k-Fuzzy Ideals of Ternary Semirings
Sathinee Malee and Ronnason Chinram
Abstract—The notion of k-fuzzy ideals of semirings was intro-
duced by Kim and Park in 1996. In 2003, Dutta and Kar introduced
a notion of ternary semirings. This structure is a generalization of
ternary rings and semirings. The main purpose of this paper is to
introduce and study k-fuzzy ideals in ternary semirings analogous to
k-fuzzy ideals in semirings considered by Kim and Park.
Keywords—k-ideals, k-fuzzy ideals, fuzzy k-ideals, ternary
The notion of ternary algebraic system was introduced by
Lehmer  in 1932. He investigated certain ternary algebraic
systems called triplexes. In 1971, Lister  characterized
additive semigroups of rings which are closed under the triple
ring product and he called this algebraic system a ternary ring.
Dutta and Kar  introduced a notion of ternary semirings
which is a generalization of ternary rings and semirings, and
they studied some properties of ternary semirings (, , ,
,  and , etc.).
The theory of fuzzy sets was first studied by Zadeh 
in 1965. Many papers on fuzzy sets appeared showing the
importance of the concept and its applications to logic, set
theory, group theory, ring theory, real analysis, topology,
measure theory, etc. Fuzzy ideals of semirings were studied
by some authors (, , , ,  and , etc.). The
notion of k-fuzzy ideals of semirings was introduced by
Kim and Park . Recently, Kavikumar, Khamis and Jun
studied fuzzy ideals, fuzzy bi-ideals and fuzzy quasi-ideals
in ternary semirings in  and . The fuzzy ideal of
ternary semirings is a good tool for us to study the fuzzy
algebraic structure. The main purpose of this paper is to study
k-fuzzy ideals in ternary semirings analogous to k-fuzzy ideals
in semirings considered by Kim and Park.
In this section, we refer to some elementary aspects of the
theory of semirings and ternary semirings and fuzzy algebraic
systems that are necessary for this paper.
Definition 2.1. A nonempty set S together with two associative
binary operations called addition and multiplication (denoted
Sathinee Malee is with the Department of Mathematics, Faculty of Science,
Prince of Songkla University, Hat Yai, Songkhla, 90110 THAILAND, e-
Ronnason Chinram is with the Department of Mathematics, Faculty of
Science, Prince of Songkla University, Hat Yai, Songkhla, 90110 THAILAND
and Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok
10400, THAILAND, e-mail:email@example.com.
Most of the work in this paper is a part of the Master thesis written by
Miss Satinee Malee under the supervision of Assistant Professor Dr.Ronnason
This research is supported by the Centre of Excellence in Mathematics, the
Commission on Higher Education, Thailand.
by + and ·, respectively) is called a semiring if (S,+) is a
commutative semigroup, (S,·) is a semigroup and multiplica-
tive distributes over addition both from the left and from the
right, i.e., a(b + c) = ab + ac and (a + b)c = ac + bc for all
a,b,c ∈ S.
Definition 2.2. A nonempty set S together with a binary
operation and a ternary operation called addition + and ternary
multiplication, respectively, is said to be a ternary semiring if
(S,+) is a commutative semigroup satisfying the following
conditions: for all a,b,c,d,e ∈ S,
(i) (abc)de = a(bcd)e = ab(cde),
(ii) (a + b)cd = acd + bcd,
(iii) a(b + c)d = abd + acd and
(iv) ab(c + d) = abc + abd.
We can see that any semiring can be reduced to a ternary
semiring. However, a ternary semiring does not necessarily
reduce to a semiring by this example. We consider Z−
the set of all non-positive integers under usual addition and
multiplication, we see that Z−
is closed under the triple multiplication but is not closed under
the binary multiplication. Moreover, Z−
but is not a semiring under usual addition and multiplication.
0is an additive semigroup which
0is a ternary semiring
Definition 2.3. Let S be a ternary semiring. If there exists an
element 0 ∈ S such that 0+x = x = x+0 and 0xy = x0y =
xy0 = 0 for all x,y ∈ S, then 0 is called the zero element or
simply the zero of the ternary semiring S. In this case we say
that S is a ternary semiring with zero.
Definition 2.4. An additive subsemigroup T of S is called a
ternary subsemiring of S if t1t2t3∈ T for all t1,t2,t3∈ T.
Definition 2.5. An additive subsemigroup I of S is called a
left [resp. right, lateral] ideal of S if s1s2i ∈ I [resp. is1s2∈
I,s1is2∈ I] for all s1,s2∈ S and i ∈ I. If I is a left, right
and lateral ideal of S, then I is called an ideal of S.
It is obvious that every ideal of a ternary semiring with zero
contains a zero element.
Definition 2.6. Let S and R be ternary semirings. A mapping
ϕ : S → R is said to be a homomorphism if ϕ(x + y) =
ϕ(x)+ϕ(y) and ϕ(xyz) = ϕ(x)ϕ(y)ϕ(z) for all x,y,z ∈ S.
Let ϕ : S → R be an onto homomorphism of ternary
semirings. Note that if I is an ideal of S, then ϕ(I) is an
ideal of R. If S and R be ternary semirings with zero 0, then
ϕ(0) = 0.
Definition 2.7. Let S be a non-empty set. A mapping f : S →
[0,1] is called a fuzzy subset of S.
Definition 2.8. Let A be a subset of a non-empty set S. The
characteristic function χAof A is a fuzzy subset of S defined
International Journal of Computational and Mathematical Sciences 4:4 2010