Study of Gamma Semigroups via its Operator Semigroups in terms of Atanassov's Intuitionistic Fuzzy Ideals

0.14 MB·English

by  Sujit Kumar Sardar

Γ
Study of -Semigroups via its Operator
1
1 Semigroups in terms of Atanassov’s
0
2 Intuitionistic Fuzzy Ideals

n
a
J Sujit Kumar Sardar1, Samit Kumar Majumder2

6
and
1
Manasi Mandal3

] Department of Mathematics, Jadavpur
M
University, Kolkata-700032, INDIA
G
[email protected]
.
h [email protected]
t
a 3manasi−[email protected]
m
[

1 Abstract
v
InthispapersomefundamentalrelationshipsofaΓ-semigroupanditsoperator
1
8 semigroups in terms of intuitionistic fuzzy subsets, intuitionistic fuzzy ideals, in-
0 tuitionistic fuzzy prime(semiprime) ideals, intuitionistic fuzzy ideal extensions are
3 obtained. Thesearethenusedtoobtainsomeimportantcharacterizationtheorems
.
1 of Γ-semigroups in terms of intuitionistic fuzzy subsets so as to highlight the role
0 ofoperatorsemigroupsinthestudyofΓ-semigroupsintermsofintuitionisticfuzzy
1
subsets.
1
:
v AMS Mathematics Subject Classification[2000]: 20N20
i
X KeyWordsand Phrases:Γ-semigroup,Intuitionisticfuzzysubset,Intuition-
istic fuzzy ideal, Intuitionistic fuzzy prime(semiprime) ideal, Intutionistic fuzzy
r
a ideal extension, Operator semigroups.
1 Introduction
This is a continuation of our paper on Atanassov’s Intutionistic Fuzzy Ideals of Γ-
semigroups[7]whereinwehaveinvestigated propertiesofAtanassov’s intutionisticfuzzy
(prime, semiprime) ideals and intutionistic fuzzy ideal extension and obtained charac-
terization of regular Γ-semigroup and of prime ideals of Γ-semigroups.
Dutta and Adhikari[2] have found operator semigroups of a Γ-semigroup to be a
very effective tool in studying Γ-semigroups. The principal objective of this paper is to
investigate as to whether the concept of operator semigroups can be made to work in
thestudyofΓ-semigroupsintermsofintuitionisticfuzzysubsets. Inordertodothiswe
deduce some fundamental relationships of a Γ-semigroup and its operator semigroups
in terms of intuitionistic fuzzy subsets, intuitionistic fuzzy ideals, intuitionistic fuzzy
1
prime(semiprime) ideals and intuitionistic fuzzy ideal extension. We then use these
relationships to obtain some characterization theorems thereby establishing the effec-
tiveness of operator semigroups in the study of Γ-semigroups in terms of intuitionistic
fuzzy subsets.
For preliminaries we refer to [7].
2 Main Results
ManyresultsofsemigroupscouldbeextendedtoΓ-semigroupsdirectlyandviaoperator
semigroups[1](left, right) of a Γ-semigroup. In this section in order to make operator
semigroups of a Γ-semigroup work in the context of IFS as it worked in the study
of Γ-semigroups[1, 2], we obtain various relationships between IFI(S) and that of its
operator semigroups. Here, among other results we obtain an inclusion preserving
bijection between the set of all IFI(S) and that of its operator semigroups. Among
other applications of this bijection we apply it to give new proofs of its ideal analogue
obtained in [2] by Dutta and Adhikari.
Definition 2.1. [2] Let S be a Γ-semigroup. Let us define a relation ρ on S ×Γ as
follows :(x,α)ρ(y,β) if and only if xαs = yβs for all s∈ S and γxα= γyβ forall γ ∈Γ.
Then ρ is an equivalence relation. Let [x,α] denote the equivalence class containing
(x,α). Let L = {[x,α] : x ∈ S,α ∈ Γ}. Then L is a semigroup with respect to the
multiplication defined by [x,α][y,β] = [xαy,β]. This semigroup L is called the left
operator semigroup of the Γ-semigroup S. Dually the right operator semigroup R of
Γ-semigroup S is defined where the multiplication is defined by [α,a][β,b] = [αaβ,b].
If there exists an element [e,δ] ∈ L([γ,f] ∈R) such that eδs = s(resp. sγf = s) for
all s∈ S then [e,δ](resp. [γ,f]) is called the left(resp. right) unity of S.
∗ ∗
Definition 2.2. For anIFS(R), A =(µ ,ν )we definean IFS(S),A = (µ ,ν ) =
A A A A
∗ ∗ ∗ ∗
(µ ,ν ) by µ (a) = inf µ ([γ,a]) and ν (a) = sup ν ([γ,a]), where a ∈ S. For
A A A γ∈Γ A A γ∈Γ A
∗′ ∗′ ∗′ ∗′
an IFS(S), B = (µ ,ν ) we define an IFS(R), B = (µ ,ν ) = (µ ,ν ) by
B B B B B B
′ ′
∗ ∗
µ ([α,a]) = inf µ (sαa) and ν ([α,a]) = sup ν (sαa), where [α,a] ∈ R. For an
B s∈S B B s∈S B
IFS(L), C = (µ ,ν ) we define an IFS(S), C+ = (µ ,ν )+ = (µ+,ν+) by µ+(a) =
C C C C C C C
inf µ ([a,γ]) and ν+(a) = sup ν ([a,γ]), where a ∈ S. For an IFS(S), D = (µ ,ν )
γ∈Γ C C γ∈Γ C D D
′ ′ ′ ′ ′
we definean IFS(L), D+ = (µ ,ν )+ = (µ+ ,ν+ ) by µ+ ([a,α]) = inf µ (aαs) and
D D D D D s∈S D
′
ν+ ([a,α]) = sup ν (aαs) where [a,α] ∈ L.
D D
s∈S
Now we recall the following propositions from [2] which were proved therein for one
sided ideals. But the results can be proved to be true for two sided ideals.
IFS,IFS(L),IFS(R),IFS(S),IFI(S)respectivelydenoteintuitionisticfuzzysubset(s),intuition-
istic fuzzy subset(s) of L, intuitionistic fuzzy subset(s) of R, intuitionistic fuzzy subset(s) of S, intu-
itionistic fuzzy ideal(s) of S.
2
Proposition 2.3. [2] Let S be a Γ-semigroup with unities and R be its right operator
semigroup. If P is a LI(R)(I(R)) then P∗ is a LI(S)(I(S)).
Proposition 2.4. [2] Let S be a Γ-semigroup with unities and R be its right operator
′
semigroup. If Q is a LI(S)(I(S)) then Q∗ is a LI(R)(I(R)).
For convenience of the readers, we may note that for a Γ-semigroup S and its left,
′ ′
right operator semigroups L,R respectively four mappings namely ()+, ()+ ,()∗,()∗
∗
occur. They are defined as follows: For I ⊆ R,I = {s ∈ S,[α,s] ∈ I∀α ∈ Γ}; for
′
P ⊆ S,P∗ = {[α,x] ∈ R : sαx ∈ P∀s∈ S}; for J ⊆ L,J+ = {s ∈ S,[s,α] ∈ J∀α ∈ Γ};
′
for Q ⊆ S,Q+ = {[x,α] ∈L :xαs ∈ Q∀s∈ S}.
Proposition 2.5. Let A= (µ ,ν ) be an IFS(R), then [U(µ ;t)]∗ = U((µ )∗;t) and
A A A A
[L(ν ;t)]∗ = L((ν )∗;t) for all t ∈ [0,1], provided the sets are non-empty.
A A
Proof. Let m ∈ S. Then m ∈ [U(µ ;t)]∗ ⇔ [γ,m] ∈ U(µ ;t)∀γ ∈ Γ ⇔ µ ([γ,m]) ≥
A A A
∗ ∗
t ∀γ ∈ Γ ⇔ inf µ ([γ,m]) ≥ t ⇔ (µ ) (m) ≥ t ⇔ m ∈ U((µ ) ;t). Again let n ∈ S.
A A A
γ∈Γ
∗
Then n ∈ [L(ν ;t)] ⇔ [γ,n] ∈ L(ν ;t)∀γ ∈ Γ ⇔ ν ([γ,n]) ≤ t ∀γ ∈ Γ ⇔ sup
A A A
γ∈Γ
∗ ∗ ∗ ∗
ν ([γ,n]) ≤ t ⇔ (ν ) (n) ≤ t ⇔ n ∈ L((ν ) ;t). Hence [U(µ ;t)] = U((µ ) ;t) and
A A A A A
∗ ∗
[L(ν ;t)] = L((ν ) ;t).
A A
Proposition 2.6. Let B = (µ ,ν ) be an IFS(S). Then [U(µ ;t)]∗′ = U((µ )∗′;t)
B B B B
and [L(ν ;t)]∗′ = L((ν )∗′;t) for all t ∈ [0,1], provided the sets under consideration
B B
are non-empty.
′
Proof. Let[α,x] ∈Randtisasmentionedinthestatement. Then[α,x] ∈ [U(µ ;t)]∗ ⇔
B
mαx ⊆ U(µ ;t) ∀m ∈ S ⇔ µ (mαx) ≥ t ∀m ∈ S ⇔ inf µ (mαx) ≥ t ⇔
B B B
m∈S
∗′ ∗′
(µ ) ([α,x]) ≥ t ⇔ [α,x] ∈ U((µ ) ;t). Again let [β,y] ∈ R and t is as mentioned
B B
′
∗
in the statement. Then [β,y] ∈ [L(ν ;t)] ⇔ nβy ⊆ L(ν ;t) ∀n ∈ S ⇔ ν (nβy) ≤
B B B
∗′ ∗′
t ∀n ∈ S ⇔ sup ν (nβy) ≤ t ⇔ (ν ) ([β,y]) ≤ t ⇔ [β,y] ∈ L((ν ) ;t). Hence
B B B
n∈S
∗′ ∗′ ∗′ ∗′
[U(µ ;t)] = U((µ ) ;t) and [L(ν ;t)] = L((ν ) ;t).
B B B B
In what follows S denotes a Γ-semigroup with unities[2], L, R be its left and right
operator semigroups respectively.
Proposition 2.7. If A = (µ ,ν ) ∈ IFI(R)(IFLI(R)), then A∗ = (µ ,ν )∗ =
A A A A
(µ∗,ν∗)∈ IFI(S)(respectively IFLI(S)).
A A
LI(R),LI(S),I(R),I(S),IFLI(R),IFLI(S),IFI(R),IFI(S) respectively denote left ideal(s) of
R,leftideal(s)ofS,ideal(s)ofR,ideal(s)ofS,intuitionisticfuzzyleftideal(s)ofR,intuitionisticfuzzy
left ideal(s) of S, intuitionistic fuzzy ideal(s) of R,intuitionistic fuzzy ideal(s) of S,
3
Proof. Suppose A = (µ ,ν ) ∈ IFI(R). Then U(µ ;t) and L(ν ;t) are I(R), ∀t ∈
A A A A
∗ ∗
[0,1]. Hence [U(µ ;t)] and [L(ν ;t)] are I(S), ∀t ∈ [0,1](cf. Proposition 2.3). Now
A A
since A = (µ ,ν ) is an IFI(R), A = (µ ,ν ) is a non-empty IFS(R). Hence
A A A A
for some [α,m] ∈ R, 0 < µ ([α,m]) + ν ([α,m]) ≤ 1. Then U(µ ;t) 6= φ and
A A A
L(ν ;t) 6= φ where t := µ ([α,m]) = ν ([α,m]). So by the same argument applied
A A A
∗ ∗ ∗
above [U(µ ;t)] 6= φ and [L(ν ;t)] 6= φ. Let u ∈ [U(µ ;t)] . Then [β,u] ∈ U(µ ;t)
A A A A
for all β ∈ Γ. Hence µ ([β,u]) ≥ t. This implies that inf µ ([β,u]) ≥ t, i.e.,
A A
β∈Γ
∗ ∗ ∗
(µ ) (u) ≥ t. Hence u ∈ U((µ ) ;t). Hence U((µ ) ;t) 6= φ. By similar argu-
A A A
∗ ∗ ∗
ment we can show that L((ν ) ;t) 6= φ. Consequently, [U(µ ;t)] = U((µ ) ;t) and
A A A
∗ ∗ ∗ ∗
[L(ν ;t)] = L((ν ) ;t)(cf. Proposition 2.5). Itfollows that U((µ ) ;t)and L((ν ) ;t)
A A A A
∗ ∗ ∗ ∗
are I(S) for all t ∈ [0,1]. Hence A = (µ ,ν ) = (µ ,ν ) is an IFI(S)(cf. Theorem
A A A A
3.10[7]). Similarly we can prove the other case also.
In a similar fashion by using Propositions 2.4, 2.6 and Theorems 3.9[7],3.10[7] we
deduce the following proposition.
′ ′
Proposition 2.8. If B = (µ ,ν ) ∈ IFI(S)(IFLI(S)), then B∗ = (µ ,ν )∗ =
B B B B
′ ′
(µ∗ ,ν∗ ) ∈ IFI(R)(respectively IFLI(R)).
B B
Remark 1. The left operator analogues of Propositions 2.3-2.8. are also true.
In view of Remark 1, we deduce the following theorem.
Theorem2.9. LetS beaΓ-semigroupwithunitiesandLbeitsleftoperator semigroup.
′
Then there exists an inclusion preserving bijection A 7→ A+ between the set of all
IFI(S)(IFRI(S)) and set of all IFI(L)(resp. IFRI(L)), where A = (µ ,ν ) is an
A A
IFI(S)(resp. IFRI(S)).
Proof. Let A= (µ ,ν ) ∈IFI(S)(IFRI(S)) and x ∈ S. Then
A A
′ ′
(µ+ )+(x) = infµ+ ([x,γ]) = inf[infµ (xγs)] ≥ µ (x).
A γ∈Γ A γ∈Γs∈S A A
Again
′ ′
(ν+ )+(x) = supν+ ([x,γ]) = sup[supν (xγs)] ≤ ν (x).
A A A A
γ∈Γ γ∈Γs∈S
′
Hence A ⊆ (A+ )+. Let [γ,f] be the right unity of S. Then xγf = x for all x ∈ S.
Then
′ ′
µ (x) = µ (xγf) ≥ inf[infµ (xαs)] = infµ+ ([x,α]) = (µ+ )+(x).
A A α∈Γs∈S A α∈Γ A A
Again
′ ′
ν (x) = ν (xγf)≤ sup[supν (xαs)] = supν+ ([x,α]) = (ν+ )+(x).
A A A A A
α∈Γs∈S α∈Γ
IFRI(L),IFRI(S) respectively denote intuitionistic fuzzy right ideal(s) of L and intuitionistic
fuzzy right ideal(s) of S.
4
′ ′
So A ⊇ (A+ )+. Hence (A+ )+ = A. Thus the said mapping is one-one. Now let
B = (µ ,ν )∈ IFI(L)(IFRI(S)). Then
B B
′
(µ+)+ ([x,α]) = infµ+(xαs) = inf[infµ ([xαs,γ])]
B s∈S B s∈S γ∈Γ B
= inf[infµ ([x,α][s,γ])] ≥ µ ([x,α]).
B B
s∈S γ∈Γ
Again
′
(ν+)+ ([x,α]) = supν+(xαs) = sup[supν ([xαs,γ])]
B B B
s∈S s∈S γ∈Γ
= sup[supν ([x,α][s,γ])] ≥ν ([x,α]).
B B
s∈S γ∈Γ
′
So B ⊆ (B+)+ . Let [e,δ] be the left unity of L. Then
µ ([x,α]) = µ ([x,α][e,δ]) ≥ inf[infµ ([x,α][s,γ])]
B B B
s∈S γ∈Γ
′
= (µ+)+ ([x,α]).
B
Again
ν ([x,α]) = ν ([x,α][e,δ]) ≤ sup[supν ([x,α][s,γ])]
B B B
s∈S γ∈Γ
′
= (ν+)+ ([x,α])
B
′ ′ ′
So B ⊇ (B+)+ and hence B = (B+)+ . Consequently, the correspondence A 7→ A+
is a bijection. Now let C = (µ ,ν ),D = (µ ,ν ) ∈ IFI(S)(IFRI(S)) be such that
C C D D
C ⊆ D, i.e.,µ ⊆ µ and ν ⊇ ν . Then for all [x,α] ∈ L,
C D C D
′ ′
µ+ ([x,α]) = infµ (xαs) ≤ infµ (xαs) = µ+ ([x,α])
C s∈S C s∈S D D
and
′ ′
ν+ ([x,α]) = supν (xαs) ≥supν (xαs) = ν+ ([x,α]).
C C D D
s∈S s∈S
′ ′ ′ ′ ′ ′ ′
Thus µ+ ⊆ µ+ and ν+ ⊇ ν+ . Consequently, C+ ⊆ D+ . Hence A 7→ A+ is an
C D C D
inclusion preserving bijection. The rest of the proof follows from Remark 1.
In a similar way by using Proposition 2.7 and Proposition 2.8 we can deduce the
following theorem.
Theorem 2.10. Let S be a Γ-semigroup with unities and R be its right operator semi-
′
group. Then there exists an inclusion preserving bijection B 7→ B∗ between the set of
all IFI(S)(IFLI(S)) and set of all IFI(R)(resp. IFLI(R)), where B = (µ ,ν ) is
B B
an IFI(resp. IFLI(S)).
5
Now to apply the above theorem for giving a new proof of Theorem 4.6[2] and its
two sided ideal analogue we deduce the following lemmas.
Lemma 2.11. Let I be a LI(R)(I(R)) of a Γ-semigroup S and P = (χ ,χc) where χ
I I I
is the characteristic function of I. Then P∗ = (χI,χcI)∗ = ((χI)∗,(χcI)∗)= (χI∗,χcI∗).
Proof. Suppose s ∈ I∗. Then [β,s] ∈ I for all β ∈ Γ. This means inf(χ ([β,s])) = 1
I
β∈Γ
and sup(χcI([β,s])) = 0. Also χI∗(s) = 1 and χcI∗(s) = 0. Now suppose s ∈/ I∗.
β∈Γ
Then there exists δ ∈ Γ such that [δ,s] ∈/ I. Hence χ ([δ,s]) = 0,χc([δ,s]) = 1 and so
I I
inf(χ ([β,s])) = 0,sup(χc([β,s])) = 1. Hence (χ )∗(s) = 0 and (χc)∗(s) = 1. Again
β∈Γ I β∈Γ I I I
(χI∗)(s) = 0 and (χcI∗)(s) = 1. Thus P∗ = (χI,χcI)∗ = ((χI)∗,(χcI)∗) =(χI∗,χcI∗).
The following lemma follows in a similar way.
Lemma 2.12. Let I be a RI(S)(I(S)), P = (χ ,χc) and R be the right operator
I I
semigroup of S. Then P∗′ = (χI,χcI)∗′ = ((χI)∗′,(χcI)∗′) = (χI∗′,χcI∗′), where χI is the
characteristic function of I.
Remark 2. By drawing an analogy we deduce results similar to the above lemmas for
′
left operator semigroup L of the Γ-semigroup S,i.e., for the functions + and + .
Now we present a new proof of the following result which is originally due to Dutta
and Adhikari[2].
Theorem 2.13. [2]Let S be a Γ-semigroup with unities. Then there exists an inclusion
preserving bijection between the set of all I(S)(LI(S)) and that of its right operator
′
semigroup R via the mapping I → I∗ .
′
Proof. Let us denote the mapping I → I∗ by φ. This is actually a mapping follows
′ ′
∗ ∗
from Proposition 2.8. Now let φ(I ) = φ(I ). Then I = I . This implies that
1 2 1 2
(χI1∗′,χcI1∗′) = (χI2∗′,χcI2∗′)(where χI is the characteristic function I). Hence by Lemma
′ ′
2.12,(χ ,χc )∗ = (χ ,χc )∗ . This together with Theorem 2.10, gives (χ ,χc ) =
I1 I1 I2 I2 I1 I1
(χ ,χc ) whence I = I . Consequently φ is one-one. Let I be a I(R)(LI(R)). Then
I2 I2 1 2
′
(χ ,χc) is an IFI(R)(IFLI(R)). Hence by Theorem 2.10, ((χ ,χc)∗)∗ = (χ ,χc).
I I I I I I
This implies that (χ(I∗)∗′,χc(I∗)∗′) = (χI,χcI) (cf. Lemma 2.11 and Lemma 2.12).
′
∗ ∗ ∗ ∗
Hence (I ) = I,i.e., φ(I ) = I. Now since I is a I(S)(LI(S))(cf. Proposition 2.3),
itfollowsthatφisonto. LetI ,I betwoI(S)(LI(S))withI ⊆ I .Thenχ ⊆ χ and
1 2 1 2 I1 I2
′ ′ ′ ′
χc ⊇ χc . Hence by Theorem 2.10, we see that (χ )∗ ⊆ (χ )∗ and (χc )∗ ⊇ (χc )∗
I1 I2 I1 I2 I1 I2
′ ′
i.e., χI1∗′ ⊆ χI2∗′ and χcI1∗′ ⊇ χcI2∗′ (cf. Lemma 2.12) which gives I1∗ ⊆ I2∗ .
6
Remark 3. Now by using a similar argument as above and with the help of lemmas
dual to the lemmas 2.11,2.12(cf. Remark 2) and Theorem 2.9 we deduce that the
′
mapping ()+ is an inclusion preserving bijection(with ()+ as the inverse) between the
set of all I(S)(RI(S)) and that of its left operator semigroup L.
In what follows S denotes a Γ-semigroup not necessarily with unities, L, R be its
left and right operator semigroups respectively.
Proposition 2.14. [2, 5]LetS beaΓ-semigroupandR beitsrightoperator semigroup.
If P is PI(R)(SPI(R)) then P∗ is PI(S)(SPI(S)).
Proposition 2.15. [2, 5]LetS beaΓ-semigroupandR beitsrightoperator semigroup.
′
If Q is PI(S)(SPI(S)) then Q∗ is PI(R)(SPI(R)).
Proposition 2.16. If A = (µ ,ν ) ∈ IFPI(R)(IFSPI(R)), then A∗ = (µ∗,ν∗) ∈
A A A A
IFPI(S)(resp. IFSPI(S)).
Proof. Let A = (µ ,ν ) ∈ IFPI(R). Then it is in IFI(R). Hence by Proposition 2.7,
A A
∗ ∗ ∗
A = (µ ,ν ) ∈ IFI(S). Since A = (µ ,ν ) ∈ IFPI(R), so U(µ ;t) and L(µ ;t)
A A A A A A
∗ ∗
are PI(R). Now by Proposition 2.14, for all t ∈ [0,1], [U(µ ;t)] and [L(ν ;t)] are
A A
∗ ∗ ∗ ∗
PI(S). By Proposition 2.5, [U(µ ;t)] = U((µ ) ;t) and [L(ν ;t)] = L((ν ) ;t). So
A A A A
∗ ∗ ∗ ∗ ∗
U((µ ) ;t) and L((ν ) ;t) are PI(S). Hence A = (µ ,ν ) ∈ IFPI(S). Similarly we
A A A A
can prove the other case also.
In a similar fashion by using Propositions 2.6,2.8 and Theorem 3.9[7],3.10[7] we
deduce the following proposition.
′ ′ ′
Proposition 2.17. If B = (µ ,ν ) ∈ IFPI(S)(IFSPI(S)), then B∗ = (µ∗ ,ν∗ ) ∈
B B B B
IFPI(R).
Remark 4. We can also deduce the following left operator analogues of Propositions
2.14-2.17.
The following theorem is on the inclusion preserving bijection between the set of
all IFPI(S) and the set of all IFPI(R). It may be noted that S need not have unities
here which was the case for the set of all IFI(cf. Theorems 2.9,2.10).
Theorem 2.18. Let S be a Γ-semigroup and R be its right operator semigroup. Then
′ ′ ′
there exist an inclusion preserving bijection B = (µ ,ν ) 7→ B∗ = (µ∗ ,ν∗ ) between
B B B B
the set of all IFPI(S)(IFSPI(S)) and set of all IFPI(R)(resp. IFSPI(R)).
Proof. Let B = (µ ,ν )∈ IFPI(R) and x ∈ S. Then
B B
′ ′
∗ ∗ ∗
(µ ) (x) = infµ ([γ,x]) = inf infµ (sγx)≥ µ (x)(since B ∈ IFI(S)).
B γ∈Γ B γ∈Γs∈S B B
PI(R),PI(S),SPI(R),SPI(S),IFPI(R),IFPI(S),IFSPI(R),IFSPI(S) respectively denote
primeideal(s) of R,primeideal(s) of S,semiprimeideal(s) of R,semiprime ideal(s) of S,intuitionistic
fuzzyprimeideal(s)ofR,intuitionisticfuzzyprimeideal(s)ofS,intuitionisticfuzzysemiprimeideal(s)
of R,intuitionistic fuzzy semiprime ideal(s) of S.
7
′
∗ ∗
Again for x ∈ S,(µ ) (x) = inf infµ (sγx)= inf infµ (sγx)
B γ∈Γs∈S B s∈Sγ∈Γ B
= inf max{µ (s),µ (x)}(since B ∈ IFPI(S))
B B
s∈S
≤ max{µ (x),µ (x)} = µ (x).
B B B
′
∗ ∗
Hence (µ ) (x) = µ (x). Also
B B
′ ′
∗ ∗ ∗
(ν ) (x) = supν ([γ,x]) = supsupµ (sγx) ≤ ν (x)(since B ∈ IFI(S)).
B B B B
γ∈Γ γ∈Γs∈S
′
∗ ∗
Again for x ∈ S,(ν ) (x) = supsupν (sγx) = supsupν (sγx)
B B B
γ∈Γs∈S s∈Sγ∈Γ
= supmin{ν (s),ν (x)}(since B ∈ IFPI(S))
B B
s∈S
≥ min{ν (x),ν (x)} =ν (x).
B B B
′ ′
∗ ∗ ∗ ∗
Hence (ν ) (x) = ν (x). Consequently, (B ) = B. Hence the mapping is one-one.
B B
Now let [α,x] ∈ R. Then
′
∗ ∗ ∗
(µ ) ([α,x]) = infµ (sαx) = inf infµ ([β,sαx])
B s∈S B s∈Sβ∈Γ B
= inf infµ ([β,s][α,x]) ≥ µ ([α,x]) ......................(∗ ).
B B 1
s∈Sβ∈Γ
′
∗ ∗ ∗
Also, (ν ) ([α,x]) = supν (sαx) = supsupν ([β,sαx])
B B B
s∈S s∈Sβ∈Γ
= supsupν ([β,s][α,x]) ≤ ν ([α,x]) ......................(∗∗ ).
B B 1
s∈Sβ∈Γ
Since B = (µ ,ν ) ∈ IFPI(R), for all α,β ∈ Γ, for all x,s ∈ S. µ ([α,x][β,s]) =
B B B
max{µ ([α,x]),µ ([β,s])} and ν ([α,x][β,s]) = min{ν ([α,x]),ν ([β,s])}∀s ∈ S,∀β
B B B B B
∈ Γ.Hencefors= xandβ = αweobtainµ ([α,x][β,s]) = µ ([α,x])andν ([α,x][β,s])
B B B
′
∗ ∗
= ν ([α,x]). This together with the relations (µ ) ([α,x]) = inf infµ ([α,x][β,s])
B B s∈Sβ∈Γ B
′ ′
∗ ∗ ∗ ∗
and (ν ) ([α,x]) = supsupν ([α,x][β,s]) give (µ ) ([α,x]) ≤ µ ([α,x])..........(∗ )
B B B B 2
s∈Sβ∈Γ
′ ′
∗ ∗ ∗ ∗
and (ν ) ([α,x]) ≥ ν ([α,x])..........(∗∗ ). By (∗ ) and (∗ ) we obtain (µ ) ([α,x]) =
B B 2 1 2 B
′
∗ ∗
µ ([α,x]) and by (∗∗ ) and (∗∗ ) we have (ν ) ([α,x]) = ν ([α,x]). Consequently,
B 1 2 B B
′
∗ ∗
(B ) = B. Hence the mapping is onto. Inclusion preserving property is similar as in
′
∗
Theorem 2.9. Hence B 7→ B is an inclusion preserving bijection.
Remark 5. (i) Similar results hold for IFSPI(S). (ii) Similar result holds for the
Γ-semigroup S and the left operator semigroup L of S.
Corollary 2.19. Let S be a Γ-semigroup and R,L be respectively its right and left
operator semigroups. Then there exists an inclusion preserving bijection between the
set of all IFPI(R)(IFSPI(R)) and the set of all IFPI(L)(IFSPI(L)).
8
Remark 6. In view of Theorem 2.18, we see that in a Γ-semigroup S with unities the
above result also holds for IFI.
NowwerevisitthefollowingtheoremwhichisoriginallyduetoDuttaandAdhikari[2]
via intutionistic fuzzy ideals by using Theorem 2.18 and applying similar argument as
applied in Theorem 2.13.
Theorem 2.20. Let S be a Γ-semigroup. Then there exists an inclusion preserving
bijection between the set of all PI(S)(SPI(S)) and that of its right operator semigroup
′
R via the mapping I → I∗ .
The definition of an IFE, < x,A > of an IFS, A = (µ ,ν ) is given in [7]. Now
A A
by routine verification we obtain the following two propositions.
Proposition 2.21. Let S be a commutative Γ-semigroup and L(R) the left(respectively
theright)operator semigroupsofS.LetA = (µ ,ν )beanIFLI(S)(IFRI(S),IFI(S))
A A
′ ′
then < x,A+ > (respectively < x,A∗ >) is an IFLI(L(R))(IFRI(L(R)),IFI(L(R)))
for all x ∈ L(R).
Proposition 2.22. (With same notation as in the above proposition) If B = (µ ,ν )
B B
is an IFLI(L(R))(IFRI(L(R)),IFI(L(R))) then < x,B+ > (respectively < x,B∗ >)
is an IFLI(S)(IFRI(S),IFI(S)) for all x ∈ S.
NowwededucethefollowingtwolemmasontherelationshipsbetweenaΓ-semigroup
and its operator semigroups in terms of IFE.
Lemma 2.23. Let A = (µ ,ν ) be an IFS(S) where S is commutative. Then for all
A A
x ∈S,
′ ′
∗ ∗
(1) < x,A > ⊆<[α,x],A > ∀α ∈Γ.
′ ′ ′
∗ ∗ ∗
(2) < x,A > = (< x,µ > ,< x,ν > )
A A
′ ′
∗ ∗
= (inf < [α,x],µ >,sup< [α,x],ν >).
α∈Γ A α∈Γ A
′
Proof. (1) Let [β,y] ∈ R. Then < x,µ >∗ ([β,y]) = inf < x,µ > (sβy) = inf inf
A A
s∈S s∈Sγ∈Γ
′ ′
∗ ∗
µ (xγsβy) = infinf µ (xγsβy). Again < [α,x],µ > ([β,y]) = µ ([α,x][β,y]) =
A γ∈Γs∈S A A A
′
∗ ∗ ∗
µ ([α,xβy]) = infµ (sαxβy) = infµ (xαsβy)(using the commutativity of S). Since
A s∈S A s∈S A
′ ′
∗ ∗ ∗ ∗
inf infµ (xγsβy) ≤ infµ (xαsβy), we obtain < x,µ > ([β,y]) ≤< [α,x],µ >
γ∈Γs∈S A s∈S A A A
′ ′
∗ ∗
([β,y]). By similar argument we can show that < x,ν > ([β,y]) ≥< [α,x],ν >
A A
′ ′
∗ ∗
([β,y]). Hence < x,A> ⊆< [α,x],A > ∀α∈ Γ.
IFE denote an intuitionsitic fuzzy extension.
9
′ ′
∗ ∗
(2) Let [β,y] ∈ R. Then inf < [α,x],µ > ([β,y]) = inf µ ([α,x][β,y]) = inf
α∈Γ A α∈Γ A α∈Γ
′ ′
∗ ∗ ∗
µ ([α,xβy]) = inf inf µ (sαxβy) = inf < x,µ > (sβy) =< x,µ > ([β,y]).
A α∈Γs∈S A s∈S A A
′ ′
∗ ∗
By applying similar argument we obtain sup < [α,x],ν > ([β,y]) =< x,ν >
A A
α∈Γ
′ ′ ′ ′
∗ ∗ ∗ ∗
([β,y]). Hence < x,A > = (< x,µ > ,< x,ν > ) = (inf < [α,x],µ >,sup <
A A α∈Γ A α∈Γ
′
∗
[α,x],ν >).
A
Lemma 2.24. If B = (µ ,ν ) is an IFS(R) then for all x ∈ S, < [β,x],B >∗⊇<
B B
∗
x,B > ∀β ∈ Γ.
Proof. Let p ∈ S. Then < [β,x],µ >∗ (p) = inf < [β,x],µ > ([γ,p]) = inf
B B
γ∈Γ γ∈Γ
∗ ∗
µ ([β,x][γ,p]) = inf µ ([β,xγp]). Again < x,µ > (p) = inf µ (xγp) = inf inf
B γ∈Γ B B γ∈Γ B γ∈Γβ∈Γ
µ ([β,xγp]) = inf inf µ ([β,xγp]). Since inf µ ([β,xγp]) ≥ inf inf µ ([β,xγp]), we
B B B B
β∈Γγ∈Γ γ∈Γ β∈Γγ∈Γ
∗ ∗ ∗
have < [β,x],µ > (p) ≥< x,µ > (p). Similarly we can show that < [β,x],ν >
B B B
∗ ∗ ∗
(p) ≤< x,ν > (p). Hence < [β,x],B > ⊇< x,B > ∀β ∈Γ.
B
′ ′
Lemma 2.25. Let I be an I(S). Then ((µI)∗ ,(νI)∗ )= (µI∗′,νI∗′).
′ ′
Proof. Let [β,y] ∈ R. Then (µ )∗ ([β,y]) = inf µ (sβy) and (ν )∗ ([β,y]) = sup
I I I
s∈S s∈S
′
∗
ν (sβy). Suppose [β,y] ∈ I . Then sβy ∈ I for all s ∈ S. Hence µ (sβy) = 1 and
I I
ν (sβy) = 0 for all s ∈ S which implies that inf µ (sβy) = 1 and sup ν (sβy) = 0
I I I
s∈S s∈S
′ ′
∗ ∗
whence (µI) ([β,y]) = 1 and (νI) ([β,y]) = 0. Also µI∗′([β,y]) = 1 and νI∗′([β,y]) =
′
∗
0. If [β,y] ∈/ I then (µ ′)([β,y]) = 0, (ν ′)([β,y]) = 1 and there exists s ∈ S such
I∗ I∗
′
∗
that sβy ∈/ I. Hence µ (sβy) = 0 and ν (sβy) = 1 whence (µ ) ([β,y]) = 0 and
I I I
′ ′ ′
∗ ∗ ∗
(νI) ([β,y]) = 1. Consequently, ((µI) ,(νI) )= (µI∗′,νI∗′).
′ ′
Lemma 2.26. [4] Let {Aα}α∈I be a family of I(S). Then ( T Aα)∗ = T (Aα)∗ .
α∈I α∈I
Lemma 2.27. LetS be aΓ-semigroup, R beitsright operator semigroup and{Ai}i∈I =
(µAi,νAi)i∈I be a family of IFS(S) such that A = (µA,νA) := ii∈nIf Ai = (iin∈If µAi,siu∈Ip
′ ′ ′ ′ ′ ′
ν ). Then A∗ := ((µ )∗ ,(ν )∗ ) = inf (A )∗ = (inf (µ )∗ ,sup (ν )∗ ).
Ai A A i∈I i i∈I Ai i∈I Ai
′ ′
Proof. Let[α,x] ∈ R.Then(µ )∗ ([α,x]) = (inf µ )∗ ([α,x]) = inf inf µ (sαx) = inf
A i∈I Ai s∈S i∈I Ai i∈I
′ ′
∗ ∗
inf µ (sαx) = inf (µ ) ([α,x]). Similarly we can show that (ν ) ([α,x]) = sup
s∈S Ai i∈I Ai A i∈I
′ ′ ′
∗ ∗ ∗
(ν ) ([α,x]). Hence A = inf (A ) .
Ai i∈I i
10