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ISSN 2229 – 5046
NOTE ON FUZZY WEAKLY COMPLETELY PRIME - IDEALS IN TERNARY SEMIGROUPS
U. NAGI REDDY
1*, Dr. G. SHOBHALATHA
21
Research scholar, Dept. of Mathematics,
SriKrishnadevaraya University, Ananthapuramu, (A.P.), India – 515003.
2
Professor, Dept. of Mathematics,
SriKrishnadevaraya University, Ananthapuramu, (A.P.), India – 515003.
(Received On: 01-03-16; Revised & Accepted On: 18-05-16)
ABSTRACT
I
deal theory in ternary semigroups is introduced by F.M. Sioson. Samit Kumar Majumder introduced and studied on fuzzy completely prime ideals in gamma semigroups. In this paper we proved some properties of prime -ideals and fuzzy weakly completely prime -ideals in ternary semigroups. It is prove that Ifµ
be a non-empty fuzzy subset of aternary semigroup
T
, then1
−
µ
is a fuzzy ternary sub semigroup of if and only ifµ
is a fuzzy weakly completely prime - ideal inT
.Key words: Ternary semigroup, fuzzy set, Fuzzy left (lateral, right) ideals, Fuzzy weakly completely prime ideals.
INTRODUCTION
The theory of ternary algebraic system was introduced by D.H. Lehmer in 1932 but earlier such structures were studied by Kasner in1904, who gave the idea of n-ary algebras. Ternary semigroups are universal algebras with one associative ternary operation. The Concept of quasi-ideals in semigroups was introduced in 1956 by O. Steinfeld. Dixit and Dewan studied quasi-ideals and biideals in ternary semigroups. The introduction of fuzzy sets by L.A. Zadeh. After N. Kuroki introduced and studied the notion of fuzzy semigroups. He also studied the concept of fuzzy bi-ideals (1976) and fuzzy quasi-ideals(1982) of semigroups. Shabir, Jun and Bano introduced and studied the prime, strongly prime, semiprime and irreducible fuzzy bi-ideals of semigroups. They characterized those semigroups for which each fuzzy bi-ideal is semiprime and also characterized those semigroups for which each fuzzy bi-ideal is strongly prime. Several researchers conducted the researches on the generalizations of the notions of fuzzy sets with huge applications in computer, logics and many branches of pure and applied mathematics. Chinram and Saelee in 2010 studied the concept of fuzzy ideals and fuzzy filters of ordered ternary semigroups.
1. BASIC DEFINITIONS AND PRELIMINARIES
1.1 Definition: A non-empty set
T
is said to be ternary semigroup if there exists a ternary operation⋅
:
T
×
T
×
T
→
T
written as(
a
,
b
,
c
)
→
a
.
b
.
c
satisfies the following identity)
.
.
.(
.
).
.
.
.(
.
).
.
.
(
a
b
c
d
e
=
a
b
c
d
e
=
a
b
c
d
e
for anya
,
b
,
c
,
d
,
e
∈
T
.1.2 Definition: A non-empty subset
A
of a ternary semigroupT
is called a ternary subsemigroup ofT
ifAAA
⊆
A
.1.3 Definition: A non-empty subset
A
of a ternary semigroupT
is called a left (right, lateral) ideal inT
if(
ATT
A
TAT
A
)
A
TTA
⊆
⊆
,
⊆
.1.4 Definition: A non-empty subset
A
of a ternary semigroupT
is called a two sided ideal ofT
if it is both left and right ideal inT
.Corresponding Author: U. Nagi Reddy
1*,
1Research scholar, Dept. of Mathematics,
1.5 Definition: A non-empty subset
A
of a ternary semigroupT
is called a ideal inT
if it is left, right and lateral ideal inT
.1.6 Definition: Let
T
be a non-empty set. A fuzzy subset of a ternary semigroupT
is a functionµ
:
T
→
[
0
,
1
]
.1.7 Definition: Let
µ
be a fuzzy subset of a non-empty setT
for anyt
∈
[
0
,
1
]
, the subsetµ
t=
{
x
∈
T
:
µ
(
x
)
≥
t
}
of
T
is called a level set ofµ
.1.8 Definition: For any two fuzzy subsets
µ
1 andµ
2of a non-empty setT
, the union and the intersection ofµ
1 andµ
2 denoted byµ
1∪
µ
2 andµ
1∩
µ
2 are fuzzy subsets ofT
and defined as)
(
)
(
}
)
(
),
(
{
max
)
(
)
(
µ
1∪
µ
2x
=
µ
1x
µ
2x
=
µ
1x
∨
µ
2x
andS
for allx
∈
T
.
where
∨
denotes maximum or supremum and∧
denotes minimum or infimum.1.9 Definition: Let
µ
1,µ
2andµ
3 are any three fuzzy sets of a ternary semigroupT
. Then their fuzzy product3 2 1
µ
µ
µ
is defined byµ
1.10 Definition: A fuzzy set
µ
of a ternary semigroupT
is called a fuzzy ternary subsemigroup ofT
if)}
(
)
(
)
(
{
)
(
xyz
µ
x
µ
y
µ
z
µ
≥
∧
∧
for allx
,
y
,
z
∈
T
.1.11 Definition: A fuzzy set
µ
of a ternary semigroupT
is called a fuzzy left (right, lateral) ideal inT
if)
(
)
(
x
y
z
µ
z
µ
≥
, (µ
(
x
y
z
)
≥
µ
(
x
)
,µ
(
x
y
z
)
≥
µ
(
y
)
) for allx
,
y
,
z
∈
T
.
.1.12 Definition: A fuzzy set
µ
of a ternary semigroupT
is a fuzzy ideal inT
if it is fuzzy left, right and lateral ideal inT
.1.13 Definition: Let A be a non-subset of a ternary semigroup
T
. Then the characteristic function of A is defined by( )
1
0
A
if
x
A
C
x
if
x
A
∈
=
∉
We denote the characteristic function
C
Tof
T
i.e.,T
=
C
T thusT
(
x
)
=
1
for allx
∈
T
.1.14 Definition: A subset
A
of a ternary semigroupT
is said to be a prime ideal inT
ifxyz
∈
A
impliesx
∈
A
orA
y
∈
orz
∈
A
.1.15 Definition: A fuzzy ideal
µ
of a ternary semigroupT
is called a fuzzy weakly completely prime ideal inT
if)
(
)
(
x
µ
x
y
z
µ
≥
orµ
(
y
)
≥
µ
(
x
y
z
)
orµ
(
z
)
≥
µ
(
x
y
z
)
for allx
,
y
,
z
∈
T
.1.16 Definition: A fuzzy ideal
µ
of a ternary semigroupT
is called a fuzzy prime ideal inT
if}
{
(
),
(
),
(
)
max
)
(
inf
µ
x
y
z
≥
µ
x
µ
y
µ
z
for allx
,
y
,
z
∈
T
.1.17 Proposition: If
µ
1µ
2µ
3µ
4 andµ
5 are fuzzy subsets of a non-empty setT
then (i)µ
1∩
(
µ
2∪
µ
3)
∩
µ
4=
(
µ
1∩
µ
2∩
µ
4)
∪
(
µ
1∩
µ
3∩
µ
4).
(ii)
(
C
A
C
T
C
T)
∩
(
C
T
C
A
C
T)
∩
(
C
T
C
T
C
A)
(iii)
µ
1
(
µ
2∪
µ
3)
µ
4=
(
µ
1
µ
2
µ
4)
∪
(
µ
1
µ
3
µ
4).
(iv)
µ
1
µ
2
(
µ
3∪
µ
4)
=
(
µ
1
µ
2
µ
3)
∪
(
µ
1
µ
2
µ
4).
1.18 Proposition: If
µ
1 andµ
2 are two fuzzy subsets of a non-empty setT
then(i)
((
µ
1∩
µ
2)
T
T
)
⊆
(
µ
1
T
T
)
∩
(
µ
2
T
T
)
(ii)(
T
(
µ
1∩
µ
2)
T
)
⊆
(
T
µ
1
T
)
∩
(
T
µ
2
T
)
(iii)(
T
T
(
µ
1∩
µ
2)
)
⊆
(
T
T
µ
1)
∩
(
T
T
µ
2)
.Proof:(i) Let
x
∈
T
. Ifx
≠
pqr
for anyp
,
q
,
r
∈
T
then=
=
∩
(
))
(
)
0
)
(
(
µ
1
T
T
µ
2
T
T
x
((
µ
1∩
µ
2)
T
T
)
(
x
).
If
x
=
pqr
for anyp
,
q
,
r
∈
T
then)
(
)
)
((
µ
1∩
µ
2
T
T
x
{(
1 2)(
p
)
T
(
q
)
T
(
r
)
}
pqr
x
∨
∩
∧
∧
=
=
µ
µ
}
1
1
)
(
)
(
{
1∧
2∧
∧
∨
=
=pqr
p
p
x
µ
µ
}
)
(
)
(
{
1p
2p
pqr
x
∨
µ
∧
µ
=
=)}}
(
{
{
1p
pqr x=
∨
µ
≤
⊆
}}
1
1
)
(
{
{
∨
1∧
∧
=
=pqr
p
x
µ
∧
{
x=∨
pqr{
µ
2(
p
)
∧
1
∧
1
}}
)}}
(
)
(
)
(
{
{
1p
T
q
T
r
pqr
x
∨
∧
∧
=
=µ
}}
)
(
)
(
)
(
{
{
2p
T
q
T
r
pqr
x
∨
∧
∧
∧
=
µ
=
(
µ
1
T
T
)
(
x
)
∧
(
µ
2
T
T
)
(
x
)
=
(
(
µ
1
T
T
)
∩
(
µ
2
T
T
))
(
x
)
)
(
)
)
((
µ
1∩
µ
2
T
T
x
≤
(
(
µ
1
T
T
)
∩
(
µ
2
T
T
))
(
x
)
Therefore
((
µ
1∩
µ
2)
T
T
)
⊆
(
(
µ
1
T
T
)
T
.Similarly we can prove that
(ii)
(
T
(
µ
1∩
µ
2)
T
)
⊆
(
T
µ
1
T
)
∩
(
T
µ
2
T
)
and (iii)(
T
T
(
µ
1∩
µ
2)
)
⊆
(
T
T
µ
1)
∩
(
T
T
µ
2)
.RESULTS
2.1 Theorem: Let be a non-empty fuzzy subset of a ternary semigroup . Then is a fuzzy ternary
subsemigroup of if and only if is a fuzzy weakly completely prime ideal in .
Proof: Let be a non-empty fuzzy subset of a ternary semigroup .
Assume that is a fuzzy ternary subsemigroup of and let . Then
i.e.,
Therefore or or
Hence
is a fuzzy weakly completely prime ideal in .
µ
T
1
−
µ
T
µ
T
µ
T
µ
µ
/=
1
−
T
x
,
y
,
z
∈
T
)}
(
),
(
),
(
min{
1
)
(
1
−
µ
x
y
z
=
−
µ
x
µ
y
µ
z
)}
(
1
),
(
1
),
(
1
min{
−
µ
x
−
µ
y
−
µ
z
≥
)}
(
),
(
),
(
max{
1
)
(
1
−
µ
x
y
z
≥
−
µ
x
µ
y
µ
z
)}
(
),
(
),
(
max{
)
(
x
y
z
µ
x
µ
y
µ
z
µ
≥
−
−
)}
(
),
(
),
(
max{
)
(
x
y
z
µ
x
µ
y
µ
z
µ
≤
)
(
)}
(
),
(
),
(
max{
µ
x
µ
y
µ
z
≥
µ
x
y
z
)
(
)
(
x
µ
x
y
z
µ
≥
µ
(
y
)
≥
µ
(
x
y
z
)
µ
(
z
)
≥
µ
(
x
y
z
)
Conversely, assume that is a fuzzy weakly completely prime ideal in . Then we have
or or .
Therefore is a fuzzy ternary subsemigroup of .
2.2 Theorem: Let
be a family of fuzzy weakly completely prime ideals in a ternary semigroup . Then is also a fuzzy weakly completely prime ideal in .
Proof: Let be a family of fuzzy weakly completely prime ideals in a ternary semigroup . Then we have
or or for all 𝑥𝑥,𝑦𝑦,𝑧𝑧 ∈ 𝑇𝑇 , . Then
or
or
Hence is fuzzy weakly ternary completely prime ideal in .
2.3 Theorem: Let be a ternary semigroup and be a non-empty fuzzy subset of . Then the following are equivalent.
(1)
is a fuzzy weakly completely prime ideal in .
(2) For any , (if it is non-empty) is a prime ideal in .
Proof: Let be a fuzzy weakly completely prime ideal in a ternary semigroup . Then we have
or or for all . Let be such that is non-empty. Let
, . Then . Since is a fuzzy weakly completely prime ideal in a ternary
semigroup , so, we have or or for all . Then
or or which implies that or
or . Hence is a prime -ideal in .
Conversely, let us suppose that is a prime ideal in a ternary semigroup . Let
µ
t is non-empty. Thenxyz
∈
µ
t⇒
µ
t(
xyz
)
≥
t
.
Since is a prime ideal in , we have or or . Then oror which implies that or or . Hence is
a fuzzy weakly completely prime ideal in a ternary semigroup .
2.4 Theorem: Let be a non-empty subset of a ternary semigroup and be the characteristic function of .
Then is a left ideal of if and only if is a fuzzy left ideal of .
µ
T
)
(
)
(
x
µ
x
y
z
µ
≥
µ
(
y
)
≥
µ
(
x
y
z
)
µ
(
z
)
≥
µ
(
x
y
z
)
)
(
)}
(
),
(
),
(
max{
µ
x
µ
y
µ
z
≥
µ
x
y
z
)
(
1
)}
(
),
(
),
(
max{
1
−
µ
x
µ
y
µ
z
≤
−
µ
x
y
z
)
(
1
)}
(
1
),
(
1
),
(
1
min{
−
µ
x
−
µ
y
−
µ
z
≤
−
µ
x
y
z
)}
(
),
(
),
(
min{
)
(
/ / / /z
y
x
z
y
x
µ
µ
µ
µ
≥
µ
µ
/=
1
−
T
}
:
{
µ
ii
∈
I
T
I i i ∈µ
T
}
:
{
µ
ii
∈
I
T
)
(
)
(
x
ix
y
z
i
µ
µ
≥
µ
i(
y
)
≥
µ
i(
x
y
z
)
µ
i(
z
)
≥
µ
i(
x
y
z
)
i
∈
I
}
:
)
(
inf{
)
(
x
y
z
ix
y
z
i
I
Ii
i
=
∈
∈
µ
µ
∴
(
x
y
z
)
inf{
i(
x
)
:
i
I
}
I i
i
≤
∈
∈
µ
µ
}
:
)
(
inf{
)
(
x
y
z
iy
i
I
I i
i
≤
∈
∈
µ
µ
}
:
)
(
inf{
)
(
x
y
z
iz
i
I
I i
i
≤
∈
∈
µ
µ
I i i ∈µ
T
T
µ
T
µ
T
]
1
,
0
[
∈
t
µ
tT
µ
T
µ
(
x
)
≥
µ
(
x
y
z
)
)
(
)
(
y
µ
x
y
z
µ
≥
µ
(
z
)
≥
µ
(
x
y
z
)
x
,
y
,
z
∈
T
t
∈
[
0
,
1
]
µ
tT
z
y
x
,
,
∈
xyz
∈
µ
iµ
(
x
y
z
)
≥
t
µ
T
µ
(
x
)
≥
µ
(
x
y
z
)
µ
(
y
)
≥
µ
(
x
y
z
)
µ
(
z
)
≥
µ
(
x
y
z
)
x
,
y
,
z
∈
T
t
x
)
≥
(
µ
µ
(
y
)
≥
t
µ
(
z
)
≥
t
x
∈
µ
ty
∈
µ
tz
∈
µ
tµ
tT
t
µ
T
t
µ
T
x
∈
µ
ty
∈
µ
tz
∈
µ
tµ
(
x
)
≥
t
t
y
)
≥
(
µ
µ
(
z
)
≥
t
µ
(
x
)
≥
µ
(
x
y
z
)
µ
(
y
)
≥
µ
(
x
y
z
)
µ
(
z
)
≥
µ
(
x
y
z
)
µ
T
A
T
C
AA
Proof: Let be non-empty subset of a ternary semigroup and be a characteristic function of .
We assume that is left ideal of then .
We have to prove that is fuzzy left ideal of . i.e.,
Let for all
Consider
Therefore is a fuzzy left ideal of .
Conversely, suppose is a fuzzy left ideal in . Then for all .
To prove that is a left ideal in . i.e.,
Let then
C
T(
x
)
=
1
,C
T(
y
)
=
1
,C
T(
z
)
=
1
Consider
Therefore is a left ideal in a ternary semigroup .
2.5 Theorem: Let be a non-empty subset of a ternary semigroup and be the characteristic function of .
Then is a right ideal (lateral ideal, ideal) in if and only if is a fuzzy right ideal (fuzzy lateral ideal, fuzzy ideal) in .
Proof: Similar to the proof of Theorem 3.3.4.
2.6 Theorem: Let be a ternary semigroup and be a non-empty subset of . Then the following are equivalent (1) is prime ideal in a ternary semigroup
(2) The characteristic function of is a fuzzy weakly completely prime ideal in .
Proof: Let be a prime ideal in a ternary semigroup and be the characteristic function of Since , so,
is non-empty. Let . Suppose Then . Since is a prime ideal in ,
A
T
C
AA
A
T
TTA
⊆
A
A
C
T
C
A(
x
y
z
)
≥
C
A(
z
)
xyz
a
=
x
,
y
,
z
∈
T
.
)
(
)
(
x
y
z
C
x
y
z
C
A≥
TTA)
)(
(
C
T
C
T
C
Ax
y
z
=
))
(
)
(
)
(
(
C
Tp
C
Tq
C
Ar
r q p z y
x
∨
∧
∧
=
=
)
)
(
)
(
)
(
(
C
Tx
∧
C
Ty
∧
C
Az
=
)
)
(
1
1
(
∧
∧
C
Az
=
)
(
z
C
A=
)
(
)
(
x
y
z
C
z
C
A≥
A∴
A
C
T
A
C
T
C
A(
x
y
z
)
≥
C
A(
z
)
x
,
y
,
z
∈
T
A
T
TTA
⊆
A
.
T
z
y
x
,
,
∈
)
(
)
(
x
y
z
C
z
C
A≥
A)
)
(
1
1
(
∧
∧
C
Az
=
)
)
(
)
(
)
(
(
C
Tx
∧
C
Ty
∧
C
Az
=
)
)(
(
C
T
C
T
C
Ax
y
z
=
=
)
(
xyz
C
A(
C
T
C
T
C
A)(
x
y
z
)
⊇
⇒
C
AC
T
C
T
C
A.
A
ATT
C
C
A⊇
TTA⇒
⊆
⇒
A
T
A
T
C
AA
A
T
C
AT
T
A
T
A
T
A
C
A
T
A
T
C
AA
A
≠
φ
A
or or which implies that or or . Hence
or or . Suppose . Then . Since be a
prime ideal in , or or which implies that or
µ
A(
y
)
=
0
or .Hence or . Consequently is a fuzzy weakly completely prime ideal
in
T
.Conversely, let the characteristic function of is a fuzzy weakly completely prime ideal in . Then is a fuzzy ideal in . By the theorem 2.1.4, is an ideal in . Let be such that . Then
. Let if possible and and . Then which implies
and and . This contradicts our assumption that is
a fuzzy weakly completely prime ideal of . Hence is prime ideal in .
REFERENCES
1. Dixit, V.N. and Dewan,S., “A note on quasi ideals and bi - ideals in ternary semigroups”, Internet, J.Math and Math.sci. vol.18, No. 3 (1995) 501-508.
2. Dutta,T.K., Kar,S. and Maity,B.K., “On ideals in regular ternary semigroups”, Discuss. Math.Gen. Algebra Appl. 28 (2008), 147-159.
3. Kar, S. and Paltu Sarka., “Fuzzy quasi ideals and fuzzy bi- ideals of ternary semigroups”, Annals of Fuzzy Mathematics and Informatics. Volume 4, No. 2, (October 2012), pp. 407-423.
4. Kuroki. N., “Fuzzy bi-ideals in Semigroups”, Comment. Math. Univ. St. Paul., 28: 1979, 17-21. 5. Lehmer, D.H., “A ternary analogue of abelian groups” , Amer. J. Math. 54 (1932) 329-338.
6. Samit Kumar Majumder, “Fuzzy completely prime ideals in gamma semigroups”, SDU Jornal of Scince (E- Journal), 2010, 5(2): 230-238.
7. Sison,F.M., “Ideal theory in ternary semigroups”, Maths. Japan, 10 (1965), 6384. 8. Zadeh, L.A., “Fuzzy sets”, Inform. Control. 8 (1965) 338-353.
Source of support: Nil, Conflict of interest: None Declared
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