(
λ,
µ
)-Fuzzy Ideals in Ordered Ternary Semigroups
T. Anitha
∗Mathematics wing, D.D.E, Annamalai University, Annamalainagar 608 002, India.
Abstract
In this paper we introduce the notion of(λ,µ)-fuzzy ternary subsemigroup (respectively left ideal, right ideal, lateral ideal, bi-ideal, interior ideal, quasi-ideal) in an ordered ternary semigroups is introduce. We have shown that f is a(λ,µ)-fuzzy ternary subsemigroup (respectively left ideal, right ideal, lateral ideal, bi-ideal, interior ideal, quasi-ideal) of an ordered ternary semigroup Tif and only if the upper level cutU(f;t)is a ternary subsemigroup (respectively left ideal, right ideal, lateral ideal, bi-ideal, interior ideal, quasi-ideal) of T.
Keywords: (λ,µ)-fuzzy ternary subsemigroups, (λ,µ)-fuzzy ideals, (λ,µ)-fuzzy bi-ideal, (λ,µ)-fuzzy generalised bi-ideal,(λ,µ)-fuzzy interior ideal,(λ,µ)-fuzzy quasi-ideal,(λ,µ)-characteristic function.
2010 MSC:06F05, 06D72, 08A72. 2012 MJM. All rights reserved.c
1
Introduction
In 1932, Lehmer introduced the concept of ternary semigroup [6]. The algebraic structures of ternary semigroups were also studied by some authors, for example, Sioson studied ideals in ternary semigroups [13]. Dixit and Dewan studied quasi-ideals and bi-ideals in ternary semigroups [2], Kar and Maity studied congruences of ternary semigroups [4] and Iampan studied minimal and maximal lateral ideals of ternary semigroups [3]. The concept of a fuzzy subset was introduced by Zadeh [15], provides a natural framework for generalizing some of the notions of classical algebraic structures. In 1971, Rosenfeld [11] defined the concept of fuzzy group. Since then many papers have been published in the field of fuzzy algebra. Several researchers conducted the researches on the generalizations of the notions of fuzzy subsets with huge applications in computer, logics and many branches of pure and applied mathematics. Fuzzy semigroups have been first considered by Kuroki [5]. Chinram and Saelee [1], studied the concept of fuzzy ideals and fuzzy filters of ordered ternary semigroups. Naveed Yahoob etal. [10] studied the concepts of roughness and fuzziness in ordered ternary semigroups. S.Lekkoksung [7, 8] studied fuzzy interior ideals and some properties of Ω-fuzzy ideals in ordered ternary semigroups. In this paper we introduce the notion of
(λ,µ)-fuzzy ternary subsemigroup (respectively left ideal, right ideal, lateral ideal, bi-ideal, interior ideal, quasi-ideal) in an ordered ternary semigroups which can be regarded as the generalization of fuzzy ternary subsemigroup (respectively left ideal, right ideal, lateral ideal, bi-ideal, interior ideal, quasi-ideal) in ordered ternary semigroups. We prove that f is a(λ,µ)-fuzzy ternary subsemigroup (respectively left ideal, right ideal, lateral ideal, bi-ideal, interior ideal, quasi-ideal) of an ordered ternary semigroupTif and only if the upper level cutU(f;t)is a ternary subsemigroup (respectively left ideal, right ideal, lateral ideal, bi-ideal, interior ideal, quasi-ideal) ofT.
2
Preliminaries
∗Corresponding author.
Definition 2.1. A non-empty set T is called a ternary semigroup if there exists a ternary operation T×T×T → T, written as(a,b,c)→abc satisfying the following identity, for any a,b,c,d,e∈T, ((abc)de)=(a(bcd)e)=(ab(cde)).
Any semigroup can be reduced to a ternary semigroup. However, Banach showed that a ternary
semigroup does not necessarily reduce to a semigroup by this example. Let T = {−i, 0,i} be a ternary semigroup while T is not a semigroup under the multiplication over complex numbers. Los showed that every ternary semigroup can be embedded in a semigroup [9].
Definition 2.2. An ordered ternary semigroup we mean a structure(T;·;≤)in which (1)(T;·)is a ternary semigroup,
(2)(T;≤)is a poset,
(3) a≤b⇒acd≤bcd,cad≤cbd and cda≤cdb, for all a,b,c,d∈T.
LetTbe an ordered ternary semigroup. For any non-empty subsets, A,BandCofT,ABC = {abc /a ∈ A, b∈Bandc∈C}. ForA⊆T, denote(A] ={t∈T/t≤hfor someh∈ A}.
Definition 2.3. A non-empty subset A of T is called a ternary subsemigroup of T if(A]⊆A and AAA⊆ A .
Definition 2.4. A non-empty subset A of T is called a left ideal of T if(A]⊆ A and TTA ⊆ A, a lateral ideal of T if
(A]⊆ A and TAT ⊆ A and a right ideal of T if(A] ⊆ A and ATT⊆ A. If A is both a left and right ideal then A is called a two-sided ideal of T. If A is a left, right and lateral ideal of T, then A is called an ideal of T.
Definition 2.5. A ternary subsemigroup B of T is called a bi-ideal of T if(B]⊆B and BTBTB⊆B.
Definition 2.6. A non-empty subset A of T is called a generalized bi-ideal of T if(A]⊆A and ATATA⊆A.
Clearly, every bi-ideal ofTis a generalized bi-ideal but not the converse.
Definition 2.7. A non-empty subset A of T is called an interior ideal of T if(A]⊆ A and TTATT⊆A.
Definition 2.8. A non-empty subset Q of T is called a quasi-ideal of T if(Q] ⊆ Q, QTT∩TQT∩TTQ ⊆ Q and QTT∩TTQTT∩TTQ⊆Q.
Definition 2.9. An ordered ternary semigroup T is said to be regular if for each a∈ T, there exists x∈ T such that a≤axa.
LetXbe a non-empty set. A map f :X→[0, 1]is called a fuzzy subset inX.
Definition 2.10. Let f , g and h be any three fuzzy subsets of an ordered ternary semigroup T. Then f∩g, f ∪g, f◦g◦h are fuzzy subsets of T defined by
(f ∩g)(x) =min{f(x), g(x)}= f(x)∧g(x),
(f ∪g)(x) =max{f(x), g(x)}= f(x)∨g(x), for all x∈T.
For x ∈T, we define Ax={(u,v,w)∈T×T×T/x ≤uvw}. The product of f◦g◦h is defined by for all x∈T,
(f◦g◦h)(x) =
W
(u,v,w)∈Ax
{min{f(u),g(v),h(w)}} , i f Ax6=φ,
0 , i f Ax=φ.
Definition 2.11. Let X be a non-empty set and let f be a fuzzy subset of X. Let0 ≤ t ≤1. Then the set U(f;t) =
{x∈X/ f(x)≥t}is called the upper level cut of X with respect to f .
Definition 2.12. Let f be a fuzzy subset of an ordered ternary semigroup T. Then f is called a fuzzy ternary subsemigroup of T if
1. x≤y implies f(x)≥ f(y)
2. f(xyz)≥min{f(x),f(y),f(z)}for all x,y,z∈T.
Definition 2.13. A fuzzy subset f of an ordered ternary semigroup T is called a fuzzy ideal of T if (i) x≤y implies f(x)≥ f(y)
(ii) f(xyz)≥ f(x)
(iii) f(xyz)≥ f(z)and
A fuzzy subset f with conditions (i) and (ii) is called a fuzzy right ideal of T. If f satisfies (i) and (iii), then it is called a fuzzy left ideal of T. If f satisfies (i) and (iv), then it is called a fuzzy lateral ideal of T. Also if f satisfies (i), (ii) and (iii), then it is called a fuzzy two sided ideal of T. It is clear that f is a fuzzy ideal of an ordered ternary semigroup T if and only if f(xyz)≥max{f(x),f(y),f(z)}for all x,y,z∈T.
Definition 2.14. A fuzzy ternary subsemigroup f of an ordered ternary semigroup T is called a fuzzy bi-ideal of T if f(xuyvz)≥min{f(x),f(y),f(z)}for all x,y,z,u,v∈T.
Definition 2.15. A fuzzy subset f of an ordered ternary semigroup T is called a fuzzy generalised bi-ideal of T if 1. x≤y implies f(x)≥ f(y)
2. f(xuyvz)≥min{f(x),f(y),f(z)}for all x,y,z,u,v∈T.
Definition 2.16. A fuzzy subset f of an ordered ternary semigroup T is called a fuzzy interior ideal of T if 1. x≤y implies f(x)≥ f(y)
2. f(rsatu)≥ f(a)for all r,s,a,t,u∈ T.
Definition 2.17. A fuzzy subset f of an ordered ternary semigroup T is called a fuzzy quasi-ideal of T if it satiesfies for all x,y∈T,
1. x≤y implies f(x)≥ f(y)
2. f(x)≥min{(f◦T◦T)(x),(T◦f◦T)(x),(T◦T◦f)(x)}
3. f(x)≥min{(f◦T◦T)(x),(T◦T◦f ◦T◦T)(x),(T◦T◦f)(x)}, whereTis the fuzzy subset of T mapping every element of T on1.
3
(
λ
,
µ
)
- Fuzzy ideals
Based on the concept of (λ,µ)-fuzzy subrings and (λ,µ)-fuzzy ideals introduced by B.Yao [14], we introduce the following concepts which are the generalization of fuzzy subsets. Throughout this paperλand µ(0≤λ<µ≤1), are arbitrary, but fixed.
Definition 3.18. A fuzzy subset f of an ordered ternary semigroup T is called a(λ,µ)-fuzzy ternary subsemigroup of T if
1. x≤y implies f(x)∨λ≥ f(y)∧µ
2. f(xyz)∨λ≥ f(x)∧ f(y)∧ f(z)∧µfor all x,y,z∈T.
Theorem 3.1. Let f be a fuzzy subset of an ordered ternary semigroup T. Then f is a(λ,µ)-fuzzy ternary subsemigroup of T if and only if U(f;t)is a ternary subsemigroup of T for all t∈(λ,µ]whenever non-empty.
Proof. Let f be a(λ,µ)-fuzzy ternary subsemigroup ofT andt ∈ (λ,µ]. Lety ∈ U(f;t)and letx ∈ T such thatx≤y. Thus f(x)∨λ≥ f(y)∧µ≥t. Hencex∈U(f;t). Next, letx,y,z∈U(f;t)then f(x)≥t, f(y)≥ t, f(z) ≥ t. Now f(xyz)∨λ ≥ f(x)∧ f(y)∧ f(z)∧µ ≥ t. Thus f(xyz) ≥ t. Hencexyz ∈ U(f;t). Hence U(f;t)is a ternary subsemigroup ofT.
Conversely, letU(f;t)be a ternary subsemigroup ofTfor allt∈(λ,µ]. Letx,y∈Tbe such thatx≤y. Choose t= f(y)∧µ. Thus,y∈U(f;t), which impliesx ∈U(f;t). Thenf(x)∨λ≥t= f(y)∧µ. Suppose if there exist x,y,z ∈ Tsuch that f(xyz)∨λ < t= f(x)∧ f(y)∧f(z)∧µthent∈ (λ,µ]andx,y,z ∈U(f;t)withxyz ∈/ U(f;t)this contradicts to thatU(f;t)is a ternary subsemigroup. Hence f(xyz)∨λ≥ f(x)∧ f(y)∧ f(z)∧µ. Thereforef is a(λ,µ)-fuzzy ternary subsemigroup ofT.
Remark 3.1. Every fuzzy ternary subsemigroup is a(λ,µ)-fuzzy ternary subsemigroup by takingλ=0andµ=1. But the converse need not be true.
Example 3.1. Let(Z−,·,≤)be an ordered ternary semigroup. Letµ:Z−→[0, 1]defined by
µ(x) =
0.9 i f x=−3 0.8 i f x∈(−∞,−4]
0 otherwise.
Definition 3.19. A fuzzy subset f of an ordered ternary semigroup T is called a(λ,µ)-fuzzy right [resp. left, lateral] ideal of T if
1. x≤y implies f(x)∨λ≥ f(y)∧µ
2. f(xyz)∨λ≥ f(x)∧µ [resp. f(xyz)∨λ≥ f(z)∧µ, f(xyz)∨λ≥ f(y)∧µ] for all x,y,z∈T.
Definition 3.20. A fuzzy subset f of an ordered ternary semigroup T is called a(λ,µ)-fuzzy ideal of T if 1. x≤y implies f(x)∨λ≥ f(y)∧µ
2. f(xyz)∨λ≥[f(x)∨f(y)∨f(z)]∧µfor all x,y,z∈T.
Definition 3.21. A(λ,µ)-fuzzy ternary subsemigroup f of an ordered ternary semigroup T is called a(λ,µ)-fuzzy bi-ideal of T if
f(xuyvz)∨λ≥min{f(x),f(y),f(z)} ∧µfor all x,y,z,u,v∈ T.
Definition 3.22. A fuzzy subset f of an ordered ternary semigroup T is called a(λ,µ)-fuzzy generalised bi-ideal of T if 1. x≤y implies f(x)∨λ≥ f(y)∧µ
2. f(xuyvz)∨λ≥min{f(x),f(y),f(z)} ∧µfor all x,y,z,u,v∈T.
Theorem 3.2. Let f be a fuzzy subset of an ordered ternary semigroup T. Then f is a(λ,µ)-fuzzy generalised bi-ideal of T if and only if U(f;t)is a generalised bi-ideal of T for all t∈(λ,µ]whenever non-empty.
Proof. Let f be a(λ,µ)-fuzzy generalised bi-ideal of T andt ∈ (λ,µ]. Lety ∈ U(f;t) and letx ∈ Tsuch that x ≤ y. Thus f(x)∨λ ≥ f(y)∧µ ≥ t. Hencex ∈ U(f;t). Now, letx,y,z ∈ U(f;t)andu,v ∈ T then f(x) ≥ t, f(y) ≥ t, f(z) ≥ t. Hence f(xuyvz)∨λ ≥ min{f(x),f(y),f(z)} ∧µ ≥ t. Thus f(xuyvz) ≥ t. Hencexuyvz∈U(f;t). HenceU(f;t)is a generalised bi-ideal ofT.
Conversely, letU(f;t)be a generalised bi-ideal ofTfor allt∈ (λ,µ]. Letx,y∈Tbe such thatx ≤y. Choose t = f(y)∧µ. Thus, y ∈ U(f;t), which implies x ∈ U(f;t). Then f(x)∨λ ≥ t = f(y)∧µ. Suppose if there existx,y,z,u,v∈ Tsuch that f(xuyvz)∨λ<t=min{f(x),f(y),f(z)} ∧µthent∈ (λ,µ]andx,y,z∈ U(f;t)withxuyvz ∈/ U(f;t)this contradicts to thatU(f;t)is a generalised bi-ideal. Hence f(xuyvz)∨λ ≥ min{f(x),f(y),f(z)} ∧µ. Therefore f is a(λ,µ)-fuzzy generalised bi-ideal ofT.
Theorem 3.3. Let f be a fuzzy subset of an ordered ternary semigroup T. Then f is a(λ,µ)-fuzzy bi-ideal of T if and only if U(f;t)is a generalised bi-ideal of T for all t∈(λ,µ]whenever non-empty.
Proof. Follows from Theorem 3.1 and Theorem 3.2.
Theorem 3.4. Let f be a fuzzy subset of an ordered ternary semigroup T. Then f is a(λ,µ)-fuzzy right (resp. left, lateral) ideal of T if and only if U(f;t)is a right (resp. left, lateral) ideal of T for all t∈(λ,µ]whenever non-empty.
Proof. The proof follows from Theorem 3.1 and using the method applied in Theorem 3.2.
Remark 3.2. Every fuzzy right (resp. left, lateral) ideal is a(λ,µ)-fuzzy right (resp. left, lateral) ideal by takingλ=0 andµ=1. But the converse need not be true.
Example 3.2. Let(Z−,·,≤)be an ordered ternary semigroup. Letµ:Z− →[0, 1]defined by
µ(x) =
0.85 i f x=−2 0.8 i f x∈(−∞,−3]
0 otherwise.
Clearlyµis a(0.3, 0.8)-fuzzy right ideal. Butµis not a fuzzy right ideal asµ(−8=−2· −2· −2) =0.8<µ(−2) = 0.85.
Theorem 3.5. A non-empty subset L of an ordered ternary semigroup T is a right (resp. left, lateral) ideal of T if and only if the fuzzy subset f defined by
f(x) =
(
t i f x∈ T−L r i f x∈ L,
Proof. SupposeLis a right ideal ofTand letx,y∈ Tbe such thatx≤y. Ify ∈T−L, then f(x)∨λ≥ f(y)∧ µ=t. Ify ∈L, impliesx∈ L, then f(x)∨λ=r= f(y)∧µ. Hence f(x)∨λ≥ f(y)∧µ. Now letx,y,z∈ T. Letx ∈ L. Thenxyz ∈L. Hence f(xyz)∨λ=r= f(x)∧µ. Ifx∈/ L, then f(x)∧µ=t≤ f(xyz)∨λ. Hence f is a(λ,µ)-fuzzy right ideal ofT. Conversely, assume that f is a(λ,µ)-fuzzy right ideal ofT. Letx≤y ∈L. Then f(x)∨λ ≥ f(y)∧µ = r, this implies that f(x) = r. Thus, x ∈ L. Now, ifx ∈ L andy,z ∈ T, then
f(xyz)∨λ≥ f(x)∧µ=r. This implies that f(xyz) =r, that isxyz∈ L. HenceLis a right ideal ofT.
Corollary 3.1. A non-empty subset I of an ordered ternary semigroup T is a right (lateral, left) ideal of T if and only if the characteristic function of I, defined by
χ∗I(x) = (
λ i f x∈T−I
µ i f x∈I
is a(λ,µ)-fuzzy right (lateral, left) ideal of T.
The following theorems can be proved in a similar way.
Theorem 3.6. A non-empty subset L of an ordered ternary semigroup T is a ternary subsemigroup of T if and only if the fuzzy subset f of T defined by
f(x) =
(
t i f x∈T−L r i f x∈L,
where t≤r and t,r∈(λ,µ], is a(λ,µ)-fuzzy ternary subsemigroup of T.
Theorem 3.7. A non-empty subset L of an ordered ternary semigroup T is a generalised bi-ideal (bi-ideal) of T if and only if the fuzzy subset f of T defined by
f(x) =
(
t i f x∈T−L r i f x∈L,
where t≤r and t,r∈(λ,µ], is a(λ,µ)-fuzzy generalised bi-ideal (bi-ideal) of T.
Corollary 3.2. A non-empty subset I of an ordered ternary semigroup T is a generalised bi-ideal (bi-ideal) of T if and only if the characteristic functionχ∗I of I is a(λ,µ)-fuzzy generalised bi-ideal (bi-ideal) of T.
Lemma 3.1. Union of any family of (λ,µ)-fuzzy right (left, lateral) ideals of an ordered ternary semigroup T is a
(λ,µ)-fuzzy right (left, lateral) ideal of T.
Proof. Let{fi}i∈I be a family of(λ,µ)-fuzzy right ideals of an ordered ternary semigroupT. Letx,y ∈ Tbe such thatx≤y.
Then,(S
i∈I
fi)(x)∨λ= S i∈I
(fi(x)∨λ)≥ S i∈I
(fi(y)∧µ) = (S i∈I
fi)(y)∧µ.
Now, letx,y,z∈T. Then,(S
i∈I
fi)(xyz)∨λ= S i∈I
(fi(xyz)∨λ)
≥ S
i∈I
(fi(x)∧µ) = (S i∈I
fi)(x)∧µ.
Hence S
i∈I
fiis a(λ,µ)-fuzzy right ideal ofT.
Lemma 3.2. Intersection of any family of(λ,µ)-fuzzy right (left, lateral) ideals of an ordered ternary semigroup T is a
(λ,µ)-fuzzy right (left, lateral) ideal of T.
Proof. Suppose let {fi}i∈I be a family of (λ,µ)-fuzzy right ideals of an ordered ternary semigroup T. Let x,y∈Tbe such thatx≤y.
Then,(T
i∈I
fi)(x)∨λ= T i∈I
(fi(x)∨λ)≥ T i∈I
(fi(y)∧µ) = (T i∈I
fi)(y)∧µ.
Now, letx,y,z∈T. Then,(T
i∈I
fi)(xyz)∨λ= T i∈I
(fi(xyz)∨λ)
≥ T
i∈I
(fi(x)∧µ) = (T i∈I
fi)(x)∧µ.
Hence T
i∈I
Definition 3.23. Let f , g and h be any three fuzzy subsets of an ordered ternary semigroup T. The (λ,µ)-fuzzy product f?g?h is defined by for all x∈T,
(f ?g?h)(x) =
W
(u,v,w)∈Ax
{min{f(u),g(v),h(w),µ} ∨λ} i f Ax6=φ,
0 i f Ax=φ.
Proposition 3.1. The (λ,µ)-fuzzy product of three (λ,µ)-fuzzy right (left, lateral) ideals of an ordered ternary semigroup T is again a(λ,µ)-fuzzy right (left, lateral) ideal of T.
Proof. Let f,gandhbe three(λ,µ)-fuzzy right ideals ofT. Letx,y ∈ Tsuch thatx ≤ y. Let(u,v,w) ∈ Ay
theny≤uvw. Sincex≤yimpliesx≤uvw, that is(u,v,w)∈ Ax. HenceAy⊆ Ax. Now
(f?g?h)(y)∧µ= "
W
(p,q,r)∈Ay
{f(p)∧g(q)∧h(r)∧µ} ∨λ #
∧µ
= W
(p,q,r)∈Ay⊆Ax
{f(p)∧g(q)∧h(r)∧µ} ∧µ∨(λ∧µ)
≤ W
(p,q,r)∈Ax
{f(p)∧g(q)∧h(r)∧µ} ∨λ
=
"
W
(p,q,r)∈Ax
{f(p)∧g(q)∧h(r)∧µ} ∨λ #
∨λ
= (f?g?h)(x)∨λ.
Sincehis a(λ,µ)-fuzzy right ideal,h(ryz)∨λ≥h(r)∧µ.
(f?g?h)(x)∧µ= "
W
(p,q,r)∈Ax
{f(p)∧g(q)∧h(r)∧µ} ∨λ #
∧µ
= W
(p,q,r)∈Ax
{f(p)∧g(q)∧h(r)∧µ} ∧µ∨(λ∧µ)
= W
(p,q,r)∈Ax
{f(p)∧g(q)∧h(r)∧µ∧µ} ∨λ
≤ W
(p,q,ryz)∈Axyz
{f(p)∧g(q)∧(h(ryz)∨λ)∧µ} ∨λ
≤ W
(p,q,ryz)∈Axyz
[{f(p)∧g(q)∧h(ryz)∧µ} ∨λ]∨λ
=
"
W
(p,q,ryz)∈Axyz
{f(p)∧g(q)∧h(ryz)∧µ} ∨λ #
∨λ
= (f?g?h)(xyz)∨λ.
If Ax = φ, then(f ?g?h)(x)∧µ = 0 ≤ (f ?g?h)(xyz)∨λ. Thus f ?g?h is a(λ,µ)-fuzzy right ideal of T.
4
(
λ
,
µ
)
- Fuzzy interior ideals
Definition 4.24. A fuzzy subset f of an ordered ternary semigroup T is called a(λ,µ)-fuzzy interior ideal of T if 1. x≤y implies f(x)∨λ≥ f(y)∧µ
2. f(rsatu)∨λ≥ f(a)∧µfor all r,s,a,t,u∈T.
Theorem 4.8. Let f be a fuzzy subset of an ordered ternary semigroup T. Then f is a(λ,µ)-fuzzy interior ideal of T if and only if U(f;t)is an interior ideal of T for all t∈(λ,µ]whenever non-empty.
Theorem 4.9. A non-empty subset L of an ordered ternary semigroup T is an interior ideal of T if and only if the fuzzy subset f of T defined by
f(x) =
(
t i f x∈T−L r i f x∈L,
where t≤r and t,r∈(λ,µ], is a(λ,µ)-fuzzy interior ideal of T.
Proof. The proof is similar to the proof of Theorem 3.5.
Corollary 4.3. A non-empty subset I of an ordered ternary semigroup T is an interior ideal of T if and only if the characteristic functionχ∗I of I is a(λ,µ)-fuzzy interior ideal of T.
Proposition 4.2. Let T be an ordered ternary semigroup and f a (λ,µ)-fuzzy two-sided ideal of T. Then f is a
(λ,µ)-fuzzy interior ideal of T.
Proof. Let r,s,a,t,u ∈ T. Since f is a (λ,µ)-fuzzy left ideal of T, we have f(rs(atu))∨λ ≥ f(atu)∧µ. Since f is a (λ,µ)-fuzzy right ideal of T, we have f(atu)∨λ ≥ f(a)∧µ. Then we have f(rsatu)∨λ =
[f(rsatu)∨λ]∨λ≥[f(atu)∧µ]∨λ= [f(atu)∨λ]∧µ≥[f(a)∧µ]∧µ= f(a)∧µ. Thus f is a(λ,µ)-fuzzy interior ideal ofT.
Proposition 4.3. Let T be an ordered ternary semigroup and f a(λ,µ)-fuzzy lateral ideal of T. Then f is a(λ,µ)-fuzzy interior ideal of T.
Proof. Letr,s,a,t,u ∈ T. Since f is a (λ,µ)-fuzzy lateral ideal of T, we have f(r(sat)u)∨λ ≥ f(sat)∧µ. Again since f is a(λ,µ)-fuzzy lateral ideal ofT, we have f(sat)∨λ≥ f(a)∧µ. Then we havef(rsatu)∨λ=
[f(rsatu)∨λ]∨λ ≥[f(sat)∧µ]∨λ= [f(sat)∨λ]∧µ≥ [f(a)∧µ]∧µ= f(a)∧µ. Thus f is a(λ,µ)-fuzzy interior ideal ofT.
Proposition 4.4. Let T be a regular ordered ternary semigroup and f a(λ,µ)-fuzzy interior ideal of T. Then f is a
(λ,µ)-fuzzy ideal of T.
Proof. Let x,y,z ∈ T. SinceT is regular and x ∈ T, there existsa ∈ T such that x ≤ xax. Then we have xyz≤xaxyzwhich impliesf(xyz)∨λ≥ f(xaxyz)∧µ. Sincef is a(λ,µ)-fuzzy interior ideal ofT,f(xaxyz)∨ λ ≥ f(x)∧µ. Now we have f(xyz)∨λ = [f(xyz)∨λ]∨λ ≥ [f(xaxyz)∧µ]∨λ = [f(xaxyz)∨λ]∧µ ≥
[f(x)∧µ]∧µ = f(x)∧µ. Thus f is a(λ,µ)-fuzzy right ideal ofT. In similar way we can prove that f is a
(λ,µ)-fuzzy left ideal ofTand f is a(λ,µ)-fuzzy lateral ideal ofT. Thusf is a(λ,µ)-fuzzy ideal ofT.
Theorem 4.10. In regular ordered ternary semigroups the concepts of(λ,µ)-fuzzy ideals and (λ,µ)-fuzzy interior ideals coincide.
Proof. The proof follows from Proposition 4.3 and Proposition 4.4.
5
(
λ
,
µ
)
- Fuzzy quasi-ideals
Definition 5.25. A fuzzy subset f of an ordered ternary semigroup T is called a (λ,µ)-fuzzy quasi-ideal of T if it satiesfies for all x,y∈T,
1. x≤y implies f(x)∨λ≥ f(y)∧µ
2. f(x)∨λ≥min{(f◦T◦T)(x),(T◦f◦T)(x),(T◦T◦f)(x)} ∧µ 3. f(x)∨λ≥min{(f◦T◦T)(x),(T◦T◦f◦T◦T)(x),(T◦T◦f)(x)} ∧µ.
Remark 5.3. Every fuzzy quasi ideal is a(λ,µ)-fuzzy quasi ideal of T by takingλ =0andµ=1. But the converse need not be true.
Proof. Let f be a(λ,µ)-fuzzy quasi-ideal ofTandt ∈ (λ,µ]. Lety ∈ U(f;t)and letx ∈ Tsuch thatx ≤ y. Thus f(x)∨λ ≥ f(y)∧µ ≥ t. Hencex ∈ U(f;t). Letx ∈ (U(f;t)TT)∩(TU(f;t)T)∩(TTU(f;t)). Then there exist a,b,d ∈ U(f;t) and s1,s2,s3,s4,s5,s6 ∈ T such that x ≤ as1s2 = s3bs4 = s5s6d. Thus f(a) ≥
t,f(b)≥t,f(d)≥t.
Thenmin{(f◦T◦T)(x),(T◦f ◦T)(x),(T◦T◦f)(x)}
=min{ W (a,s1,s2)∈Ax
{f(a)}, W (s3,b,s4)∈Ax
{f(b)}, W (s5,s6,d)∈Ax
{f(d)}} ≥t.
Nowmin{(f ◦T◦T)(x),(T◦f◦T)(x),(T◦T◦f)(x)} ∧µ≥min{t,µ}=t.
Since f is a(λ,µ)-fuzzy quasi-ideal ofT, f(x)∨λ≥min{(f ◦T◦T)(x),(T◦f ◦T)(x),(T◦T◦f)(x)} ∧µ≥ t>λ. Then f(x)≥tandx∈U(f;t). Hence(U(f;t)TT)∩(TU(f;t)T)∩(TTU(f;t))⊆U(f;t). Next, letx ∈
(U(f;t)TT)∩(TTU(f;t)TT)∩(TTU(f;t)). Then there exista,c,d ∈U(f;t)ands1,s2,s3,s4,s5,s6,s7,s8 ∈ T
such thatx≤as1s2=s3s4cs5s6=s7s8d. Thusf(a)≥t,f(c)≥t,f(d)≥t.
Thenmin{(f◦T◦T)(x),(T◦T◦f ◦T◦T)(x),(T◦T◦f)(x)}
=min{ W (a,s1,s2)∈Ax
{f(a)}, W (s3,s4,c,s5,s6)∈Ax
{f(c)}, W (s7,s8,d)∈Ax
{f(d)}} ≥t.
Nowmin{(f ◦T◦T)(x),(T◦T◦f◦T◦T)(x),(T◦T◦f)(x)} ∧µ≥min{t,µ}=t.
Sincef is a(λ,µ)-fuzzy quasi-ideal ofT,f(x)∨λ≥min{(f◦T◦T)(x),(T◦T◦f◦T◦T)(x),(T◦T◦f)(x)} ∧ µ ≥ t > λ. Then f(x) ≥ tand x ∈ U(f;t). Hence(U(f;t)TT)∩(TTU(f;t)TT)∩(TTU(f;t)) ⊆ U(f;t). ThereforeU(f;t)is a quasi-ideal inT.
Conversely let us assume thatU(f;t), t∈(λ,µ]is a quasi-ideal inT. Letx,y∈Tbe such thatx≤y. Choose t= f(y)∧µ. Thus,y ∈U(f;t), which impliesx ∈U(f;t). Then f(x)∨λ≥t= f(y)∧µ. Letx ∈Tbe such thatmin{(f ◦T◦T)(x),(T◦f ◦T)(x),(T◦T◦ f)(x)} ∧µ ≥ t > f(x)∨λ. Then(f ◦T◦T)(x) ≥ timplies x∈U(f;t)TT,(T◦f◦T)(x)≥timpliesx∈TU(f;t)T,(T◦T◦f)(x)≥timpliesx ∈TTU(f;t). This implies thatx ∈ (U(f;t)TT)∩(TU(f;t)T)∩(TTU(f;t)) ⊆ U(f;t). This implies that f(x)≥ tthat is f(x)∨λ≥ t, which is a contradiction. Hence,min{(f ◦T◦T)(x),(T◦f ◦T)(x),(T◦T◦f)(x)} ∧µ≤ f(x)∨λ. Thus f is a(λ,µ)-fuzzy quasi-ideal ofT.
Theorem 5.12. A non-empty subset Q of an ordered ternary semigroup T is a quasi-ideal of T if and only if the fuzzy subset f of T defined by
f(x) =
(
t i f x∈T−Q r i f x∈Q,
where t≤r and t,r∈(λ,µ], is a(λ,µ)-fuzzy quasi-ideal of T.
Proof. Assume thatQis a quasi-ideal ofTand letx,y ∈ Tbe such thatx ≤y. Ify∈ T−Q, then f(x)∨λ≥ f(y)∧µ = t. Ify ∈ Q, impliesx ∈ Q, then f(x)∨λ =r = f(y)∧µ. Hence f(x)∨λ ≥ f(y)∧µ. Now let x∈T. Ifx ∈Qthen f(x)∨λ=r≥min{(f ◦T◦T)(x),(T◦f◦T)(x),(T◦T◦f)(x)} ∧µ
and f(x)∨λ=r≥min{(f◦T◦T)(x),(T◦T◦f ◦T◦T)(x),(T◦T◦f)(x)} ∧µ.
Ifx ∈/ Qthen f(x)∨λ =t. SinceQis a quasi-ideal,x ∈/ Qimpliesx ∈/ (QTT∩TQT∩TTQ)which implies x∈/QTT,x∈/TQTandx∈/TTQ. Supposemin{(f◦T◦T)(x),(T◦f◦T)(x),(T◦T◦f)(x)} ∧µ=rthen(f◦
T◦T)(x) =r,(T◦f◦T)(x) =r,(T◦T◦f)(x) =rwhich implies sup
(u,t1,t2)∈Ax
{f(u)}=r, sup
(t3,v,t4)∈Ax
{f(v)}=r
and sup
(t5,t6,w)∈Ax
{f(w)} = rfor somet1,t2,t3,t4,t5,t6,u,v,w ∈ T. Since sup (u,t1,t2)∈Ax
{f(u)} = rimpliesu ∈ Q
that isx≤ut1t2∈QTTwhich is a contradiction tox∈/QTT. Thusmin{(f◦T◦T)(x),(T◦f ◦T)(x),(T◦T◦
f)(x)} ∧µ=t.
Similarlymin{(f◦T◦T)(x),(T◦T◦f◦T◦T)(x),(T◦T◦f)(x)} ∧µ=t. Hencefis a(λ,µ)-fuzzy quasi-ideal ofT.
Conversely, assume that f is a(λ,µ)-fuzzy quasi-ideal ofT. Letx≤y∈Q. Then f(x)∨λ≥ f(y)∧µ=r, this implies that f(x) = r. Thus, x ∈ Q. Let a ∈ (QTT∩TQT∩TTQ)then min{(f ◦T◦T)(a),(T◦ f ◦
T)(a),(T◦T◦f)(a)} ∧µ=r. Since f is a(λ,µ)-fuzzy quasi-ideal ofT,
min{(f ◦T◦T)(a),(T◦f ◦T)(a),(T◦T◦ f)(a)} ∧µ = r ≤ f(a)∨λ. Which implies a ∈ Q. Similarly, if a∈QTT∩TTQTT∩TTQimpliesa∈Q. ThusQis a quasi-ideal ofT.
Theorem 5.13. Let f be a fuzzy subset of an ordered ternary semigroup T. If f is a(λ,µ)-fuzzy lateral (left, right) ideal of T then f is a(λ,µ)-fuzzy quasi-ideal of T.
Proof. Let f be a(λ,µ)-fuzzy lateral (left, right) ideal ofT. Letx,a,b,c,d,s1,s2,s3,s4,s5,s6,s7,s8 ∈ T. Then
f(s3bs4)∨λ≥ f(b)∧µ. Nowmin{(f◦T◦T)(x),(T◦f◦T)(x),(T◦T◦f)(x)} ∧µ
=min{ "
W
(a,s1,s2)∈Ax
min{f(a),T(s1),T(s2)}
#
,
"
W
(s3,b,s4)∈Ax
min{T(s3),f(b),T(s4)}
#
,
"
W
(s5,s6,d)∈Ax
min{T(s5),T(s6),f(d)}
# } ∧µ
=min ("
W
(a,s1,s2)∈Ax
{f(a)} #
, "
W
(s3,b,s4)∈Ax
{f(b)} #
, "
W
(s5,s6,d)∈Ax
{f(d)} #)
∧µ
≤min (
1, W
(s3,b,s4)∈Ax
{f(b)}, 1 )
∧µ
= W
(s3,b,s4)∈Ax
{f(b)} ∧µ= W
(s3,b,s4)∈Ax
{f(b)∧µ} ≤ f(s3bs4)∨λ≤ f(x)∨λ. Sincef is a(λ,µ)-fuzzy lateral ideal, thenf(s3s4cs5s6)∨λ≥ f(c)∧µ. Nowmin{(f ◦T◦T)(x),(T◦T◦f◦T◦T)(x),(T◦T◦f)(x)} ∧µ
=min{ "
W
(a,s1,s2)∈Ax
min{f(a),T(s1),T(s2)}
#
,
"
W
(s3,s4,c,s5,s6)∈Ax
min{T(s3),T(s4),f(c),T(s5),T(s6)}
#
,
"
W
(s7,s8,d)∈Ax
min{T(s7),T(s8),f(d)}
#
} ∧µ
=min (
W
(a,s1,s2)∈Ax
{f(a)}, W (s3,s4,c,s5,s6)∈Ax
{f(c)}, W (s7,s8,d)∈Ax
{f(d)} )
∧µ
≤min (
1, W
(s3,s4,c,s5,s6)∈Ax
{f(c)}, 1 )
∧µ
= W
(s3,s4,c,s5,s6)∈Ax
{f(c)} ∧µ= W
(s3,s4,cs5,s6)∈Ax
{f(c)∧µ} ≤ f(s3s4cs5s6)∨λ ≤ f(x)∨λ. Thusf is a(λ,µ)-fuzzy quasi-ideal ofT.
Theorem 5.14. Every(λ,µ)-fuzzy quasi-ideal of T is a(λ,µ)-fuzzy bi-ideal of T.
Proof. Let f be a(λ,µ)-fuzzy quasi-ideal ofT. Letx,y,z,u,v∈T. Then f(xyz)∨λ≥min{(f◦T◦T)(xyz),(T◦f ◦T)(xyz),(T◦T◦f)(xyz)} ∧µ
=min{ "
W
(a,b,c)∈Axyz
min{f(a),T(b),T(c)} #
,
"
W
(p,q,r)∈Axyz
min{T(p),f(q),T(r)} #
,
"
W
(l,m,n)∈Axyz
min{T(l),T(m),f(n)} #
} ∧µ
≥min{min{f(x),T(y),T(z)},min{T(x),f(y),T(z)}, min{T(x),T(y),f(z)}} ∧µ
=min{f(x),f(y),f(z)} ∧µ.
Thus f is a ternary subsemigroup ofT.
Also f(xuyvz)∨λ≥min{(f◦T◦T)(xuyvz),(T◦T◦f ◦T◦T)(xuyvz),(T◦T◦f)(xuyvz)} ∧µ
=min{ "
W
(a,b,c)∈Axuyvz
min{f(a),T(b),T(c)} #
"
W
(p,q,r,s,t)∈Axuyvz
min{T(p),T(q),f(r),T(s),T(t)} #
,
"
W
(i,j,k)∈Axuyvz
min{T(i),T(j),f(k)} #
} ∧µ
≥min{min{f(x),T(uyv),T(z)},min{T(x),T(u),f(y),T(v),T(z)}, min{T(x),T(uyv),f(z)}} ∧µ
=min{f(x),f(y),f(z)} ∧µ.
Hence f is a(λ,µ)-fuzzy bi-ideal ofT.
Corollary 5.5. Every(λ,µ)-fuzzy left (lateral, right) ideal of an ordered ternary semigroup T is a(λ,µ)-fuzzy bi-ideal of T.
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Received: July 10, 2015;Accepted: September 23, 2015
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