for eachn- fold fuzzy positive implicative idealµofX,Im(µ)⊆Simplies that there exists a natural numberksuch thatIm(µ)⊆ {t1, t2, . . . , tk}∪{0}.

Proof. Assume that X is a PIⁿ-Noetherian BCK-algebra and letµ be an n-fold fuzzy positive implicative ideal ofX. Then byTheorem 4.4we know that Im(µ)is a well-ordered subset of[0,1]and so the condition is necessary.

Conversely, suppose that the condition is satisfied. Assume that X is not PIⁿ- Noetherian. Then there exists a strictly ascending chain of n-fold positive implica- tive ideals

A1⊊A2⊊A3⊊···. (4.14)

Define a fuzzy setµinXby

µ(x)=











t1 ifx∈A1,

tk ifx∈Ak\Ak−1, k=2,3, . . . , 0 ifx∈X\∪^∞k=1Ak.

(4.15)

Since 0∈A1, we haveµ(0)=t1≥µ(x)for allx∈X. If either(x∗yⁿ⁺¹)∗z orz belongs toX\∪^∞k=1Ak,then eitherµ((x∗yⁿ⁺¹)∗z)orµ(z)is equal to 0 and hence

µ x∗yⁿ

≥0=min µ

x∗yⁿ⁺¹

∗z , µ(z)

. (4.16)

If(x∗yⁿ⁺¹)∗z∈A1andz∈A1, thenx∗yⁿ∈A1and thus µ

x∗yⁿ

=t1=min µ

x∗yⁿ⁺¹

∗z , µ(z)

. (4.17)

If(x∗yⁿ⁺¹)∗z∈Ak\Ak−1andz∈Ak\Ak−1, thenx∗yⁿ∈Ak. Hence µ

x∗yⁿ

≥tk=min µ

x∗yⁿ⁺¹

∗z , µ(z)

. (4.18)

Assume that(x∗yⁿ⁺¹)∗z∈A1andz∈Ak\Ak−1fork=2,3, . . . .Thenx∗yⁿ∈Ak

and therefore µ

x∗yⁿ

≥tk=min µ

x∗yⁿ⁺¹

∗z , µ(z)

. (4.19)

Similarly for(x∗yⁿ⁺¹)∗z∈Ak\A_k−1andz∈A1, k=2,3, . . . ,we obtain µ

x∗yⁿ

≥tk=min µ

x∗yⁿ⁺¹

∗z , µ(z)

. (4.20)

Consequently,µ is ann-fold fuzzy positive implicative ideal of X. This contradicts our assumption.

5. Normalizations ofn-fold fuzzy positive implicative ideals

Definition5.1. Ann-fold fuzzy positive implicative idealµ ofX is said to be normalif there existsx∈Xsuch thatµ(x)=1.

Example5.2. Let= {0, a, b}be a BCK-algebra inExample 3.3. Then the fuzzy setµ inXdefined byµ(0)=1,µ(a)=0.8, andµ(b)=0.5 is a normaln-fold fuzzy positive implicative ideal ofX.

Note that ifµis a normaln-fold fuzzy positive implicative ideal ofX, then clearly µ(0)=1, and henceµis normal if and only ifµ(0)=1.

Proposition5.3. Given ann-fold fuzzy positive implicative idealµofXletµ⁺be a fuzzy set inXdefined byµ⁺(x)=µ(x)+1−µ(0)for allx∈X. Thenµ⁺is a normal n-fold fuzzy positive implicative ideal ofXwhich containsµ.

Proof. We haveµ⁺(0)=µ(0)+1−µ(0)=1≥µ(x)for allx∈X. For anyx, y, z∈ X, we have

min µ⁺

x∗yⁿ⁺¹

∗z , µ⁺(z)

=min µ

x∗yⁿ⁺¹

∗z

+1−µ(0), µ(z)+1−µ(0)

=min µ

x∗yⁿ⁺¹

∗z , µ(z)

+1−µ(0)

≤µ x∗yⁿ

+1−µ(0)=µ⁺ x∗yⁿ

.

(5.1)

Hence µ⁺ is a normal n-fold fuzzy positive implicative ideal of X, and obviously µ⊆µ⁺.

Noticing thatµ⊆µ⁺, we have the following corollary.

Corollary5.4. If there isx∈Xsuch thatµ⁺(x)=0, thenµ(x)=0.

UsingProposition 3.10, we know that for anyn-fold positive implicative idealAof X, the characteristic functionχ_A ofAis a normaln-fold fuzzy positive implicative ideal ofX. It is clear thatµis a normaln-fold fuzzy positive implicative ideal ofXif and only ifµ⁺=µ.

Proposition 5.5. If µ is an n-fold fuzzy positive implicative ideal of X, then (µ⁺)⁺=µ⁺.

Proof. The proof is straightforward.

Corollary5.6. Ifµis a normaln-fold fuzzy positive implicative ideal ofX, then (µ⁺)⁺=µ.

Proposition5.7. Letµ andν be n-fold fuzzy positive implicative ideals ofX. If µ⊆νandµ(0)=ν(0), thenXµ⊆Xν.

Proof. Ifx∈Xµ, thenν(x)≥µ(x)=µ(0)=ν(0)and soν(x)=ν(0),that is, x∈Xν. ThereforeXµ⊆Xν.

Proposition5.8. Letµbe ann-fold fuzzy positive implicative ideal ofX. If there is ann-fold fuzzy positive implicative idealνofXsatisfyingν⁺⊆µ, thenµis normal.

Proof. Assume that there is ann-fold fuzzy positive implicative idealνofXsuch thatν⁺⊆µ. Then 1=ν⁺(0)≤µ(0), and soµ(0)=1. Henceµis normal.

Given ann-fold fuzzy positive implicative ideal, we construct a new normaln-fold fuzzy positive implicative ideal.

Theorem5.9. Letµ be an n-fold fuzzy positive implicative ideal ofX and letf: [0, µ(0)]→[0,1]be an increasing function. Let µf :X →[0,1] be a fuzzy set inX defined byµf(x)=f (µ(x))for allx∈X. Thenµfis ann-fold fuzzy positive implicative ideal of X. In particular, iff (µ(0))=1 then µf is normal; and if f (t)≥t for all t∈[0, µ(0)], thenµ⊆µf.

Proof. Sinceµ(0)≥µ(x)for allx∈Xand sincef is increasing, we haveµf(0)= f (µ(0))≥f (µ(x))=µf(x)for allx∈X. For anyx, y, z∈Xwe get

min µf

x∗yⁿ⁺¹

∗z , µf(z)

=min f

µ

x∗yⁿ⁺¹

∗z , f

µ(z)

=f min

µ

x∗yⁿ⁺¹

∗z , µ(z)

≤f µ

x∗yⁿ

=µf

x∗yⁿ .

(5.2)

Henceµf is ann-fold fuzzy positive implicative ideal ofX. Iff (µ(0))=1, then clearly µf is normal. Assume thatf (t)≥tfor allt∈[0, µ(0)]. Thenµf(x)=f (µ(x))≥µ(x) for allx∈X, which provesµ⊆µf.

Letᏺ(X)denote the set of all normaln-fold fuzzy positive implicative ideals ofX.

Theorem5.10. Letµ∈ᏺ(X)be nonconstant such that it is a maximal element of the poset(ᏺ(X),⊆). Thenµtakes only the values0and1.

Proof. Sinceµis normal, we haveµ(0)=1. Letx∈Xbe such thatµ(x)=1. It is sufficient to show thatµ(x)=0. If not, then there existsa∈Xsuch that 0< µ(a) <1.

Define a fuzzy setνinXbyν(x)=(1/2){µ(x)+µ(a)}for allx∈X. Clearly,νis well defined, and we get

ν(0)=1 2

µ(0)+µ(a)

=1 2

1+µ(a)

≥1 2

µ(x)+µ(a)

=ν(x) ∀x∈X. (5.3) Letx, y, z∈X. Then

ν x∗yⁿ

=1 2

µ x∗yⁿ

+µ(a)

≥1 2

min µ

x∗yⁿ⁺¹

∗z , µ(z)

+µ(a)

=min 1

2 µ

x∗yⁿ⁺¹

∗z +µ(a)

,1 2

µ(z)+µ(a)

=min ν

x∗yⁿ⁺¹

∗z , ν(z)

.

(5.4)

Hence ν is an n-fold fuzzy positive implicative ideal of X. By Proposition 5.3, ν⁺ is a maximal n-fold fuzzy positive implicative ideal of X, whereν⁺ is defined by ν⁺(x)=ν(x)+1−ν(0)for allx∈X. Note that

ν⁺(a)=ν(a)+1−ν(0)=1 2

µ(a)+µ(a) +1−1

2

µ(0)+µ(a)

=1 2

µ(a)+1

> µ(a)

(5.5)

and ν⁺(a) <1=ν⁺(0). It follows that ν⁺ is nonconstant, andµ is not a maximal element of(ᏺ(X),⊆). This is a contradiction.

Definition5.11. Ann-fold fuzzy positive implicative idealµ ofXis said to be fuzzy maximalifµis nonconstant andµ⁺is a maximal element of the poset(ᏺ(X),⊆).

For any positive implicative idealIofXletµIbe a fuzzy set inXdefined by

µI(x)=





1 ifx∈I,

0 otherwise. (5.6)

Theorem5.12. Letµbe ann-fold fuzzy positive implicative ideal ofX. Ifµis fuzzy maximal, then

(i) µis normal,

(ii) µtakes only the values0and1, (ii) µ_Xµ=µ,

(iv) Xµis a maximaln-fold positive implicative ideal ofX.

Proof. Letµ be an n-fold fuzzy positive implicative ideal ofX which is fuzzy maximal. Thenµ⁺is a nonconstant maximal element of the poset(ᏺ(X),⊆). It follows fromTheorem 5.10thatµ⁺takes only the values 0 and 1. Note thatµ⁺(x)=1 if and only ifµ(x)=µ(0),andµ⁺(x)=0 if and only ifµ(x)=µ(0)−1. By Corollary 5.4, we haveµ(x)=0, and soµ(0)=1. Henceµis normal andµ⁺=µ. This proves (i) and (ii).

(iii) Obviouslyµ_Xµ⊂µandµ_Xµ takes only the values 0 and 1. Letx∈X. Ifµ(x)=0, thenµ⊆µ_Xµ. Ifµ(x)=1, thenx∈Xµand soµ_Xµ(x)=1. This shows thatµ⊆µ_Xµ.

(iv) Sinceµis nonconstant,Xµis a propern-fold positive implicative ideal ofX. Let Jbe ann-fold positive implicative ideal ofXcontainingXµ. Thenµ=µ_Xµ⊆µ_J. Since µandµ_J are normaln-fold fuzzy positive implicative ideals ofXand sinceµ=µ⁺is a maximal element ofᏺ(X), we have that eitherµ=µ_J orµ_J =1where1:X→[0,1]

is a fuzzy set defined by1(x)=1 for allx∈X. The later case implies thatJ=X. If µ=µ_J, thenXµ=Xµ_J =J. This shows thatXµis a maximaln-fold positive implicative ideal ofX. This completes the proof.

Acknowledgement. The first author was supported by Korea Research Founda- tion Grant (KRF-2000-005-D00003).

References

[1] Y. Huang and Z. Chen,On ideals in BCK-algebras, Math. Japon.50(1999), no. 2, 211–226.

CMP 1 718 851. Zbl 938.06018.

[2] Y. B. Jun, S. M. Hong, J. Meng, and X. L. Xin,Characterizations of fuzzy positive implica- tive ideals in BCK-algebras, Math. Japon.40(1994), no. 3, 503–507.CMP 1 305 546.

Zbl 814.06012.

[3] J. Meng,On ideals in BCK-algebras, Math. Japon.40(1994), no. 1, 143–154.MR 95e:06049.

Zbl 807.06012.

Young Bae Jun: Department of Mathematics Education, Gyeongsang National Uni- versity, Jinju660-701, Korea

E-mail address:[email protected]

Kyung Ho Kim: Department of Mathematics, Chungju National University, Chungju 380-702, Korea

E-mail address:[email protected]