Tripolar Fuzzy Ideals In CV Algebra

1. Introduction

It was Zadeh who introduced the notion of fuzzy set in 1965, as a generalization of the concept of characteristic function of a set. Fuzzy set is identified as a better tool for the scientific study of uncertainty, and came as a boost to the researchers working in the field of uncertainty. Many extensions and generalizations of fuzzy set was conceived by a number of researchers and a large number of real-life applications were developed in a variety of areas like, logic, finite state machines, automata theory, artificial intelligence, computer science, control engineering and so on. In addition to this, parallel analysis of the classical results of many branches of Mathematics was undertaken in the fuzzy settings. One such abstract area in the branch of fuzzy algebra. It was initiated by Rosenfeld, who coined the idea of fuzzy subgroup of a group in 1971 and studied some basic properties of this structure. Iseki and Tanaka have introduced the theory of BCK-algebras. Iseki has introduced BCI-algebras. Hu and Li introduced BCH-algebras. The class of BCI-algebras is a proper subclass of the class of BCH-algebras.

Zhang introduced the concept of bipolar fuzzy set as an extension of fuzzy set whose membership degree range is [-1,1]. In a bipolar valued fuzzy set the membership degree 0 means that elements are irrelevant to the corresponding property, the membership degrees on (0,1] indicate that elements somewhat satisfy the property, and the membership degrees on [-1,0) indicate that elements somewhat satisfy the implicit counter property. Kim and Kim introduced the notion of a BE-algebra. Ahn and So introduced the notion of ideals in BE-algebras. Kim has studied the notion of essence in BE algebra. The monograph by Chinnadurai gives a detailed discussion on fuzzy ideals in algebraic structures. Senapti et al. introduced and investigated some of their properties of fuzzy dot subalgebras, fuzzy normal dot subalgebras and fuzzy dot ideals of B-algebras. The notion of cubic intuitionistic q-ideals in BCI-algebras is introduced by Senapti et al. Murali Krishna Rao introduced tripolar fuzzy interior ideals of Γ- semigroup. Jana et al initiated the concept of bipolar fuzzy soft subalgebras and ideals of BCK/BCI-algebras baesd on bipolar fuzzy points. Mostafa1 and Ghanem introduced tripolar fuzzy sub implicative ideals of KU-Algebras. In this research work, we introduce the concept of CV -algebra and tripolar fuzzy ideals in CV -algebra.

2. CV -algebra

Definition 2.1. A non-empty set U with a binary operation ⋆ satisfying the conditions

(i) u ⋆ 0 = 0 = 0 ⋆ u

(ii) u ⋆ u = u

(iii) u ⋆ (v ⋆ w) = w ⋆ (u ⋆ v), for all u, v, w ∈ U is called CV -algebra.

Example 2.1. The set U = {0, 1, 2, 3} with the binary operation ⋆, with the following Cayley table becomes CV -algebra.

⋆ 0 1 2 3 0 0 0 0 0 1 0 1 2 1 2 0 1 2 3 3 0 1 1 3

Definition 2.2. A CV -algebra (U, ⋆, 0) is said to be self-distributive if u ⋆ (v ⋆ w) = (u ⋆ v) ⋆ (u ⋆ w) for all u, v, w ∈ U.

Example 2.2. The set U = {0, 1, 2, 3} with the binary operation ⋆, with the following Cayley tables become self-distributive CV -algebra.

⋆ 0 1 2 3 ⋆ 0 1 2 3 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 and 1 0 1 2 3 2 0 1 2 3 2 0 1 2 3 3 0 2 2 3 3 0 1 2 3

Definition 2.3. If a non-empty subset E of the CV -algebra U satisfies the condition U ⋆ E = E, then E is called an essence of U.

Example 2.3. In Example 2.2. {0} and U itself are essences of U. E1 = {0, 1}, E2 = {0, 1, 2}, E3 = {0, 1, 3} are also an essences of U but E4 = {0, 2}, E5 = {0, 3} are not an essences of U.

Lemma 2.1. If (U, ⋆, 0) be CV -algebra, then u ⋆ (v ⋆ u) = v ⋆ u.

Proof: u ⋆ (v ⋆ u) = v ⋆ (u ⋆ u) = v ⋆ u, by (iii) and (ii) of Definition 2.1.

Lemma 2.2. If (U, ⋆, 0) be CV -algebra, then (i) u ⋆ (0 ⋆ u) = 0 and (ii) u ⋆ (w ⋆ u) = w ⋆ u.

Proof:

(i) u ⋆ (0 ⋆ u) = u ⋆ 0 = 0, by (i) and (ii) of Definition 2.1.

(ii) u ⋆ (w ⋆ u) = w ⋆ (u ⋆ u) = w ⋆ u, by (iii) and (ii) of Definition 2.1.

Lemma 2.2. Every essence contains the element 0.

Proof: Let E be an essence of U. Then ∅ ̸= E = U ⋆ E, and so there exists a ∈ E such that and thus 0 = U ⋆ 0 = U ⋆ E = E. Hence every essence contains the element 0.

Theorem 2.1. Every essence is a sub algebra of U.

Proof: Let E be an essence of U and let u, v ∈ E. Then u ⋆ v ∈ E ⋆ E ⊆ U ⋆ E = E. And so E is a sub algebra of U.

But the converse is not true. In Example 2.1. the set {0, 2} is a sub algebra but not an essence of U.

Theorem 2.2. Let E and F be essences of U. Then E ∩ F and E ∪ F are an essences of U.

Proof: Let R = E ∩ F. Then 0 ⋆ R ⊆ U ⋆ R = U ⋆ (E ∩ F) = (U ⋆ E) ∩ (U ⋆ F) = E ∩ F.

So U ⋆ R = R is an essence of U.

Now let S = E ∪ F. Then 0 ⋆ S ⊆ U ⋆ S = U ⋆ (E ∪ F) = (U ⋆ E) ∪ (U ⋆ F) = E ∪ F.

So U ⋆ S = S is an essence of U.

Lemma 2.3. Let E be an essence of CV -algebra of U. If 0 ∈ F ⊆ U, then F ⋆ E = E.

Proof: Let E be an essence of CV -algebra of U. Then 0 ⋆ E ⊆ F ⋆ E ⊆ U ⋆ E = E. Therefore F ⋆ E = E.

Lemma 2.4. Let U be a CV -algebra. If 0 ∈ E ⊆ U, then F is contained in E ⋆ F for every sub set F of U.

Proof: Let f ∈ F. Then f = 0 ⋆ f ∈ E ⋆ F and so F is contained in E ⋆ F.

Theorem 2.3. Let U be a CV -algebra. If E is an essence of U and F is a sub algebra of U and E ⊆ F, then E ∪ F and E ∩ F are the essences of F.

Proof: F ⋆ (E ∩ F) = (F ⋆ E) ∩ (F ⋆ F) ⊆ (U ⋆ E) ∩ F ⊆ E ∩ F.

Obviously, E ∩ F ⊆ F ⋆ (E ∩ F). Therefore F ⋆ (E ∩ F) = E ∩ F.

Hence E ∩ F is an essence of F.

Similarly E ∪ F is also an essence of F if E ⊆ F.

Definition 2.4. Let U and V be CV -algebras. A mapping φ : U −→ V is called a homomorphism if φ(u ⋆ v) = φ(u) ⋆ φ(v) for all u, v ∈ U. Note that φ(0) = 0.

Lemma 2.5. For any sub sets E, F and G of CV -algebra U, we have

(1) E ⊆ F ⇒ E ⋆ G ⊆ F ⋆ G, G ⋆ E ⊆ G ⋆ F,

(2) (E ∩ F) ⋆ G ⊆ (E ⋆ G) ∩ (F ⋆ G),

(3) G ⋆ (E ∩ F) ⊆ (G ⋆ E) ∩ (G ⋆ F),

(4) (E ∪ F) ⋆ G = (E ⋆ G) ∪ (F ⋆ G),

(5) G ⋆ (E ∪ F) = (G ⋆ E) ∪ (G ⋆ F).

Proof:

(1) Let u ∈ E ⋆ G. Then u = e ⋆ g for some e ∈ E and g ∈ G. Since E ⊆ F, it follows that u = e ⋆ g for some e ∈ F and ginG. So that u ∈ F ⋆ G. Which implies that E ⋆ G ⊆ F ⋆ G. Similarly we can obtain G ⋆ E ⊆ G ⋆ F.

(2) Since E ∩ F ⊆ E and E ∩ F ⊆ F. From u ∈ E ⋆ G, we get (E ∩ F) ⋆ G ⊆ E ⋆ G and (E ∩ F) ⋆ G ⊆ F ⋆ G, and so (E ∩ F) ⋆ G ⊆ (E ⋆ G) ∩ (F ⋆ G).

(3) From (1)and (2), we can write G ⋆ (E ∩ F) ⊆ (G ⋆ E) ∩ (G ⋆ F).

(4) Since E ⊆ E ∪ F and F ⊆ E ∪ F. We get E ⋆ G ⊆ (E ∪ F) ⋆ G and F ⋆ G ⊆ (E ∪ F) ⋆ G, and so (E ⋆ G) ∪ (F ⋆ G) ⊆ (E ∪ F) ⋆ G. If u ∈ (E ∪ F) ⋆ G, then u = v ⋆ g for some v ∈ E ∪ F and g ∈ G. It follows that u = v ⋆ g for some v ∈ E and g ∈ G or u = v ⋆ g for some v ∈ F and g ∈ G. So that u = v ⋆ g ∈ E ⋆ G or u = v ⋆ g ∈ F ⋆ G. Hence u ∈ (E ⋆ G) ∪ (F ⋆ G), which shows that (E ∪ F) ⋆ G ⊆ (E ⋆ G) ∪ (F ⋆ G). Thus (E ∪ F) ⋆ G = (E ⋆ G) ∪ (F ⋆ G).

(5) From (1)and (4), we can write G ⋆ (E ∪ F) = (G ⋆ E) ∪ (G ⋆ F).

Lemma 2.6. For any subsets E, F of CV -algebra U, we have the following results

(1) E = E,

c c c c c c

(2) (E ∪ F) = E ∩ F and (E ∩ F) = E ∪ F ,

(3) E ∪ F = F ∪ E and E ∩ F = F ∩ E,

(4) E ∪ (F ∪ G) = (E ∪ F) ∪ G and E ∩ (F ∩ G) = (E ∩ F) ∩ G,

(5) E ∪ ∅ = E and E ∩ U = E,

(6) E ∪ E = U and E ∩ E = ∅.

Proof:

c c c c

c c c c c c

(1) Let u ∈ E ⇒ u / ∈ E ⇒ u ∈ E ⇒ E ⊆ E . Now let u ∈ E ⇒ u / ∈ E ⇒ u ∈ E ⇒ E ⊆ E. Hence we get E = E .

c c c c c

(2) Let u ∈ (E ∪ F) ⇔ u / ∈ E ∪ F ⇔ u / ∈ E and u / ∈ F ⇔ u ∈ E and u ∈ F ⇔ u ∈ E ∩ F . Then

c c c c c c c c

(E ∪ F) = E ∩ F . And let u ∈ (E ∩ F) ⇔ u / ∈ E ∩ F ⇔ u / ∈ E or u / ∈ F ⇔ u ∈ E or u ∈ F ⇔ u ∈ E ∪ F .

c c c

Then (E ∩ F) = E ∪ F .

(3) E ∪ F = {u : u ∈ E or u ∈ F} = {u : u ∈ F or u ∈ E} = F ∪ E and E ∩ F = {u : u ∈ E and u ∈ F} = {u : u ∈ F and u ∈ E} = F ∩ E.

(4) Let u ∈ E ∪ (F ∪ G) ⇔ u ∈ E or u ∈ F ∪ G ⇔ u ∈ E or u ∈ F or u ∈ G ⇔ u ∈ (E or F) or u ∈ G ⇔ u ∈ (E ∪ F) ∪ G. Then E ∪ (F ∪ G) = (E ∪ F) ∪ G. And also let u ∈ E ∩ (F ∩ G) ⇔ u ∈ E and u ∈ F ∩ G ⇔ u ∈ E and u ∈ F and u ∈ G ⇔ u ∈ (E and F) and u ∈ G ⇔ u ∈ (E ∩ F) ∩ G. Then E ∩ (F ∪ G) = (E ∩ F) ∩ G.

(5) Let u ∈ E ∪ ∅ ⇔ u ∈ E or u ∈ ∅ ⇔ u ∈ E. Then E ∪ ∅ = E. And also let u ∈ E ∩ U ⇔ u ∈ E and u ∈ U ⇔ u ∈ E. Then E ∩ U = E.

c c c c

(6) Let u ∈ E ∪ E ⇔ u ∈ E or u ∈ E ⇔ u ∈ U. Then E ∪ E = U. And also let u ∈ E ∩ E ⇔ u ∈ E and u ∈ E ⇔ u ∈ ∅. Then E ∩ E = ∅.

Theorem 2.4. Let φ : U −→ V be a homomorphism of CV -algebra.

(1) If φ is onto and E is an essence of U, then φ(E) is an essence of V ,

(2) If F is an essence of V , then φ (F) is an essence of U.

Proof:

(1) Suppose that φ is onto and E is an essence of U.

Let f ∈ φ(E) and v ∈ V . Then f = φ(e) and v = φ(u) for some e ∈ E and u ∈ U.

Thus v ⋆ f = φ(u) ⋆ φ(e) = φ(u ⋆ e) ∈ φ(U ⋆ E) = φ(E) and so V ⋆ φ(E) ⊆ φ(E). Obviously φ(E) ⊆ V ⋆ φ(E). Therefore V ⋆ φ(E) = φ(E).

Hence φ(E) is an essence of V .

(2) Let e ∈ φ (F) and u ∈ U. Then φ(e) ∈ F and φ(u) ∈ V . φ(u ⋆ e) = φ(u) ⋆ φ(e) ∈ V ⋆ F = F. We have

−1 −1 −1 −1 −1

u ⋆ e ∈ φ (F). That is, U ⋆ φ (F) ⊆ φ (F). Obviously φ (F) ⊆ U ⋆ φ (F).

−1 −1

Therefore U ⋆ φ (F) = φ (F).

Hence φ (F) is an essence of U.

Theorem 2.5. Let E be a CV -algebra, then E ∪ E is also a CV -algebra.

Venkatesan | Opt. 2(1) (2025) 30-42

c c c

Proof: Given E is a CV -algebra, then there exists 0 ∈ E. But 0 ∈ / E . Definitely 0 ∈ E ∪ E . Hence E ∪ E is also a CV -algebra.

Lemma 2.6. Let E, F be essences of CV -algebra of U. Then U ⋆ E = U ⋆ F iff E = F.

Proof: Since by Definition 2.5. if U ⋆ E = U ⋆ F, then E = F.

If we assume E = F, then E ⊆ F and F ⊆ E ⇒ U ⋆ E ⊆ U ⋆ F and U ⋆ F ⊆ U ⋆ E ⇒ U ⋆ E = U ⋆ F.

3. On ideals in CV -algebra

Definition 3.1. A non-empty subset I of U is called an ideal of U if

(i) If u ∈ U and l ∈ I, then u ⋆ l ∈ I, i.e., U ⋆ I ⊆ I,

(ii) If u ∈ U and l, m ∈ I, then (l ⋆ (m ⋆ u)) ⋆ u ∈ I.

Example 3.1. In Example 2.2. {0} and U are the only ideals, since (1 ⋆ (1 ⋆ 2)) ⋆ 2 = 2 ̸= {0, 1} and (2 ⋆ (2 ⋆ 3)) ⋆ 3 = 3 ̸= {0, 1, 2}.

Lemma 3.1. Every ideal of U is an essence of U.

Proof: In Example 2.2. {0} and U are the only ideals of U that are also an essence of U. But every essence of U need not be an ideal of U.

Theorem 3.1. Every ideal of U contains 0.

Proof: Let I ̸= ∅ be an ideal of U. There exists u ∈ I. Then 0 = 0 ⋆ u ∈ U ⋆ I ⊆ I.

Hence 0 ∈ I.

Theorem 3.2. If I is an ideal of U, then (l ⋆ u) ⋆ u ∈ I for all l ∈ I and u ∈ U.

Proof: Let m = u in Definition 3.1(ii). Thus, (l ⋆ (u ⋆ u)) ⋆ u = (l ⋆ u) ⋆ u ∈ I.

Lemma 3.2. Let I be an ideal of U. If l ∈ I and l ≥ u, then u ∈ I.

Proof: Let l ∈ I, u ∈ U with l ≥ u. Then l ⋆ u = u. Hence u = u ⋆ u = (l ⋆ u) ⋆ u ∈ I. Therefore u ∈ I.

Definition 3.2. Let I be an ideal of U. Define It by It = {u ∈ U|t ⋆ u ∈ I} for any t ∈ U.

Lemma 3.3. Let U be a self-distributive CV -algebra and I an ideal of U. Then It is a sub algebra of the CV -algebra U.

Proof: Let l, m ∈ It. Then t ⋆ l ∈ I and t ⋆ m ∈ I. By self-distributive property t ⋆ (l ⋆ m) = (t ⋆ l) ⋆ (t ⋆ m) ⊆ I ⋆ I ⊆ U ⋆ I ⊆ I.

Then l ⋆ m ∈ It.

Therefore It is a sub algebra of the CV -algebra U.

Theorem 3.3. Let U be a self-distributive CV -algebra and I is an ideal of U. Then It is an ideal of the CV -algebra U.

Proof: Let u ∈ U and l ∈ It. We have t ⋆ l ∈ I, and so t ⋆ (u ⋆ l) = (t ⋆ u) ⋆ (t ⋆ l) ∈ U ⋆ I ⊆ I.

Then u ⋆ l ∈ It. 99K (1) Also let l, m ∈ It and u ∈ U. We obtain t ⋆ l ∈ I and t ⋆ m ∈ I.

Then t⋆((l⋆(m⋆u))⋆u) = (t⋆(l⋆(m⋆u)))⋆(t⋆u) = ((t⋆l)⋆(t⋆(m⋆u)))⋆(t⋆u) = ((t⋆l)⋆((t⋆m)⋆(t⋆u)))⋆t⋆u) ∈ I. Then (l ⋆ (m ⋆ u)) ⋆ u ∈ It. 99K (2) From (1) and (2) It is an ideal of the CV - algebra U.

4. Fuzzy ideals in CV -algebra

Definition 4.1. A fuzzy set µ in U is called a fuzzy ideal of U if it satisfies

(i) µ(u ⋆ v) ≥ µ(u), ∀u, v ∈ U,

(ii) µ((u ⋆ (v ⋆ w)) ⋆ w) ≥ min{µ(u), µ(v)}, ∀u, v, w ∈ U.

Example 4.1. Define a fuzzy set µ : U −→ [0, 1] by µ(0) = 0.7, µ(1) = 0.5, µ(2) = 0.5, µ(3) = 0.5 in Example 2.2. Then we have a fuzzy ideal of the CV -algebra.

Lemma 4.1. Every fuzzy ideal µ of U satisfies the inequality: µ(0) ≥ µ(u), ∀u ∈ U.

Proof: By the definition of CV -algebra and fuzzy ideal in U,

µ(0) = µ(u ⋆ 0) ≥ µ(u).

Lemma 4.2. If µ is a fuzzy ideal of U, then µ((u ⋆ 0) ⋆ v) ≥ µ(u), ∀u, v ∈ U.

Proof: Since by the definition of fuzzy ideals in U and by Lemma 4.3.

µ((u ⋆ 0) ⋆ v) = µ((u ⋆ (0 ⋆ v)) ⋆ v) ≥ min{µ(u), µ(0)} = µ(u).

For every l, m ∈ U, let µ be a fuzzy set in U defined by

m α, if l ⋆ (m ⋆ u) = 0 µl (u) = β, otherwise

for all u ∈ U and α, β ∈ [0, 1] with α > β.

Example 4.2. Let U = {0, l, m, n} be the CV -algebra with binary operation as defined by the cay-ley table,

⋆ 0 l m n 0 0 0 0 0 l 0 l m l m 0 l m n n 0 l l n

Then µ0 is not a fuzzy ideal of U.

Proof: µ0((l ⋆ (l ⋆ m)) ⋆ m) = µ0((l ⋆ m) ⋆ m) =µ0(m ⋆ m) =µ0(m)

=µ0(l) < min{µ0(l), µ0(l)}.

Therefore µ0 is not a fuzzy ideal of U.

Theorem 4.1. If U is self-distributive, then the fuzzy set µ in U is a fuzzy ideal of U for all l, m ∈ U.

Proof: Let l, m ∈ U. For any u, v ∈ U, if (l ⋆ (m ⋆ v)) ̸= 0, then µ (u) = β ≤ µ (u ⋆ v).

Assume that (l ⋆ (m ⋆ u)) = 0. Then, l ⋆ (m ⋆ (u ⋆ v)) = l ⋆ ((m ⋆ u) ⋆ (m ⋆ v)) = (l ⋆ (m ⋆ u)) ⋆ (l ⋆ (m ⋆ v)) = 0 ⋆ (l ⋆ (m ⋆ v)) = 0, and so, µ (u ⋆ v) = α = µ (u).

Hence µ (u ⋆ v) ≥ µ (u) for all u, v ∈ U.

Venkatesan | Opt. 2(1) (2025) 30-42

Now for every u, v, w ∈ U, if l ⋆ (m ⋆ u) ̸= 0 or l ⋆ (m ⋆ v) ̸= 0, then µ (u) = β or µ (v) = β.

m m m

Thus µ ((u ⋆ (v ⋆ w)) ⋆ w) ≥ β = min{µ (u), µ (v)}.

l l l

Suppose that (l ⋆ (m ⋆ u)) = 0 and (l ⋆ (m ⋆ v)) = 0, then l ⋆ (m ⋆ (u ⋆ (v ⋆ w)) ⋆ w)) = l ⋆ (m ⋆ (u ⋆ (v ⋆ w)) ⋆ (m ⋆ w))) = l ⋆ ((m ⋆ u) ⋆ (m ⋆ (v ⋆ w)) ⋆ (m ⋆ w)) = (l ⋆ (m ⋆ u)) ⋆ (l ⋆ (m ⋆ (v ⋆ w))) ⋆ (l ⋆ (m ⋆ w)) = 0 ⋆ ((l ⋆ (m ⋆ v) ⋆ (l ⋆ (m ⋆ w)) ⋆ (l ⋆ (m ⋆ w)) = 0 ⋆ 0 ⋆ (l ⋆ (m ⋆ w)) ⋆ (l ⋆ (m ⋆ w)) = 0.

m m m

Which implies that, µ ((u ⋆ (v ⋆ w)) ⋆ w) = α = min{µ (u), µ (v)}.

l l l m m m

Therefore µ ((u ⋆ (v ⋆ w)) ⋆ w) ≥ min{µ (u), µ (v)}.

l l l

Hence µ is a fuzzy ideal of U, ∀ l, m ∈ U.

Theorem 4.2. The intersection of any set of fuzzy ideals of CV -algebra U is also a fuzzy ideal.

Proof: Let µi be a family of fuzzy ideals of CV -algebra U, then for any u, v, w ∈ U,

(i)(∩µi)(u ⋆ v) = inf(µi(u ⋆ v)) ≥ inf(µi(u)) = (∩µi)(u).

(ii)(∩µi)((u ⋆ (v ⋆ w)) ⋆ w) = inf µi(((u ⋆ (v ⋆ w)) ⋆ w) ≥ inf(min{µi(u), µi(v)}) = min{inf(µi(u), inf(µi(v))} = min{(∩µi(u)), (∩µi(v))}.

Hence the intersection of any set of fuzzy ideals of CV -algebra U is also a fuzzy ideal.

5. Tripolar fuzzy ideals in CV -algebra

Definition 5.1. A tripolar fuzzy set A = {(u, µ (u), µ (u), λ (u)|u ∈ U} in U is called a tripolar fuzzy ideal of U if it satisfies

+ + − − + +

(i) µ (u ⋆ v) ≥ µ (u), µ (u ⋆ v) ≤ µ (u) and λ (u ⋆ v) ≤ λ (u) ∀u, v ∈ U,

+ + + − − −

(ii) µ ((u ⋆ (v ⋆ w)) ⋆ w) ≥ min{µ (u), µ (v)}, µ ((u ⋆ (v ⋆ w)) ⋆ w) ≤ max{µ (u), µ (v)} and

+ + +

λ ((u ⋆ (v ⋆ w)) ⋆ w) ≤ max{λ (u), λ (v)}, ∀u, v, w ∈ U.

Example 5.1. The set U = {0, 1, 2, 3} with the binary operation ⋆, with the following Cayley table becomes tripolar fuzzy ideal of CV -algebra.

⋆ 0 1 2 3 µ 0.8 0.5 0.5 0.5 µ 0.3 0.5 0.6 0.8 λ 0.2 0.4 0.6 0.7

+ + − −

Lemma 5.1. Every tripolar fuzzy ideal A of U satisfies the inequality: µ (0) ≥ µ (u), µ (0) ≤ µ (u), λ (0) ≤ λ (u) ∀u ∈ U.

Proof: By the definition of CV -algebra and tripolar fuzzy ideal in U,

+ + + − − − + + +

µ (0) = µ (u ⋆ 0) ≥ µ (u), µ (0) = µ (u ⋆ 0) ≤ µ (u), λ (0) = λ (u ⋆ 0) ≤ λ (u).

+ + − − + +

Hence A satisfies µ (0) ≥ µ (u), µ (0) ≤ µ (u), λ (0) ≤ λ (u) ∀u ∈ U.

+ + − −

Lemma 5.2. If A is a tripolar fuzzy ideal of U, then µ ((u ⋆ 0) ⋆ v) ≥ µ (u), µ ((u ⋆ 0) ⋆ v) ≤ µ (u), λ ((u ⋆ 0) ⋆ v) ≤ λ (u), ∀u, v ∈ U.

Proof: Since by the definition of CV -algebra, tripolar fuzzy ideals in U and by Lemma 5.3.

µ ((u ⋆ 0) ⋆ v) = µ ((u ⋆ (0 ⋆ v)) ⋆ v) ≥ min{µ (u), µ (0)} = µ (u).

µ ((u ⋆ 0) ⋆ v) = µ ((u ⋆ (0 ⋆ v)) ⋆ v) ≤ max{µ (u), µ (0)} = µ (u).

λ ((u ⋆ 0) ⋆ v) = λ ((u ⋆ (0 ⋆ v)) ⋆ v) ≤ max{λ (u), λ (0)} = λ (u).

Theorem 5.1. The intersection of any set of tripolar fuzzy ideals of CV -algebra U is also a tripolar fuzzy ideal.

Proof: Let Ai = {(u, µi (u), µi (u), λi (u)|u ∈ U} be a family of tripolar fuzzy ideals of CV -algebra U, then for any u, v, w ∈ U,

(i)(∩µi )(u ⋆ v) = inf(µi (u ⋆ v)) ≥ inf(µi (u)) = (∩µi )(u)

(ii)(∩µi )(u ⋆ v) = inf(µi (u ⋆ v)) ≤ inf(µi (u)) = (∩µi )(u)

(iii)(∩λi )(u ⋆ v) = inf(λi (u ⋆ v)) ≤ inf(λi (u)) = (∩λi )(u)

(iv)(∩µi )((u ⋆ (v ⋆ w)) ⋆ w) = inf µi (((u ⋆ (v ⋆ w)) ⋆ w) ≥ inf(min{µi (u), µi (v)}) = min{inf(µi (u), inf(µi (v))} = min{(∩µi (u)), (∩µi (v))}

(v)(∩µi )((u ⋆ (v ⋆ w)) ⋆ w) = inf µi (((u ⋆ (v ⋆ w)) ⋆ w) ≤ inf(max{µi (u), µi (v)}) = max{inf(µi (u), inf(µi (v))} = max{(∩µi (u)), (∩µi (v))}

(vi)(∩λi )((u ⋆ (v ⋆ w)) ⋆ w) = inf λi (((u ⋆ (v ⋆ w)) ⋆ w) ≤ inf(max{λi (u), λi (v)}) = max{inf(λi (u), inf(λi (v))} = max{(∩λi (u)), (∩λi (v))}.

Hence the intersection of any set of tripolar fuzzy ideals of CV -algebra U is also a tripolar fuzzy ideal.

Theorem 5.2. The union of any set of tripolar fuzzy ideals of CV -algebra U is also a tripolar fuzzy ideal.

Proof: Let Ai = {(u, µi (u), µi (u), λi (u)|u ∈ U} be a family of tripolar fuzzy ideals of CV -algebra U, then for any u, v, w ∈ U,

(i)(∪µi )(u ⋆ v) = sup(µi (u ⋆ v)) ≥ sup(µi (u)) = (∪µi )(u)

(ii)(∪µi )(u ⋆ v) = sup(µi (u ⋆ v)) ≤ sup(µi (u)) = (∪µi )(u) (iii)(∪λi )(u ⋆ v) = sup(λi (u ⋆ v)) ≤ sup(λi (u)) = (∪λi )(u)

(iv)(∪µi )((u ⋆ (v ⋆ w)) ⋆ w) = sup µi (((u ⋆ (v ⋆ w)) ⋆ w) ≥ sup(min{µi (u), µi (v)}) = min{sup(µi (u), sup(µi (v))} = min{(∪µi (u)), (∪µi (v))}

(v)(∪µi )((u ⋆ (v ⋆ w)) ⋆ w) = sup µi (((u ⋆ (v ⋆ w)) ⋆ w) ≤ sup(max{µi (u), µi (v)}) = max{sup(µi (u), sup(µi (v))} = max{(∪µi (u)), (∪µi (v))}

(vi)(∪λi )((u ⋆ (v ⋆ w)) ⋆ w) = sup λi (((u ⋆ (v ⋆ w)) ⋆ w) ≤ sup(max{λi (u), λi (v)}) = max{sup(λi (u), sup(λi (v))} = max{(∪λi (u)), (∪λi (v))}.

Hence the union of any set of tripolar fuzzy ideals of CV -algebra U is also a tripolar fuzzy ideal.

6. Product of Tripolar fuzzy ideal of CV -algebra

Definition 6.1. Let A and B be two tripolar fuzzy sets in U then the product A × B : U × U −→ [0, 1] is defined by (A × B)(u, v) = min{A(u), B(v)}, for all u, v ∈ U.

+ − + + − +

Definition 6.2. Let A = {(u, µ1 (u), µ1 (u), λ1 (u))|u ∈ U} and B = {(u, µ2 (u), µ2 (u), λ2 (u))|u ∈ U}

+ + − − + +

be two tripolar fuzzy sets of U, then the cartesian product A × B = {µ1 × µ2 , µ1 × µ2 , λ1 × λ2 } is defined by

+ + + + − − − −

(µ1 × µ2 )(u, v) = min{µ1 (u), µ2 (v)}, (µ1 × µ2 )(u, v) = max{µ1 (u), µ2 (v)} and

+ + + +

(λ1 × λ2 )(u, v) = max{λ1 (u), λ2 (v)}.

+ + − − + +

Where µ1 × µ2 : U × U −→ [0, 1], µ1 × µ2 : U × U −→ [−1, 0], and λ1 × λ2 : U × U −→ [0, 1], for all u, v ∈ U.

+ − + + − +

Theorem 6.1. Let A1 = {(u, µ1 (u), µ1 (u), λ1 (u)|u ∈ U1}, A2 = {(u, µ2 (u), µ2 (u), λ2 (u)|u ∈ U2}, · · · , An = {(u, µn (u), µn (u), λn (u)|u ∈ Un} be n- tripolar fuzzy sets of U, then A1 × A2 × · · · × An is a tripolar fuzzy ideal of U1 × U2 × · · · × Un.

Proof: Let (u1, u2, ..., un), (v1, v2, ..., vn), (w1, w2, ..., wn) ∈ U1 × U2 × ... × Un, we have

+ + + +

(i) (µ1 × µ2 × ... × µn )((u1, u2, ..., un) ⋆ (v1, v2, ..., vn)) = µ ((u1, u2, ..., un) ⋆ (v1, v2, ..., vn)) = µ ((u1 ⋆ v1), (u2 ⋆ v2), ..., (un ⋆ vn))

+ + +

= (µ1 × µ2 × ... × µn )((u1 ⋆ v1), (u2 ⋆ v2), ..., (un ⋆ vn))

+ + +

≥ min{µ1 (u1 ⋆ v1), µ2 (u2 ⋆ v2), ..., µn (un ⋆ vn)}

+ + +

= min{µ1 (u1), µ2 (u2), ..., µn (un)}

+ + +

= (µ1 × µ2 × ... × µn )(u1, u2, ..., un),

− − − −

(ii) (µ1 × µ2 × ... × µn )((u1, u2, ..., un) ⋆ (v1, v2, ..., vn)) = µ ((u1, u2, ..., un) ⋆ (v1, v2, ..., vn)) = µ ((u1 ⋆ v1), (u2 ⋆ v2), ..., (un ⋆ vn))

− − −

= (µ1 × µ2 × ... × µn )((u1 ⋆ v1), (u2 ⋆ v2), ..., (un ⋆ vn))

− − −

≤ max{µ1 (u1 ⋆ v1), µ2 (u2 ⋆ v2), ..., µn (un ⋆ vn)}

− − −

= max{µ1 (u1), µ2 (u2), ..., µn (un)}

− − −

= (µ1 × µ2 × ... × µn )(u1, u2, ..., un),

+ + + +

(iii) (λ1 × λ2 × ... × λn )((u1, u2, ..., un) ⋆ (v1, v2, ..., vn)) = λ ((u1, u2, ..., un) ⋆ (v1, v2, ..., vn)) = λ ((u1 ⋆ v1), (u2 ⋆ v2), ..., (un ⋆ vn))

+ + +

= (λ1 × λ2 × ... × λn )((u1 ⋆ v1), (u2 ⋆ v2), ..., (un ⋆ vn))

+ + +

≤ max{λ1 (u1 ⋆ v1), λ2 (u2 ⋆ v2), ..., λn (un ⋆ vn)} + + +

= max{λ1 (u1), λ2 (u2), ..., λn (un)}

+ + +

= (λ1 × λ2 × ...λn )(u1, u2, ..., un),

+ + +

(iv) (µ1 × µ2 × ... × µn )(((u1, u2, ..., un) ⋆ ((v1, v2, ..., vn) ⋆ (w1, w2, ..., wn))) ⋆ (w1, w2, ..., wn)) = µ (((u1, u2, ..., un) ⋆ ((v1, v2, ..., vn) ⋆ (w1, w2, ..., wn))) ⋆ (w1, w2, ..., wn)) = µ (((u1, u2, ..., un) ⋆ ((v1 ⋆ w1), (v2 ⋆ w2), ..., (vn ⋆ wn))) ⋆ (w1, w2, ..., wn)) = µ (((u1 ⋆ (v1 ⋆ w1)), (u2 ⋆ (v2 ⋆ w2)), ..., (un ⋆ (vn ⋆ wn))) ⋆ (w1, w2, ..., wn)) = µ (((u1 ⋆ (v1 ⋆ w1)) ⋆ w1), ((u2 ⋆ (v2 ⋆ w2)) ⋆ w2), ..., ((un ⋆ (vn ⋆ wn)) ⋆ wn))

+ + +

= (µ1 × µ2 × ... × µn )(((u1 ⋆ (v1 ⋆ w1)) ⋆ w1), ((u2 ⋆ (v2 ⋆ w2)) ⋆ w2), ..., ((un ⋆ (vn ⋆ wn)) ⋆ w1))

+ + +

= min{µ1 ((u1 ⋆ (v1 ⋆ w1)) ⋆ w1), µ2 ((u2 ⋆ (v2 ⋆ w2)) ⋆ w2), ..., µn ((un ⋆ (vn ⋆ wn)) ⋆ wn)}

+ + + + + +

≥ min{min{µ1 (u1), µ1 (v1)}, min{µ2 (u2), µ2 (v2)}, ..., min{µn (un), µn (vn)}}

+ + + + + +

= min{min{µ1 (u1), µ2 (u2), ..., µn (un)}, min{µ1 (v1), µ2 (v2), ..., µn (vn)}}

+ + + + + +

=min{(µ1 × µ2 × ... × µn )(u1, u2, ..., un), (µ1 × µ2 × ... × µn )(v1, v2, ..., vn)},

− − −

(v) (µ1 × µ2 × ... × µn )(((u1, u2, ..., un) ⋆ ((v1, v2, ..., vn) ⋆ (w1, w2, ..., wn))) ⋆ (w1, w2, ..., wn)) = µ (((u1, u2, ..., un) ⋆ ((v1, v2, ..., vn) ⋆ (w1, w2, ..., wn))) ⋆ (w1, w2, ..., wn)) = µ (((u1, u2, ..., un) ⋆ ((v1 ⋆ w1), (v2 ⋆ w2), ..., (vn ⋆ wn))) ⋆ (w1, w2, ..., wn)) = µ (((u1 ⋆ (v1 ⋆ w1)), (u2 ⋆ (v2 ⋆ w2)), ..., (un ⋆ (vn ⋆ wn))) ⋆ (w1, w2, ..., wn)) = µ (((u1 ⋆ (v1 ⋆ w1)) ⋆ w1), ((u2 ⋆ (v2 ⋆ w2)) ⋆ w2), ..., ((un ⋆ (vn ⋆ wn)) ⋆ wn))

− − −

= (µ1 × µ2 × ... × µn )(((u1 ⋆ (v1 ⋆ w1)) ⋆ w1), ((u2 ⋆ (v2 ⋆ w2)) ⋆ w2), ..., ((un ⋆ (vn ⋆ wn)) ⋆ wn))

− − −

= min{µ1 (((u1 ⋆ (v1 ⋆ w1)) ⋆ w1), µ2 ((u2 ⋆ (v2 ⋆ w2), ..., µn (((un ⋆ (vn ⋆ wn)) ⋆ wn)) ⋆ wn))}

− − − − − −

≤ min{max{µ1 (u1), µ1 (v1)}, max{µ2 (u2), µ2 (v2)}, ..., max{µn (un), µn (vn)}}

− − − − − −

= min{max{µ1 (u1), µ2 (u2), ..., µn (un)}, max{µ1 (v1), µ2 (v2), ..., µn (vn)}}

− − − − − −

= max{min{µ1 (u1), µ2 (u2), ..., µn (un)}, min{µ1 (v1), µ2 (v2), ..., µn (vn)}}

− − − − − −

=max{(µ1 × µ2 × ... × µn )(u1, u2, ..., un), (µ1 × µ2 times... × µn )(v1, v2, ..., vn)}

+ + +

(vi) (λ1 × λ2 × ... × λn )(((u1, u2, ..., un) ⋆ ((v1, v2, ..., vn) ⋆ (w1, w2, ..., wn))) ⋆ (w1, w2, ..., wn)) = λ (((u1, u2, ..., un) ⋆ ((v1, v2, ..., vn) ⋆ (w1, w2, ..., wn))) ⋆ (w1, w2, ..., wn)) = λ (((u1, u2, ..., un) ⋆ ((v1 ⋆ w1), (v2 ⋆ w2), ..., (vn ⋆ wn))) ⋆ (w1, w2, ..., wn)) = λ (((u1 ⋆ (v1 ⋆ w1)), (u2 ⋆ (v2 ⋆ w2)), ..., ((un ⋆ (vn ⋆ wn))) ⋆ (w1, w2, ..., wn)) = λ (((u1 ⋆ (v1 ⋆ w1)) ⋆ w1), ((u2 ⋆ (v2 ⋆ w2)) ⋆ w2), ..., ((un ⋆ (vn ⋆ wn)) ⋆ wn))

+ + +

= (λ1 × λ2 × ... × λn )(((u1 ⋆ (v1 ⋆ w1)) ⋆ w1), ((u2 ⋆ (v2 ⋆ w2)) ⋆ w2), ..., ((un ⋆ (vn ⋆ wn)) ⋆ wn))

+ + +

= min{λ1 (((u1 ⋆ (v1 ⋆ w1)) ⋆ w1), λ2 ((u2 ⋆ (v2 ⋆ w2)) ⋆ w2), ..., λn (((un ⋆ (vn ⋆ wn)) ⋆ wn))}

+ + + + + +

≤ min{max{λ1 (u1), λ1 (v1)}, max{λ2 (u2), λ2 (v2)}, ..., max{λn (un), λn (vn)}}

+ + + + + +

= min{max{λ1 (u1), λ2 (u2), ..., λn (un)}, max{λ1 (v1), λ2 (v2), ..., λn (vn)}}

+ + + + + +

= max{min{λ1 (u1), λ2 (u2), ..., λn (un)}, min{λ1 (v1), λ2 (v2), ..., λn (vn)}}

+ + + + + +

=max{(µ1 × µ2 × ... × µn )(u1, u2, ..., un), (µ1 × µ2 × ... × µn )(v1, v2, ..., vn)}.

Hence A1 × A2 × · · · × An is a tripolar fuzzy ideal of U1 × U2 × · · · × Un.

7. Homomorphism of Tripolar fuzzy ideal of CV -algebra

Definition 7.1. Let (U, ⋆, 0) and (V, ⋆ , 0 ) be CV -algebras. A mapping φ : U −→ V is said to be a homomorphism if φ(u ⋆ v) = φ(u) ⋆ φ(v) for all u, v ∈ U. Note that φ(0) = 0.

Definition 7.2. Let φ : U −→ V be a homomorphism of CV -algebras for any tripolar fuzzy set

+ − + + − +

A = {(u, µ (u), µ (u), λ (u))|u ∈ U} in U and we define Aφ = {(u, µφ (u), µφ (u), λφ (u))|u ∈ U} in V by

+ + − − + +

µφ (u) = µ (φ(u)), µφ (u) = µ (φ(u)) and λφ (u) = λ (φ(u)), for all u ∈ U.

Theorem 7.1. Let φ : U −→ V be a homomorphism of CV -algebras. If A = {(u, µ (u), µ (u), λ (u))|u ∈ U} is a tripolar fuzzy ideal of U, then Aφ = {(u, µφ (u), µφ (u), λφ (u))|u ∈ U} is a tripolar fuzzy ideal of V . Proof: Let u, v, w ∈ U and A is a tripolar fuzzy ideal of U, we have

(i) µφ (u ⋆ v) = µ (φ(u ⋆ v)) = µ (φ(u) ⋆ φ(v)) ≥ µ (φ(u)) = µφ (u)

(ii) µφ (u ⋆ v) = µ (φ(u ⋆ v)) = µ (φ(u) ⋆ φ(v)) ≤ µ (φ(u)) = µφ (u)

(iii) λφ (u ⋆ v) = λ (φ(u ⋆ v)) = λ (φ(u) ⋆ φ(v)) ≤ λ (φ(u)) = λφ (u)

(iv) µφ ((u ⋆ (v ⋆ w)) ⋆ w) = µ (φ(((u ⋆ (v ⋆ w)) ⋆ w))) = µ ((φ((u ⋆ (v ⋆ w))) ⋆ φ(w)) = µ ((φ(u) ⋆ (φ(v ⋆ w))) ⋆ φ(w)) = µ ((φ(u) ⋆ (φ(v) ⋆ φ(w))) ⋆ φ(w)) ≥ min{µ (φ(u)), µ (φ(v))} = min{µφ (u), µφ (v)}

(v) µφ ((u ⋆ (v ⋆ w)) ⋆ w) = µ (φ(((u ⋆ (v ⋆ w)) ⋆ w))) = µ ((φ((u ⋆ (v ⋆ w))) ⋆ φ(w)) = µ ((φ(u) ⋆ (φ(v ⋆ w))) ⋆ φ(w)) = µ ((φ(u) ⋆ (φ(v) ⋆ φ(w))) ⋆ φ(w)) ≤ max{µ (φ(u)), µ (φ(v))} = max{µφ (u), µφ (v)}

(vi) λφ ((u ⋆ (v ⋆ w)) ⋆ w) = λ (φ(((u ⋆ (v ⋆ w)) ⋆ w))) = λ ((φ((u ⋆ (v ⋆ w))) ⋆ φ(w)) = λ ((φ(u) ⋆ (φ(v ⋆ w))) ⋆ φ(w)) = λ ((φ(u) ⋆ (φ(v) ⋆ φ(w))) ⋆ φ(w)) ≤ max{λ (φ(u)), λ (φ(v))} = max{λφ (u), λφ (v)}.

Hence Aφ is a tripolar fuzzy ideal of V .

Theorem 7.2. Let φ : U −→ V be an epimorphism of CV -algebras. If Aφ = {(u, µφ (u), µφ (u), λφ (u))|u ∈ U} is a tripolar fuzzy ideal of V , then A = {(u, µ (u), µ (u), λ (u))|u ∈ U} is a tripolar fuzzy ideal of U.

Proof: For any x, y, z ∈ V , there exists u, v, w ∈ U such that φ(x) = u, φ(y) = v and φ(z) = w.

(i) µ (u ⋆ v) = µ (φ(x) ⋆ φ(y)) = µ (φ(x ⋆ y)) = µφ (x ⋆ y) ≥ µφ (x) = µ (φ(x)) = µ (u)

(ii) µ (u ⋆ v) = µ (φ(x) ⋆ φ(y)) = µ (φ(x ⋆ y)) = µφ (x ⋆ y) ≤ µφ (x) = µ (φ(x)) = µ (u)

(iii) λ (u ⋆ v) = λ (φ(x) ⋆ φ(y)) = λ (φ(x ⋆ y)) = λφ (x ⋆ y) ≤ λφ (x) = λ (φ(x)) = λ (u)

(iv) µ ((u ⋆ (v ⋆ w)) ⋆ w) = µ ((φ(x) ⋆ (φ(y) ⋆ φ(z))) ⋆ φ(z)) = µ ((φ(x) ⋆ (φ(y ⋆ z))) ⋆ φ(z)) = µ (φ(x ⋆ (y ⋆ z)) ⋆ φ(z)) = µ (φ(x ⋆ (y ⋆ z)) ⋆ z)) = µφ ((x ⋆ (y ⋆ z)) ⋆ z) ≥ min{µφ (x), µφ (y)} = min{µ (φ(x)), µ (φ(y))} = min{µ (u), µ (v)}

(v) µ ((u ⋆ (v ⋆ w)) ⋆ w) = µ ((φ(x) ⋆ (φ(y) ⋆ φ(z))) ⋆ φ(z)) = µ ((φ(x) ⋆ (φ(y ⋆ z))) ⋆ φ(z)) = µ (φ(x ⋆ (y ⋆ z)) ⋆ φ(z)) = µ (φ(x ⋆ (y ⋆ z)) ⋆ z)) = µφ ((x ⋆ (y ⋆ z)) ⋆ z) ≤ max{µφ (x), µφ (y)} = max{µ (φ(x)), µ (φ(y))} = max{µ (u), µ (v)}

(vi) λ ((u ⋆ (v ⋆ w)) ⋆ w) = λ ((φ(x) ⋆ (φ(y) ⋆ φ(z))) ⋆ φ(z)) = λ ((φ(x) ⋆ (φ(y ⋆ z))) ⋆ φ(z)) = λ (φ(x ⋆ (y ⋆ z)) ⋆ φ(z)) = λ (φ(x ⋆ (y ⋆ z)) ⋆ z)) = λφ ((x ⋆ (y ⋆ z)) ⋆ z) ≤ max{λφ (x), λφ (y)} = max{λ (φ(x)), λ (φ(y))} = max{λ (u), λ (v)}.

Hence A is a tripolar fuzzy ideal of U.

8. Application of Tripolar fuzzy ideal of CV -algebra

Let U be a set consisting of five students u, v, w, x, y ie., U = {u, v, w, x, y}. They have a result consisting of three aspects, the result of u is A(u) =< 0.76, 0.82, 0.52 >, where 0.5 represents ok or notbad. Suppose A(v) =< 0.86, 0.81, 0.54 >, A(w) =< 0.68, 0.72, 0.51 >, A(x) =< 0.79, 0.83, 0.57 >, A(y) =< 0.64, 0.89, 0.50 >. Then we obtain tripolar fuzzy ideal of CV -algebra by the Cayley table,

⋆ u v w x y µ 0.76 0.86 0.68 0.79 0.64 µ 0.82 0.81 0.72 0.83 0.89 λ 0.52 0.54 0.0.51 0.57 0.50

9. Conclusion

In this paper we have introduced CV -algebra. We have proved some of its properties like self-distributiveness and essence. We have given the condition for an existence of ideals, fuzzy ideals and tripolar fuzzy ideals in CV -algebra and we have proved some of its results. We have defined the product and homomorphism of tripolar fuzzy ideal of CV -algebra.