On intuitionistic fuzzy idempotent,prime, strongly irreducible and t-pure ideals of semirings

Abstract

We define intuitionistic fuzzy idempotent, prime, strongly irreducible and t-pure ideals on a semiring, and investigate some properties of such ideals. We show that all intuitionistic fuzzy ideals are idempotent in a fully idempotent semirings, and right weakly regular semirings. We establish that an intuitionistic fuzzy set of a semiring is an intuitionistic fuzzy ideal if and only if (α, β)-cut is an ideal. We prove that if all intuitionistic fuzzy ideals of a semiring are t-pure, then an intuitionistic fuzzy ideal is prime if and only if it is strongly irreducible. We also establish that for a fully idempotent semirings, an intuitionistic fuzzy ideal is prime if and only if it is strongly irreducible.

Keywords

Intuitionistic fuzzy idempotent ideals intuitionistic fuzzy t-pure ideals intuitionistic fuzzy prime ideals intuitionistic fuzzy strongly irreducible ideals fully idempotent semirings weakly regular semirings

1 Introduction

There are algebraic structures which generalizes an associative ring (R,+,.), some of them, in particular, the semirings were found useful for solving problems in different areas of mathematical sciences. Since the structure of a semiring provides an algebraic framework for modeling, and studying the key factors in these areas. This structure play an important role in mathematics with wide ranging applications in many disciplines such as graph theory, fuzzy computation, automata and formal language theory, analysis of computer program, coding theory, and so on. The notion of semiring was first introduced by Vandiver [29] in the year 1935. In the year 1950, Brown and Mc coy [10] introduced the weakly regular semiring. Ahsan and Khan [1] introduced the fully idempotent semiring, and the fully idempotent semirings were discussed by Ahsan [5]. Later Ramamurthy [24], Camillo and Xiao [11] investigate these rings. These studies motivate us to investigates further on these types of structures on fuzzy setting.

Fuzzy set theory which has the tolerant of imprecision and uncertainty was introduced by Zadeh [30]. Its algebraic structures play a prominent role in mathematics with wide applications in many other branches such as computer sciences, control engineering, information science, coding theory, tropical space, optimization and decision making, neural networking, coding theory etc. Later, in the year 1983 [7, 8], Atanassov introduced the intuitionistic fuzzy set. The knowledge and semantic representation of intuitionistic fuzzy set becomes more tolerance of imprecision, uncertainty, partial truth, and approximation over fuzzy sets. While fuzzy set deals with the degree of membership of an element in a given set, intuitionistic fuzzy sets give both degree of membership and non-membership with an addition that sum of degrees of membership and nonmembership should not exceeds 1. Fuzzy prime ideals, and fuzzy k-ideals of semiring were discussed in the year 1994 [14, 15]. Kuroki [18] was given some properties of fuzzy ideals, and fuzzy k-ideals of semiring were investigated by Ghosh [16]. Wang-Jin Lice introduced, and developed some basic properties concerning the notions of fuzzy sub-ring as well as fuzzy ideals of a ring. The properties of fuzzy ideals and fuzzy prime ideals of a semiring have been studied by many researchers: Mukherjee and Sen [20], Swamy and Swamy [26], Malik and Mordeson [19]. Ahsan et al. [2 –4] initiated to study some of the interesting result of fuzzy semiring and semi-modules. In the year 1989, Biswas [9] introduced the intuitionistic fuzzy subgroups, and intuitionistic fuzzy k-ideals of a semiring were studied by Dheena and Mohanaaj [13]. Rahman and Saikia [21–23 , 25] was introduced some special types of intuitionistic fuzzy submodules, and established some interesting properties therein. The intuitionistic fuzzy bi-ideals of a semi group are introduced in [17]. Akram and Dudek [6] has investigated the notion of intuitionistic fuzzy left k-ideals in semiring, and produced some useful results. We note that fuzzy hyperfilters of ordered semihypergroups [27], lattice valued fuzzy lattice [28] and rough soft hemirings [31] are few remarkable applications of fuzzy set in algebraic structure.

In this paper, we apply the concept of intuitionistic fuzzy set to semirings. We introduced the notion of intuitionistic fuzzy prime ideal, intuitionistic fuzzy irreducible ideal, intuitionistic fuzzy t-pure ideal, idempotent intuitionistic fuzzy ideals in a fully idempotent semiring and right weakly semirings, and investigate some properties of such ideals.

2 Basic definitions and notations

A semiring is a non-empty set R together with two binary operations, namely, addition (+) and multiplication (·) such that (R, +) is a commutative semigroup, and (R, ·) is a semigroup, where both algebraic structures are connected by the two distributive laws: a (b + c) = ab + ac and (a + b) c = ac + bc. A subset S of R is called subsemiring if (S, + _s, · _s) is a semiring. A non-empty subset I of a semiring R is said to be a left (resp. right) ideal of R if (I, +) is a subsemigroup of (R, +) and ra ∈ I (resp. ar ∈ I) for all a ∈ I, r ∈ R. A subset I of R is said to be an ideal if it is both left and right ideal of R. A proper ideal of a semiring R is said to be prime ideal of R if AB ⊆ P for any two ideals A and B of R implies that either A ⊆ P or B ⊆ P. An ideal of a semiring R is said to be irreducible ideal of R if AB = I for any two ideals A and B of R implies that either A = I or B = I. An ideal A of a semiring R is said to be idempotent ideal of R if A² = A. If all ideals of R is idempotent, then R is called fully idempotent. A semiring R is a regular if for each x ∈ R there exists a ∈ R such that x = xax. R is called weakly regular if x ∈ (xR) ² for each x ∈ R.

A fuzzy set μ of a set X is a mapping from X to [0, 1]. For each fuzzy set μ of X, and every α ∈ [0, 1], we define two sets U (μ, α) ={ x ∈ X : μ (x) ≥ α }, L (μ, α) ={ x ∈ X : μ (x) ≤ α }, which are called an upper level cut and lower level cut of μ, respectively.

A fuzzy subset λ of a semiring R is called a fuzzy sub-semiring of R if for every x, y ∈ R the following conditions are satisfied:

λ (x + y) ≥ λ (x) ∧ λ (y)

λ (xy) ≥ λ (x) ∧ λ (y)

where ∧ represents the min operator.

A fuzzy set λ of a semiring R is called a fuzzy left (resp. right) ideal of R if for every x, y ∈ R the following conditions are satisfied:

λ (x + y) ≥ λ (x) ∧ λ (y)

λ (xy) ≥ λ (y) (resp. λ (yx) ≥ λ (y)).

A fuzzy subset λ of a semiring R is called fuzzy ideal of R if it is both left and right fuzzy ideal of R.

Let us recall the definition of intuitionistic fuzzy set. An intuitionistic fuzzy set on a set X was introduced by Atanassov in the year 1983 as an object of the form = { (x, μ_A (x), ν_A (x)) : x ∈ X }, where μ_A and ν_A are fuzzy sets on X, and denote the degree of membership (namely μ_A (x)) and degree of non-membership (namely ν_A (x)) for each x ∈ X to the set A, respectively. Additionally, 0 ≤ μ_A (x) + ν_A (x) ≤1 for all x ∈ X. We shall use the notation A = (μ_A, ν_A) for the above expression.

Let R be a semiring. An intuitionistic fuzzy subset A of R is said to an intuitionistic fuzzy subsemiring (IFSSR) of R if it satisfies the following conditions:

μ_A (x+ y) ≥ { μ_A (x) ∧ μ_A (y) }

μ_A (xy)≥ { μ_A (x) ∧ μ_A (y) }

ν_A (x+ y) ≤ { ν_A (x) ∨ ν_A (y) }

ν_A (xy)≤ { ν_A (x) ∨ ν_A (y) }

where ∨ represent the max operator.

An intuitionistic fuzzy set A = (μ_A, ν_A) of a semiring R is called an intuitionistic fuzzy left (resp. right) ideal of R if A satisfies the following conditions:

μ_A (x+ y) ≥ { μ_A (x) ∧ μ_A (y) }

ν_A (x+ y) ≤ { ν_A (x) ∨ ν_A (y) }

μ_A (rx) ≥ μ_A (x) and ν_A (rx) ≤ ν_A (x) (resp. μ_A (xr) ≥ μ_A (x) and ν_A (xr) ≤ ν_A (x)).

Definition 2.1. [23] Let A = (μ_A, ν_A) and B = (μ_B, ν_B) be two intuitionistic fuzzy sets in a semiring R. Then the sum A + B of A and B is defined to be the intuitionistic fuzzy set $A + B = (μ_{A} \hat{+} μ_{B}, ν_{A} \tilde{+} ν_{B})$ , where $μ_{A} \hat{+} μ_{B}, ν_{A} \tilde{+} ν_{B}$ are given by $\begin{matrix} μ_{A} \hat{+} μ_{B} (x) = μ_{A + B} (x) \\ = {\begin{matrix} ⋁_{x = y + z} {μ_{A} (y) \land μ_{B} (z)}, & if x = y + z \\ 0, & otherwise \end{matrix} \\ ν_{A} \tilde{+} ν_{B} (x) = ν_{A + B} (x) \\ = {\begin{matrix} ⋀_{x = y + z} {ν_{A} (y) \lor ν_{B} (z)}, & if x = y + z \\ 1, & otherwise . \end{matrix} \end{matrix}$

Definition 2.2. [23] Let A = (μ_A, ν_A) and B = (μ_B, ν_B) be two intuitionistic fuzzy sets in a semiring R. Then the intrinsic product of A and B is defined to be the intuitionistic fuzzy set A ⊛ B = (μ_A⊛B, ν_A⊗B) = (μ_A ⊛ μ_B, ν_A ⊗ ν_B), where μ_A ⊛ μ_B is $⋁_{x = \sum_{i = 1}^{n} y_{i} z_{i}} [⋀_{i = 1}^{n} {μ_{A} (y_{i}) \land μ_{B} (z_{i})}]$ when $x = \sum_{i = 1}^{n} y_{i} z_{i}$ and 0 otherwise, and ν_A ⊗ ν_B is $⋀_{x = \sum_{i = 1}^{n} y_{i} z_{i}} [⋁_{i = 1}^{n} {ν_{A} (y_{i}) \lor ν_{B} (z_{i})}]$ when $x = \sum_{i = 1}^{n} y_{i} z_{i}$ and 1 otherwise.

Definition 2.3. An intuitionistic fuzzy ideal A = (μ_A, ν_A) of a semiring R is called an intuitionistic fuzzy prime ideal of R if for any two intuitionistic fuzzy ideals B = (μ_B, ν_B) and C = (μ_C, ν_C) of R such that B ⊛ C ⊆ A implies that B ⊆ A or C ⊆ A, i.e., μ_(B⊛C) ⊆ μ_A and ν_B⊗C ⊇ ν_A implies that μ_B ⊆ μ_A and ν_B ⊇ ν_A or μ_C ⊆ μ_A and ν_C ⊇ ν_A.

Definition 2.4. An intuitionistic fuzzy ideal A = (μ_A, ν_A) of a semiring R is called an intuitionistic fuzzy irreducible ideal of R if for any two intuitionistic fuzzy ideals B = (μ_B, ν_B) andC = (μ_C, ν_C) of R such that B ∩ C = A implies that B = A or C = A, i.e., μ_(B∩C) = μ_A and ν_B∩C = ν_A. implies that μ_B = μ_A and ν_B = ν_A or μ_C = μ_A and ν_C = ν_A.

Definition 2.5. An intuitionistic fuzzy ideal A = (μ_A, ν_A) of a semiring R is called an intuitionistic fuzzy strongly irreducible ideal of R if for any two intuitionistic fuzzy ideals B = (μ_B, ν_B) and C = (μ_C, ν_C) of R such that B ∩ C ⊆ A implies that B ⊆ A or C ⊆ A, i.e., μ_B∩C ⊆ μ_A and ν_B∩C ⊇ ν_A gives μ_B ⊆ μ_A and ν_B ⊇ ν_A or μ_C ⊆ μ_A and ν_C ⊇ ν_A.

A fuzzy ideal λ of a semiring R is called a t-pure fuzzy ideal of R if μ ∧ λ = μ ⨀ λ for each fuzzy ideal of μ of R.

Definition 2.6. An intuitionistic fuzzy ideal A = (μ_A, ν_A) of a semiring R is called a t-pure if A ∩ B = A ⊛ B (i. e., μ_A ∧ μ_B = μ_A ⊛ μ_B and ν_A ∨ ν_B = ν_A ⊗ ν_B) for each intuitionistic fuzzy ideal B = (μ_B, ν_B) of R.

3 Main results

Theorem 3.1. For a right weakly regular semiring R, all the intuitionistic fuzzy right ideals of R are idempotent.

Proof. Let A = (μ_A, ν_A) be an intuitionistic fuzzy right ideal of R and let x ∈ R. Then from Definitions 2.1 and 2.2 $\begin{matrix} μ_{A ⊛ A} (x) & = & ⋁_{x = \sum_{i = 1}^{n} y_{i} z_{i}} [⋀_{i = 1}^{n} {μ_{A} (y_{i}) \land μ_{A} (z_{i})}] \\ \leq & ⋁_{x = \sum_{i = 1}^{n} y_{i} z_{i}} {μ_{A} (y_{i}) \land μ_{A} (z_{i})} \\ \leq & ⋁_{x = \sum_{i = 1}^{n} y_{i} z_{i}} {μ_{A} (y_{i})} \\ \leq & ⋁_{x = \sum_{i = 1}^{n} y_{i} z_{i}} {μ_{A} (y_{i} z_{i})} \\ \leq & ⋁_{x = \sum_{i = 1}^{n} y_{i} z_{i}} {μ_{A} (\sum_{i = 1}^{n} y_{i} z_{i})} \\ = & μ_{A} (\sum_{i = 1}^{n} y_{i} z_{i}) \\ = & μ_{A} (x) . \end{matrix}$

This implies that μ_A⊛A (x) ≤ μ_A (x). Also, $\begin{matrix} ν_{A \otimes A} (x) \\ = ⋀_{x = \sum_{i = 1}^{n} y_{i} z_{i}} [⋁_{i = 1}^{n} {ν_{A} (y_{i}) \lor ν_{A} (z_{i})}] \\ \geq ⋀_{x = \sum_{i = 1}^{n} y_{i} z_{i}} {ν_{A} (y_{i}) \lor ν_{A} (z_{i})} \\ \geq ⋀_{x = \sum_{i = 1}^{n} y_{i} z_{i}} {ν_{A} (y_{i})} \\ \geq ⋀_{x = \sum_{i = 1}^{n} y_{i} z_{i}} {ν_{A} (y_{i} z_{i})} \\ \geq ⋀_{x = \sum_{i = 1}^{n} y_{i} z_{i}} {ν_{A} (\sum_{i = 1}^{n} y_{i} z_{i})} \\ = ν_{A} (\sum_{i = 1}^{n} y_{i} z_{i}) \\ = ν_{A} (x) . \end{matrix}$

Therefore, we have ν_A⊛A (x) ≥ ν_A (x). If $x \neq \sum_{i = 1}^{n} y_{i} z_{i}$ , then μ_A⊛A (x) =0 ≤ μ_A (x), and ν_A⊗A (x) =1 ≥ ν_A (x).

Since R is a right weakly regular, it follows that there exists r_i, s_i ∈ R such that $x = \sum_{i = 1}^{n} {xr}_{i} {xs}_{i}$ . Now, μ_A (x) = μ_A (x) ∧ μ_A (x) ≤ μ_A (xr_i) ∧ μ_A (xs_i) for all i = 1, 2, ..., n.

It follows from the above that $\begin{matrix} μ_{A} (x) & \leq & ⋀_{i = 1}^{n} {μ_{A} ({xr}_{i}) \land μ_{A} ({xs}_{i})} \\ \leq & ⋁_{\sum_{i = 1}^{n} {xr}_{i} {xs}_{i}} {⋀_{i = 1}^{n} (μ_{A} ({xr}_{i}) \land μ_{A} ({xs}_{i}))} \\ = & μ_{A ⊛ A} (x) . \end{matrix}$

Thus, μ_A (x) ≤ μ_A⊛A (x). Moreover, ν_A (x) = ν_A (x) ∨ ν_A (x) ≥ ν_A (xr_i) ∨ ν_A (xs_i) for all i = 1, 2, ..., n. It follows that $\begin{matrix} ν_{A} (x) & \geq & ⋁_{i = 1}^{n} {ν_{A} ({xr}_{i}) \lor ν_{A} ({xs}_{i})} \\ \geq & ⋀_{\sum_{i = 1}^{n} {xr}_{i} {xs}_{i}} {⋁_{i = 1}^{n} (ν_{A} ({xr}_{i}) \lor ν_{A} ({xs}_{i}))} \\ = & ν_{A \otimes A} (x) . \end{matrix}$

This gives ν_A (x) ≥ ν_A⊗A (x). Therefore, we conclude that μ_A (x) = μ_A⊛A (x), and ν_A (x) = ν_A⊗A (x) whence A ⊛ A = (μ_A⊛A, ν_A⊗A) = (μ_A, ν_A) = A, and hence A is idempotent intuitionistic fuzzy ideal. □

Example 3.1. Consider the semiring R ={ 0, 1, 2 } defined by the followings tables.

+	0	1	2
0	0	1	2
1	1	1	2
2	2	2	2

.	1	2
0	0	0
1	1	1
2	1	2

The ideals of R are {0},{0, 1},{0, 1, 2}. Clearly, all ideals of R are idempotent, and hence R is a fully idempotent semiring. Let R₁ ={ 0, x }, and consider A = (μ_A, ν_A) is a intuitionistic fuzzy set defined as $μ_{A} (x) = {\begin{matrix} 0.2, & if x \in R_{1} \\ 0.1, & if x \notin R_{1} \end{matrix}$ and $ν_{A} (x) = {\begin{matrix} 0.3, & if x \in R_{1} \\ 0.5, & if x \notin R_{1} . \end{matrix}$

Then we can easily verify that A is an idempotent intuitionistic fuzzy ideal. This example motivates us to state the following theorem.

Theorem 3.2. If a semiring R is fully idempotent, then all its intuitionistic fuzzy ideals of R are idempotent.

Proof. Let A = (μ_A, ν_A) be an intuitionistic fuzzy ideal of R and let x ∈ R. Since each ideal of R is idempotent, it follows that <x> = < x > ², < x > denotes the principal ideal of R generated by x for each x ∈ R. Thus, there exist $r_{i}, r_{i}^{'}, s_{i}, s_{i}^{'} \in R$ such that $x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'} .$ Now, μ_A (x) = μ_A (x) ∧ μ_A (x) $\leq μ_{A} (r_{i} {xr}_{i}^{'}) \land μ_{A} (s_{i} {xs}_{i}^{'})$ for all i = 1, 2, ..., n. It follows that $\begin{matrix} μ_{A} (x) \leq ⋀_{i = 1}^{n} {μ_{A} (r_{i} {xr}_{i}^{'}) \land μ_{A} (s_{i} {xs}_{i}^{'})} \\ \leq ⋁_{\sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {⋀_{i = 1}^{n} (μ_{A} (r_{i} {xr}_{i}^{'}) \land μ_{A} (s_{i} {xs}_{i}^{'}))} \\ = μ_{A ⊛ A} (x) . \end{matrix}$

Thus, μ_A (x) ≤ μ_A⊛A (x). Also, ν_A (x) = ν_A (x) ∨ ν_A (x) $\geq ν_{A} (r_{i} {xr}_{i}^{'}) \lor ν_{A} (s_{i} {xs}_{i}^{'})$ for all i = 1, 2, ..., n. This gives $\begin{matrix} ν_{A} (x) \geq ⋁_{i = 1}^{n} {ν_{A} (r_{i} {xr}_{i}^{'}) \lor ν_{A} (s_{i} {xs}_{i}^{'})} \\ \geq ⋀_{\sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {⋁_{i = 1}^{n} (ν_{A} (r_{i} {xr}_{i}^{'}) \lor ν_{A} (s_{i} {xs}_{i}^{'}))} \\ = ν_{A \otimes A} (x) . \end{matrix}$

This implies that ν_A (x) ≥ ν_A⊗A (x). Also, if $x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}$ , then, from the Definitions 2.1 and 2.2, we have $\begin{matrix} μ_{A ⊛ A} (x) \\ = ⋁_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} [⋀_{i = 1}^{n} {μ_{A} (r_{i} {xr}_{i}^{'}) \land μ_{A} (s_{i} {xs}_{i}^{'})}] \\ \leq ⋁_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {μ_{A} (r_{i} {xr}_{i}^{'}) \land μ_{A} (s_{i} {xs}_{i}^{'})} \\ \leq ⋁_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {μ_{A} (r_{i} {xr}_{i}^{'})} \\ \leq ⋁_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {μ_{A} (r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'})} \\ \leq ⋁_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {μ_{A} (\sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'})} \\ = μ_{A} (\sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}) \\ = μ_{A} (x) . \end{matrix}$

Therefore, we obtain μ_A⊛A (x) ≤ μ_A (x). And $\begin{matrix} ν_{A \otimes A} (x) \\ = ⋀_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} [⋁_{i = 1}^{n} {ν_{A} (r_{i} {xr}_{i}^{'}) \lor ν_{A} (s_{i} {xs}_{i}^{'})}] \\ \geq ⋀_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {ν_{A} (r_{i} {xr}_{i}^{'}) \lor ν_{A} (s_{i} {xs}_{i}^{'})} \\ \geq ⋀_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {ν_{A} (r_{i} {xr}_{i}^{'})} \\ \geq ⋀_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {ν_{A} (r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'})} \\ \geq ⋀_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {ν_{A} (\sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'})} \\ = ν_{A} (\sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}) \\ = ν_{A} (x) . \end{matrix}$

Thus ν_A⊛A (x) ≥ ν_A (x). If $x \neq \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}$ , then μ_A⊛A (x) =0 ≤ μ_A (x), and ν_A⊗A (x) =1 ≥ ν_A (x). Therefore, we have μ_A (x) = μ_A⊛A (x) and ν_A (x) = ν_A⊗A (x) whence A ⊛ A = (μ_A⊛A, ν_A⊗A) = (μ_A, ν_A) = A. Thus, A is an idempotent intuitionistic fuzzy ideal. □

Corollary 3.3. Every intuitionistic fuzzy t-pure ideal of a semiring is idempotent.

Proof. Let A = (μ_A, ν_A) be an intuitionistic fuzzy t-pure ideal and let B = (μ_B, ν_B) be any intuitionistic fuzzy ideals of a semiring R. Then from Definition 2.6, A ⊛ B = A ∩ B. Setting A = B in the previous relation, we get A ⊛ A = A ∩ A = A. Hence, A is idempotent. □

Theorem 3.4. If A = (μ_A, ν_A) and B = (μ_B, ν_B) are two intuitionistic fuzzy ideals of a fully idempotent semiring R, then A ⊛ B = A ∩ B.

Proof. Let A = (μ_A, ν_A) and B = (μ_B, ν_B) be two intuitionistic fuzzy ideals of a fully idempotent semiring R. Since each ideal of a fully idempotent semiring is idempotent, it follows that <x > = < x > ², where <x> denotes the principal ideal of R generated by x for each x ∈ R. Hence, there exist $r_{i}, r_{i}^{'}, s_{i}, s_{i}^{'} \in R$ such that $x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}$ . Now,

$\begin{matrix} (μ_{A} ⊛ χ_{R}) (x) \\ = ⋁_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} [⋀_{i = 1}^{n} {μ_{A} (r_{i} {xr}_{i}^{'}) \land χ_{R} (s_{i} {xs}_{i}^{'})}], \\ where χ_{R} is the chracteristic function \\ = ⋁_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} [⋀_{i = 1}^{n} {μ_{A} (r_{i} {xr}_{i}^{'}) \land 1}] \\ = ⋁_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} [⋀_{i = 1}^{n} μ_{A} (r_{i} {xr}_{i}^{'})] \\ \leq ⋁_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} [⋀_{i = 1}^{n} μ_{A} (r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'})] \\ \leq ⋁_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} [μ_{A} (r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'})] \\ \leq ⋁_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} [μ_{A} \sum_{i = 1}^{n} (r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'})] \\ = μ_{A} (x) . \end{matrix}$

This gives (μ_A ⊛ χ_R) (x) ≤ μ_A (x). Since χ_R (x) ≥ μ_B (x), it follows that (μ_A ⊛ μ_B) (x) ≤ (μ_A ⊛ χ_R) (x) ≤ μ_A (x). Similarly, (μ_A ⊛ μ_B) (x) ≤ (χ_R ⊛ μ_B) (x) ≤ μ_B (x). Hence (μ_A ⊛ μ_B) (x) ≤ (μ_A ∩ μ_B) (x) implies that μ_(A⊛B) (x) ≤ μ_(A∩B) (x).

Let φ_R be denote the null function on R.Then $\begin{matrix} (φ_{R} \otimes ν_{A}) (x) \\ = ⋀_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} [⋁_{i = 1}^{n} {φ_{R} (r_{i} {xr}_{i}^{'}) \lor ν_{A} (s_{i} {xs}_{i}^{'})}] \\ = ⋀_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} [⋁_{i = 1}^{n} {0 \lor ν_{A} (s_{i} {xs}_{i}^{'})}] \\ = ⋀_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {⋁_{i = 1}^{n} ν_{A} (s_{i} {xs}_{i}^{'})} \\ \geq ⋀_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {⋁_{i = 1}^{n} ν_{A} (r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'})} \\ \geq ⋀_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} ν_{A} (r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}) \\ \geq ⋀_{x = \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {ν_{A} (\sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'})} \\ \geq ν_{A} (x) . \end{matrix}$

Therefore, we have (φ_R ⊗ ν_A) (x) ≥ ν_A (x). Since φ_R (x) ≤ ν_B (x), it follows that (ν_A ⊗ ν_B) (x) ≥ (ν_A ⊗ φ_R) (x) ≥ ν_A (x). Similarly, (ν_A ⊗ ν_B) (x) ≥ (φ_R ⊗ ν_B) (x) ≥ ν_B (x). Hence, (ν_A ⊗ ν_B) (x) ≥ (φ_R ⊗ ν_B) (x) ≥ ν_A (x) ∨ ν_B (x) = ν_A∩B (x). Now, μ_A (x) ∧ μ_B (x) $\leq μ_{A} (r_{i} {xr}_{i}^{'}) \land μ_{B} (s_{i} {xs}_{i}^{'})$ for all i = 1, 2, ..., n. This implies that $\begin{matrix} μ_{A \cap B} (x) \leq ⋀_{i = 1}^{n} {μ_{A} (r_{i} {xr}_{i}^{'}) \land μ_{B} (s_{i} {xs}_{i}^{'})} \\ \leq ⋁_{\sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {⋀_{i = 1}^{n} (μ_{A} (r_{i} {xr}_{i}^{'}) \land μ_{B} (s_{i} {xs}_{i}^{'}))} \\ = μ_{A ⊛ B} (x) . \end{matrix}$ Also, we have $ν_{A} (x) \lor ν_{B} (x) \geq ν_{A} (r_{i} {xr}_{i}^{'}) \lor ν_{B} (s_{i} {xs}_{i}^{'})$ for all i = 1, 2, ..., n. This implies that $\begin{matrix} ν_{A \cap B} (x) \geq ⋁_{i = 1}^{n} {ν_{A} (r_{i} {xr}_{i}^{'}) \lor ν_{B} (s_{i} {xs}_{i}^{'})} \\ \geq ⋀_{\sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}} {⋁_{i = 1}^{n} (ν_{A} (r_{i}^{'} {xr}_{i}) \lor ν_{B} (s_{i} {xs}_{i}^{'}))} \\ = ν_{A \otimes B} (x) . \end{matrix}$

Again, if $x \neq \sum_{i = 1}^{n} r_{i} {xr}_{i}^{'} s_{i} {xs}_{i}^{'}, then μ_{A ⊛ B} (x) = 0 \leq μ_{A \cap B} (x), and ν_{A \otimes B} (x) = 1 \geq ν_{A \cap B} (x) .$ Thus, μ_A∩B (x) ≤ μ_A⊛B (x), and ν_A∩B (x) ≥ ν_A⊗B (x). Therefore, μ_A⊛B (x) = μ_A∩B (x) and ν_A⊗B (x) = ν_A∩B (x), and hence A ⊛ B = (μ_A⊛B (x), ν_A⊗B (x)) = (μ_A∩B (x), ν_A∩B (x)) = A ∩ B. □

As a consequences of Theorem 3.4, we have the following result.

Theorem 3.5. All intuitionistic fuzzy ideals of a fully idempotent semiring R are t-pure.

Theorem 3.6. If for all intuitionistic fuzzy ideals A = (μ_A, ν_A) and B = (μ_B, ν_B) of a semiring R satisfy A ⊛ B = A ∩ B, then R is fully idempotent semiring.

Proof. Let I be any ideal of a semiring R. Then A = (χ_I, 1 - χ_I) is an intuitionistic fuzzy ideal of R (χ is the characteristic function). Now, A ⊛ A = A ∩ A implies that A ⊛ A = A. It follows that χ_I ⊛ χ_I = χ_{I
²} = χ_I. Thus, I² = I. Hence, I is fully idempotent ideal of R. □

As a consequence of the above theorem, we have the following result.

Theorem 3.7. If all intuitionistic fuzzy ideals of a semiring R are t-pure, then R is fully idempotent.

Example 3.2. Consider the semiring R ={ 0, x, 1 } defined by the followings tables.

+	0	x	1
0	0	x	1
x	x	x	x
1	1	x	1

·	x	1
0	0	0
x	x	x
1	x	1

The ideals of R are {0},{0, x},{0, x, 1}. Clearly, all ideals of R are idempotent, and hence R is a fully idempotent semiring. Take R₁ ={ 0, x } and consider A = (μ_A, ν_A), B = (μ_B, ν_B) and C = (μ_C, ν_C) are three intuitionistic fuzzy sets defined as: $\begin{matrix} μ_{A} (x) & = & {\begin{matrix} 0.4, & if x \in R_{1} \\ 0.2, & if x \notin R_{1} \end{matrix} \\ ν_{A} (x) & = & {\begin{matrix} 0.5, & if x \in R_{1} \\ 0.7, & if x \notin R_{1} \end{matrix} \\ μ_{B} (x) & = & {\begin{matrix} 0.3, & if x \in R_{1} \\ 0.2, & if x \notin R_{1} \end{matrix} \\ ν_{B} (x) & = & {\begin{matrix} 0.6, & if x \in R_{1} \\ 0.7, & if x \notin R_{1} \end{matrix} \end{matrix}$ and $\begin{matrix} μ_{C} (x) & = & {\begin{matrix} 0.2, & if x \in R_{1} \\ 0.1, & if x \notin R_{1} \end{matrix} \\ ν_{C} (x) & = & {\begin{matrix} 0.3, & if x \in R_{1} \\ 0.6, & if x \notin R_{1} . \end{matrix} \end{matrix}$

Then they form intuitionistic fuzzy ideals of R. It can be observed that it satisfies the distributive law: (A ∩ B) + C = (A + C) ∩ (B + C). This observations inspires us to investigate the following theorem.

Theorem 3.8. If R is a fully idempotent semiring, then the set of all intuitionistic fuzzy ideals of R form a distributive lattice under the sum and intersection of intuisinistic fuzzy ideals.

Proof. Let L_R be the set of all intuitionistic fuzzy ideals of R. Then clearly L_R is a lattice under the sum and intersection of intuitionistic fuzzy ideals. Since R is fully idempotent, it follows that A ⊛ B = A ∩ B for each pair of intuitionistic fuzzy ideals A, B of R. To show that L_R is distributive. Let A = (μ_A, ν_A), B = (μ_B, ν_B) and C = (μ_C, ν_C) be any three intuitionistic fuzzy ideals of R and let x ∈ R. Then $\begin{matrix} μ_{((A \cap B) \hat{+} C)} (x) \\ = ⋁_{x = a + b} {μ_{(A \cap B)} (a) \land μ_{C} (b)} \\ = ⋁_{x = a + b} {(μ_{A} (a) \land μ_{B} (a)) \land μ_{C} (b)} \\ = ⋁_{x = a + b} {(μ_{A} (a) \land μ_{C} (b)) \land (μ_{B} (a) \land μ_{C} (b))} \\ \leq ⋁_{x = a + b} [{⋁_{x = a + b} (μ_{A} (a) \land μ_{C} (b))} \\ \land {⋁_{x = a + b} (μ_{B} (a) \land μ_{C} (b))}] \\ = ⋁_{x = a + b} {μ_{A \hat{+} C} (x) \land μ_{B \hat{+} C} (x)} \\ = (μ_{A \hat{+} C} \cap μ_{B \hat{+} C}) (x) \\ = μ_{(A \hat{+} C) \cap (B \hat{+} C)} (x) . \end{matrix}$

And $\begin{matrix} ν_{((A \cap B) \tilde{+} C)} (x) \\ = ⋀_{x = a + b} {ν_{(A \cap B)} (a) \lor μ_{C} (b)} \\ = ⋀_{x = a + b} {(ν_{A} (a) \lor ν_{B} (a)) \lor ν_{C} (b)} \\ = ⋀_{x = a + b} {(ν_{A} (a) \lor ν_{C} (b)) \lor (ν_{B} (a) \lor ν_{C} (b))} \\ \geq ⋀_{x = a + b} [{⋀_{x = a + b} (ν_{A} (a) \lor ν_{C} (b))} \\ \lor {⋀_{x = a + b} (ν_{B} (a) \lor ν_{C} (b))}] \\ = ⋀_{x = a + b} {ν_{A \tilde{+} C} (x) \lor ν_{B \tilde{+} C} (x)} \\ = (ν_{A \tilde{+} C} \cap ν_{B \tilde{+} C}) (x) \\ = ν_{(A \tilde{+} C) \cap (B \tilde{+} C)} (x) . \end{matrix}$

Again, if x ≠ a + b, then $μ_{((A \cap B) \hat{+} C)} (x) = 0 \leq μ_{(A \hat{+} C) \cap (B \hat{+} C)} (x),$ and $ν_{((A \cap B) \tilde{+} C)} (x) = 1 \geq ν_{(A \tilde{+} C) \cap (B \tilde{+} C)} (x)$ . Thus, $μ_{((A \cap B) \hat{+} C)} (x) \leq μ_{(A \hat{+} C) \cap (B \hat{+} C)} (x)$ , $ν_{((A \cap B) \tilde{+} C)} (x) \geq ν_{(A \tilde{+} C) \cap (B \tilde{+} C)} (x) .$ Again, since R is a fully idempotent, it follows that $\begin{matrix} μ_{(A \hat{+} C) \cap (B \hat{+} C)} (x) \\ = μ_{(A \hat{+} C) ⊛ (B \hat{+} C)} (x) \\ = ⋁_{x = \sum_{i = 1}^{n} x_{i} y_{i}} {⋀_{i = 1}^{n} (μ_{A \hat{+} C} (x_{i}) \land μ_{B \hat{+} C} (y_{i}))} \\ = ⋁_{x = \sum_{i = 1}^{n} x_{i} y_{i}} [⋀_{i = 1}^{n} {⋁_{x_{i} = x_{i}^{'} + y_{i}^{'}} (μ_{A} (x_{i}^{'}) \land μ_{C} (y_{i}^{'}))} \\ \land {⋁_{y_{i} = x_{i}^{″} + y_{i}^{″}} (μ_{B} (x_{i}^{″}) \land μ_{C} (y_{i}^{″}))}] \\ = ⋁_{x = \sum_{i = 1}^{n} x_{i} y_{i}} [⋀_{i = 1}^{n} {⋁_{x_{i} = x_{i}^{'} + y_{i}^{'}, y_{i} = x_{i}^{″} + y_{i}^{″}} (μ_{A} (x_{i}^{'}) \\ \land μ_{C} (y_{i}^{'})) \land (μ_{B} (x_{i}^{″}) \land μ_{C} (y_{i}^{″}))}] \\ \leq ⋁_{x = \sum_{i = 1}^{n} x_{i} y_{i}} [⋀_{i = 1}^{n} {⋁_{x_{i} = x_{i}^{'} + y_{i}^{'}, y_{i} = x_{i}^{″} + y_{i}^{″}} (μ_{A} (x_{i}^{'} x_{i}^{″}) \\ \land μ_{C} (y_{i}^{'} x_{i}^{″}) \land μ_{C} (y_{i}^{'} y_{i}^{″})) \land (μ_{B} (x_{i}^{'} x_{i}^{″}) \\ \land μ_{C} (y_{i}^{″} x_{i}^{'}))}] \\ = ⋁_{x = \sum_{i = 1}^{n} x_{i} y_{i}} [⋀_{i = 1}^{n} {⋁_{x_{i} = x_{i}^{'} + y_{i}^{'}, y_{i} = x_{i}^{″} + y_{i}^{″}} \\ (μ_{A} (x_{i}^{'} x_{i}^{″}) \land μ_{B} (x_{i}^{'} x_{i}^{″})) \land (μ_{C} (y_{i}^{'} x_{i}^{″}) \\ \land μ_{C} (y_{i}^{'} y_{i}^{″})) \land μ_{C} (y_{i}^{″} x_{i}^{'}))}] \\ = ⋁_{x = \sum_{i = 1}^{n} x_{i} y_{i}} ⋀_{i = 1}^{n} [{⋁_{x_{i} = x_{i}^{'} + y_{i}^{'}, y_{i} = x_{i}^{″} + y_{i}^{″}} \\ μ_{A \cap B} (x_{i}^{'} x_{i}^{″}) \land μ_{C} (y_{i}^{'} x_{i}^{″} + y_{i}^{'} y_{i}^{″} + y_{i}^{″} x_{i}^{'})}] \\ \leq ⋁_{x = \sum_{i = 1}^{n} x_{i} y_{i}} [{⋁_{x_{i} = x_{i}^{'} + y_{i}^{'}, y_{i} = x_{i}^{″} + y_{i}^{″}} \\ (μ_{A \cap B} (x_{i}^{'} x_{i}^{″}) \land μ_{C} (y_{i}^{'} x_{i}^{″} + y_{i}^{'} y_{i}^{″} + y_{i}^{″} x_{i}^{'})}] \\ = ⋁_{x = \sum_{i = 1}^{n} x_{i} y_{i}} {μ_{(A \cap B) \hat{+} C} (x)} \\ = μ_{(A \cap B) \hat{+} C} (x) . \end{matrix}$

This gives $μ_{(A \hat{+} C) \cap (B \hat{+} C)} (x) \leq μ_{(A \cap B) \hat{+} C} (x)$ . $\begin{matrix} Also ν_{(A \tilde{+} C) \cap (B \tilde{+} C)} (x) \\ = ν_{(A \tilde{+} C) \otimes (B \tilde{+} C)} (x) \\ = ⋀_{x = \sum_{i = 1}^{n} x_{i} y_{i}} {⋁_{i = 1}^{n} (ν_{A \tilde{+} C} (x_{i}) \lor ν_{B \tilde{+} C} (y_{i}))} \\ = ⋀_{x = \sum_{i = 1}^{n} x_{i} y_{i}} [⋁_{i = 1}^{n} {⋀_{x_{i} = x_{i}^{'} + y_{i}^{'}} (ν_{A} (x_{i}^{'}) \lor ν_{C} (y_{i}^{'}))} \\ \lor {⋀_{y_{i} = x_{i}^{″} + y_{i}^{″}} (ν_{B} (x_{i}^{″}) \lor ν_{C} (y_{i}^{″}))}] \\ = ⋀_{x = \sum_{i = 1}^{n} x_{i} y_{i}} [⋁_{i = 1}^{n} {⋀_{x_{i} = x_{i}^{'} + y_{i}^{'}, y_{i} = x_{i}^{″} + y_{i}^{″}} (ν_{A} (x_{i}^{'}) \\ \lor ν_{C} (y_{i}^{'})) \lor (ν_{B} (x_{i}^{″}) \lor ν_{C} (y_{i}^{″}))}] \\ \geq ⋀_{x = \sum_{i = 1}^{n} x_{i} y_{i}} [⋁_{i = 1}^{n} {⋀_{x_{i} = x_{i}^{'} + y_{i}^{'}, y_{i} = x^{″} + y^{″}} (ν_{A} (x_{i}^{'} x_{i}^{″}) \\ \lor ν_{C} (y_{i}^{'} x_{i}^{″}) \lor ν_{C} (y_{i}^{'} y_{i}^{″})) \lor (ν_{B} (x_{i}^{'} x_{i}^{″}) \\ \lor ν_{C} (y_{i}^{″} x_{i}^{'}))}] \\ = ⋀_{x = \sum_{i = 1}^{n} x_{i} y_{i}} [⋁_{i = 1}^{n} {⋀_{x_{i} = x_{i}^{'} + y_{i}^{'}, y_{i} = x_{i}^{″} + y_{i}^{″}} (ν_{A} (x_{i}^{'} x_{i}^{″}) \\ \lor ν_{B} (x_{i}^{'} x_{i}^{″})) \lor (ν_{C} (y_{i}^{'} x_{i}^{″}) \lor ν_{C} (y_{i}^{'} y_{i}^{″})) \\ \lor ν_{C} (y_{i}^{″} x_{i}^{'}))}] \\ = ⋀_{x = \sum_{i = 1}^{n} x_{i} y_{i}} [⋀_{i = 1}^{n} {⋁_{x_{i} = x_{i}^{'} + y_{i}^{'}, y_{i} = x_{i}^{″} + y_{i}^{″}} \\ ν_{A \cap B} (x_{i}^{'} x_{i}^{″}) \land ν_{C} (y_{i}^{'} x_{i}^{″} + y_{i}^{'} y_{i}^{″} + y_{i}^{″} x_{i}^{'})}] \\ \geq ⋀_{x = \sum_{i = 1}^{n} x_{i} y_{i}} [{⋀_{x_{i} = x_{i}^{'} + y_{i}^{'}, y_{i} = x_{i}^{″} + y_{i}^{″}} \\ (ν_{A \cap B} (x_{i}^{'} x_{i}^{″}) \lor ν_{C} (y_{i}^{'} x_{i}^{″} + y_{i}^{'} y_{i}^{″} + y_{i}^{″} x_{i}^{'})}] \\ = ⋀_{x = \sum_{i = 1}^{n} x_{i} y_{i}} {ν_{(A \cap B) \tilde{+} C} (x)} \\ = ν_{(A \cap B) \tilde{+} C} (x) . \end{matrix}$

Thus, we have $ν_{(A \tilde{+} C) \cap (B \tilde{+} C)} (x) \geq ν_{(A \cap B) \tilde{+} C} (x)$ . Therefore, we obtain $μ_{(A \hat{+} C) \cap (B \hat{+} C)} (x) = μ_{(A \cap B) \hat{+} C} (x)$ and $ν_{(A \tilde{+} C) \cap (B \tilde{+} C)} (x) =$ $ν_{(A \cap B) \tilde{+} C} (x)$ . Hence, (A + C) ∩ (B + C) = (A ∩ B) + C, and hence the L_R is a distributive lattice. □

Theorem 3.9. If all intuitionistic fuzzy ideals of a semiring R are t-pure, then the followings are equivalent.

A is an intuitionistic fuzzy prime ideal of R.

A is an intuitionistic fuzzy strongly irreducible ideal of R.

Proof. (1 ⇒ 2). Assume A = (μ_A, ν_A) is an intuitionistic fuzzy prime ideal of R. Let B = (μ_B, ν_B) and C = (μ_C, ν_C) be any intuitionistic fuzzy ideals of R such that B ∩ C ⊆ A, i.e., (μ_B ∩ μ_C, ν_B ∪ ν_C) ⊆ (μ_A, ν_A). It follows that μ_B ∩ μ_C ⊆ μ_A and ν_B ∪ ν_C ⊇ ν_A. Since all intuitioistic fuzzy ideals are t-pure, B ∩ C = B ⊛ C. This implies that (μ_B∩C, ν_B∩C) = (μ_B⊛C, ν_B⊗C), i.e., μ_B∩C = μ_B⊛C and ν_B∩C = ν_B⊗C. Now, μ_A ⊇ μ_B∩C = μ_B⊛C and ν_A ⊆ ν_B∩C = ν_B⊗C together imply that μ_B ≤ μ_A and ν_B ≥ ν_A or μ_C ≤ μ_A and ν_C ≥ ν_A. Therefore, we conclude that (μ_A, ν_A) ⊇ (μ_B, ν_B) or (μ_A, ν_A) ⊇ (μ_C, ν_C), i.e., A ⊇ B or A ⊇ C. Hence, A an intuitionistic fuzzy strongly irreducible ideal of R.

(2 ⇒ 1). Assume A = (μ_A, ν_A) an intuitionistic fuzzy strongly irreducible ideal of R. Let B = (μ_B, ν_B) and C = (μ_C, ν_C) be any pair intuitionistic fuzzy ideals of R such that B ⊛ C ⊆ A, i.e., μ_B⊛C ⊆ μ_A and ν_B⊗C ⊇ ν_A. Since all intuitionistic fuzzy ideals of R are t-pure, it follows that B ∩ C = B ⊛ C, i.e., μ_B∩C = μ_B⊛C and ν_B∩C = ν_B⊗C. Since μ_B⊛C ⊆ μ_A and ν_B⊗C ⊇ ν_A, we have μ_B∩C ⊆ μ_A and ν_B∩C ⊇ ν_A. It follows that μ_B ∩ μ_C ⊆ μ_A and ν_B ∪ ν_C ⊇ ν_A. We claim that $μ_{A} \supseteq (μ_{B} \cap μ_{C}) \hat{+} μ_{A}$ and $ν_{A} \subseteq (ν_{B} \cup ν_{C}) \tilde{+} ν_{A} .$ Now, μ_B ∩ μ_C ⊆ μ_A implies that (μ_B ∩ μ_C) (x) ≤ μ_A (x) and ν_B ∪ ν_C ⊇ ν_A gives (ν_B ∪ ν_C) (x) ≥ ν_A (x). By Definitions 2.1 and 2.2, we have

$\begin{matrix} {(μ_{B} \cap μ_{C}) \hat{+} μ_{A}} (x) \\ = ⋁_{x = y + z} {(μ_{B} \cap μ_{C}) (y) \land μ_{A} (z)} \\ \leq ⋁_{x = y + z} {(μ_{B} \cap μ_{C}) (y + z) \land μ_{A} (y + z)} \\ = ⋁_{x = y + z} {(μ_{B} \cap μ_{C}) (x) \land μ_{A} (x)} \\ \leq ⋁_{x = y + z} μ_{A} (x) \\ = μ_{A} (x) . \end{matrix}$

And $\begin{matrix} {(ν_{B} \cup ν_{C}) \tilde{+} ν_{A}} (x) \\ = ⋀_{x = y + z} {(ν_{B} \cup ν_{C}) (y) \lor ν_{A} (z)} \\ \geq ⋀_{x = y + z} {(ν_{B} \cup μ_{C}) (y + z) \lor ν_{A} (y + z)} \\ = ⋀_{x = y + z} {(ν_{B} \cup ν_{C}) (x) \lor ν_{A} (x)} \\ \geq ⋀_{x = y + z} ν_{A} (x) \\ = ν_{A} (x) . \end{matrix}$

Therefore, we obtain $(μ_{B} \cap μ_{C}) \hat{+} μ_{A} \subseteq μ_{A}$ and $(ν_{B} \cup ν_{C}) \tilde{+} ν_{A} \supseteq ν_{A}$ . Now, $μ_{A} \supseteq (μ_{B} \cap μ_{C}) \hat{+} μ_{A} = (μ_{B} \hat{+} μ_{A}) \cap (μ_{A} \hat{+} μ_{C})$ and $ν_{A} \subseteq (ν_{B} \cup ν_{C}) \tilde{+} ν_{A} = (ν_{B} \tilde{+} ν_{A}) \cup (ν_{A} \tilde{+} ν_{C}) .$ Hence, $μ_{A} \supseteq (μ_{B} \hat{+} μ_{A})$ or $μ_{A} \supseteq (μ_{A} \hat{+} μ_{C})$ ; $ν_{A} \subseteq (ν_{B} \tilde{+} ν_{A})$ or $ν_{A} \subseteq (ν_{A} \tilde{+} ν_{C}) .$ This gives μ_B ⊆ μ_A or μ_C ⊆ μ_A and ν_B ⊇ ν_A or ν_C ⊇ ν_A. Hence, A = (μ_A, ν_A) is a prime ideal. □

As a consequence of Theorem 3.9, we have the following result.

Corollary 3.10. For an intuitionistic fuzzy ideal A = (μ_A, ν_A) of a fully idempotent semiring R, the followings are equivalent.

A is an intuitionistic fuzzy prime ideal.

A is an intuitionistic fuzzy strongly irreducible ideal.

Theorem 3.11. An intuitionistic fuzzy set A = (μ_A, ν_A) of a semiring R is an intuitionistic fuzzy ideal if and only if each nonempty (α, β) level cut set A^(α,β) = {x|x ∈ R, μ_A (x) ≥ α & ν_A (x) ≤ β} is ideal of R for every (α, β) ∈ Im (μ_A) × Im (ν_A) such that α + β ≤ 1.

Proof. Let A = (μ_A, ν_A) be an intuitionistic fuzzy ideal of a semiring R and let x, y ∈ A^(α,β). This gives μ_A (x) ≥ α, ν_A (x) ≤ β and μ_A (y) ≥ α, ν_A (y) ≤ β. These imply that μ_A (x) ∧ μ_A (y) ≥ α and ν_A (x) ∨ ν_A (y) ≤ β. Therefore, we have μ_A (x + y) ≥ α and ν_A (x + y) ≤ β. Hence, x + y ∈ A^(α,β). Again for any r ∈ R, μ_A (xr) ≥ μ_A (x) ≥ α and ν_A (xr) ≤ ν_A (x) ≤ β. It follows that xr ∈ A^(α,β). Similarly, rx ∈ A^(α,β). Thus, A^(α,β) is an ideal. For the converse, suppose that x, y ∈ R. Take α = μ_A (x) ∧ μ_A (y), β = ν_A (x) ∨ ν_A (y). Then, μ_A (x) ≥ α, μ_A (y) ≥ α and ν_A (x) ≤ β, ν_A (y) ≤ β. That is, x, y ∈ A^(α,β). Since A^(α,β) is an ideal of R, it follows that x + y ∈ A^(α,β). Therefore, we have μ_A (x + y) ≥ α = μ_A (x) ∧ μ₍y) and ν_A (x + y) ≤ β = ν_A (x) ∨ ν_A (y). Hence, μ_A (x + y) ≥ μ_A (x) ∧ μ₍y) and ν_A (x + y) ≤ ν_A (x) ∨ ν_A (y). Again, let r, x ∈ R. Take α = μ_A (x), β = ν_A (x). Then x ∈ A^(α,β). Since A^(α,β) is an ideal of R, it follows that rx ∈ A^(α,β). Thus, μ_A (rx) ≥ α = μ (x), ν (rx) ≤ ν_A (x). Hence, A is an intuitionistic fuzzy ideal of R. □

Theorem 3.12. An intuitionistic fuzzy ideal A = (μ_A, ν_A) of a fully idempotent semiring R is prime if and only if the ideals A^(α,β) are prime for every (α, β) ∈ Im (μ_A) × Im (ν_A) such that α + β ≤ 1.

Proof. We assume that A = (μ_A, ν_A) be an intuitionistic fuzzy prime ideal. To show A^(α,β) is a prime ideal. Let I and J be ideals of R such that IJ ⊆ A^(α,β). We construct two IF sets B and C whose membership and nonmembership functions are as follows: $\begin{matrix} μ_{B} (x) & = & {\begin{matrix} α, & if x \in I \\ 0, & if x \notin I \end{matrix}, ν_{B} (x) = {\begin{matrix} β, & if x \in I \\ 1, & if x \notin I, \end{matrix} \\ μ_{C} (x) & = & {\begin{matrix} α, & if x \in J \\ 0, & if x \notin J \end{matrix}, ν_{C} (x) = {\begin{matrix} β, & if x \in J \\ 1, & if x \notin J . \end{matrix} \end{matrix}$

Then B and C are ideals of R. Since R is fully idempotent, it follows that $\begin{matrix} μ_{B ⊛ C} (x) = μ_{B \cap C} (x) \\ = μ_{B} (x) \land μ_{C} (x) = {\begin{matrix} α, & if x \in I \cap J \\ 0, & if x \notin I \cap J \end{matrix} \\ ν_{B \otimes C} (x) = ν_{B \cap C} (x) \\ = ν_{B} (x) \lor ν_{C} (x) = {\begin{matrix} β, & if x \in I \cap J \\ 1, & if x \notin I \cap J . \end{matrix} \end{matrix}$

Since R is fully idempotent, it follows that I ∩ J = IJ. Again, since IJ ⊆ A^(α,β), we have I ∩ J ⊆ A^(α,β), whence μ_A (x) ≥ α & ν_A (x) ≤ β for all x ∈ I ∩ J. It follows that μ_A (x) ≥ μ_B⊛C (x) & ν_A (x) ≤ ν_B⊛C (x) for all x ∈ R. This implies that B ⊛ C ⊆ A. Since A is prime, it follows that B ⊆ A or C ⊆ A. This implies that $B^{(α, β)} \subseteq A^{(α, β)} or C^{(α, β)} \subseteq A^{(α, β)} .$

It is clear from the construction of B and C that B^(α,β) = I and C^(α,β) = J. Therefore, we obtain I ⊆ A^(α,β) or J ⊆ A^(α,β). Hence, A^(α,β) is prime.

For the converse, assume that A^(α,β) is prime. In order to show A = (μ_A, ν_A) is prime. Let B = (μ_B, ν_B), C = (μ_C, ν_C) be two intuitionistic fuzzy ideals such that B ⊛ C ⊆ A, i.e., μ_B⊛C ⊆ μ_A and ν_B⊗C ⊇ ν_A. This implies μ_B ∩ μ_C ⊆ μ_A and ν_B ∪ ν_C ⊇ ν_A. Now, we have $\begin{matrix} {x | x \in R, μ_{B} (x) \geq α & ν_{B} (x) \leq β} \\ \cap {x | x \in R, μ_{C} (x) \geq α & ν_{C} (x) \leq β} \\ = {x | x \in R, μ_{B \cap C} (x) \geq α & ν_{B \cap C} (x) \leq β} \\ \subseteq {x | x \in R, μ_{A} (x) \geq α & ν_{A} (x) \leq β} . \end{matrix}$

Therefore, we have B^(α,β) ∩ C^(α,β) ⊆ A^(α,β). Since R is fully idempotent, it follows that (B^(α,β)) (C^(α,β)) = B^(α,β) ∩ C^(α,β) ⊆ A^(α,β). Since the ideals A^(α,β) are prime for every (α, β) ∈ Im (μ_A) × Im (ν_A) such that α + β ≤ 1, it follows from the previous statement that either B^(α,β) ⊆ A^(α,β) or C^(α,β) ⊆ A^(α,β), whence B ⊆ A or C ⊆ A. Hence, A is prime. □

Theorem 3.13. If R is a fully idempotent semiring and the set of all intuitionistic fuzzy ideals of R is totally ordered, then every intuitionistic fuzzy ideal of R is an intuitionistic fuzzy primeideal.

Proof. Let A = (μ_A, ν_A), B = (μ_B, ν_B) and C = (μ_C, ν_C) be intuitionistic fuzzy ideals of a fully idempotent semiring R such that B ⊛ C ⊆ A. Since the set of all intuitionistic fuzzy ideals of R is totally ordered, it follows that B ⊆ C or C ⊆ B. If B ⊆ C, B = B² ⊆ B ∩ C = B ⊛ C ⊆ A. This implies that B ⊆ A. Again, if C ⊆ B, C = C² ⊆ B ∩ C = B ⊛ C ⊆ A, and so C ⊆ A. Hence, A is a primeideal. □

4 Conclusion

In this article, we have investigated some properties of intuitionistic fuzzy ideals on a weakly regular semiring and fully idempotent semiring. Intuitionistic fuzzy t-pure ideals were defined, and established some results of intuitionistic fuzzy t-pure ideals, intuitionistic fuzzy strongly irreducible and intuitionistic fuzzy prime ideals. We also investigated some properties of fully idempotent semirings in consonance with intuitionistic fuzzy strongly irreducible and intuitionistic fuzzy prime ideals.

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