On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra

Definition 2.1.

⁵ Let J be a non-empty subset of PUP algebra X. It is termed a PUP ideal of X, if it satisfies the following conditions, for all x, y, z ∈ X.

Lemma 2.2.

⁴ In a pseudo-UP algebra X the following holds, for each x ∈ X,

Definition 2.3.

⁹ A fuzzy subset κ is called fuzzy PUP subalgebra of X, if it satisfies the following axioms:

Definition 2.4.

⁹ A mapping κ: X → [0, 1] is a fuzzy PUP ideal of X, if

Definition 2.5.

⁸ Let κ and η be fuzzy subset of a set X and Y, respectively, the Cartesian product of κ and η is define by (κ × η)(x, y) = min{κ(x), η(y)}, for (x, y) ∈ X × Y .

Definition 2.6.

⁸ If κ is a fuzzy subset of X, the strongest fuzzy relation on X, that is a fuzzy relation on κ is ξ defined by ξ(x, y) = min{κ(x), κ(y)}, for each x, y ∈ X.

Theorem 2.7.

⁹ Let f: X → Y be a homomorphism of a PUP algebra. If κ be a fuzzy PUP ideal of Y . Then f ⁻¹(κ) is a fuzzy PUP ideal of X

Theorem 2.8.

⁹ Let (X, ·, ٨, 0) and (Y, ·, ٨, 0) be a PUP algebra. f: X → Y is surjective and κ is a fuzzy PUP ideals of X. Then f(κ) is a fuzzy pseudo-UP ideals of Y, provided that the sup property holds.

Remark 2.9.

Let X and Y be pseudo-UP algebras, we define on X ×Y by, For every (x, y), (u, v) ∈ X × Y , (x, y) · (u, v) = (x · u, y · v), (x, y) ٨ (u, v) = (x٨, u, y ٨ v), then clearly, (X × Y, ·, ٨, (0, 0)) is a pseudo-UP algebra.

Theorem 3.1.

If κ is a fuzzy PUP ideal of a PUP algebra X and define a fuzzy subset κ^α(x) = min{α, κ(x)}, for all x ∈ X and α ∈ [0, 1] is a fuzzy PUP ideal of X.

Proof.

Let κ be a fuzzy PUP ideal of X and α ∈ [0, 1]. Then we need to show that κ^α is a fuzzy pseudo-UP ideal of X. Since κ is a fuzzy PUP ideal, we have κ(0) ≥ κ(x) ⇒ κ^α(0) = min{α, κ(0)} ≥ min{α, κ(x)} = κ^α(x), for all α ∈ [0, 1]. Also, κ is a fuzzy PUP ideal of gives the

κ(x · z) ≥ min{κ(x · (y ٨ z)), κ(y)} , for all x, y, z ∈ X.

And, we have κ(x ٨ z) ≥ min{κ(x ٨ (y · z)), κ(y)} , for all x, y, z ∈ X.

This shows that, κ^α is a fuzzy PUP ideal of X. Since this is true for all α ∈ [0, 1], κ^α is a fuzzy PUP ideal of X. □

Remark 3.2.

The converse of above theorem need not be necessarily true. This fact is shown by the following example.

Example 3.3.

Let X = {0, a, b, c} be a set with binary operations “·” and “٨” defined by the following cayley table:

See⁴ (X, $\cdot ,\ast ,0$ ) is PUP algebra.

Define the fuzzy set $\mathrm{\kappa }\left(\mathrm{x}\right)=\{\begin{array}{ll}0.7,& \mathit{\text{if}}\mathrm{x}=0\\ 0.8,& \mathit{\text{if}}\mathrm{x}=\mathrm{a}\\ 0.5,& \mathit{\text{if}}\mathrm{x}=\mathrm{b}\\ 0.4,& \mathit{\text{if}}\mathrm{x}=\mathrm{c}.\end{array}$

Since κ(0) = 0.7 < 0.8 = κ(a). It follows that κ is not a fuzzy PUP ideal of a PUP algebra X since it does not meet the requirements of the Definition 2.4 (i). If α = 0.4, then κ^α(x) = 0.4, for all x ∈ X. Accordingly, using routine calculations, we determine κ^α is a fuzzy pseudo-UP ideal of a pseudo-UP algebra X.

Theorem 3.4.

If κ^c is a fuzzy PUP ideal of a PUP algebra X and a fuzzy subset κ^ν(x) = min{1 −κ(x), 1 − ν}, for all x ∈ X and ν ∈ [0, 1] is a fuzzy PUP ideal of X.

Proof.

Let κ^c be a fuzzy pseudo-UP ideal of X. Define κ^ν(x) = min{1 − κ(x), 1 − ν}. Since κ^c is a fuzzy PUP ideal of X, we have κ^c(0) ≥ κ^c(x), for all x ∈ X.

Then κ^ν(0) = min{1 − κ(0), 1 − ν} ≥ min{1 − κ(x), 1 − ν} = κ^ν(x), for all x ∈ X and ν ∈ [0, 1]. Next, suppose that κ^c is a fuzzy PUP ideal of X that is κ^c(x · z) ≥ min{κ^c(x · (y ٨ z)), κ^c(y)} and κ^c(x ٨ z) ≥ min{κ^c(x ٨ (y · z)), κ^c(y)} , for all x, y, z ∈ X. Now,

Moreover, κ^ν(x ٨ z) ≥ min{κ^ν(x ٨ (y · z)), κ^ν(y)} , for all x, y, z ∈ X and ν ∈ [0, 1].

This implies that κ^ν is a fuzzy PUP ideal of a PUP algebra X, for ν ∈ [0, 1]. □

Remark 3.5.

This fact is evidenced by the following example, showing that the converse of the above theorem need not be true.

Example 3.6.

From Example 3.3, we define the fuzzy subset

Since κ^c(0) = 0.2 < 0.3 = κ^c(a). It follows that κ^c is not a fuzzy pseudo-UP ideal of X due to its failure to comply with the Definition 2.4 (i). If ν = 0.8, then κ^ν(x) = 0.2, for all x ∈ X. Thus, through standard calculations, we obtain κ^ν is a fuzzy pseudo-UP ideal of a pseudo-UP algebra X.

Theorem 3.7.

Let κ^α and ξ^α be a fuzzy PUP ideals of X. Then κ^α ∩ ξ^α is a fuzzy PUP ideal of X.

Proof.

Let κ^α and ξ^α be a fuzzy PUP ideals of the universe X. First we need to show that (κ ∩ ξ)^α = κ^α ∩ ξ^α. Now,

Then,

Next,

Moreover,

Consequently, κ^α ∩ ξ^α is a fuzzy PUP ideal of X. □

Corollary 3.8.

Let {κ^α/i ∈ Λ} be a family of fuzzy PUP ideals of X. Then ∩ _i∈Λκ^α is also fuzzy iiz PUP ideal of X.

Remark 3.9.

Let κ^α and ξ^α be fuzzy PUP ideals of X. Then κ^α ∪ ξ^α may not qualify as a fuzzy PUP ideal of X.

Example 3.10.

Let X = {0, 1, 2, 3} be a set with a binary operation “·” and “*” defined by the following cayley table:

See⁴ (X, $\cdot ,\ast ,0$ ) is PUP algebra.

Clearly, κ and ξ are fuzzy PUP ideals of X. Thus by Theorem 3.1 κ^α and ξ^α are fuzzy PUP ideals of X, for α ∈ [0, 1]. If α = 0.7, then κ^α and ξ^α are given by

Now, (κ^α ∪ ξ^α)(1 · 3) = max{κ^α(1 · 3), ξ^α(1 · 3)} = max{κ^α(3), ξ^α(3)} = max{0.4, 0.1} = 0.1.

From (٨) we get 0.1 ≥ 0.7 which contradicts the Definition 2.4.

Theorem 3.11.

Any PUP subalgebra of a PUP algebra X can be realized as a level PUP subalgebra of some fuzzy PUP subalgebra of X.

Theorem 3.12.

Let κ be a fuzzy PUP subalgebra of a PUP algebra X. Two upper level PUP subalgebra U(κ, t) and U(κ, s) (with t < s) of a fuzzy PUP subalgebra are equal if and only if there is no x ∈ X such that t ≤ κ(x) < s.

Proof.

Let κ be a fuzzy PUP subalgebra of X. Suppose that U(κ, t) = U(κ, s). Then we claim that there is no x ∈ X such that t ≤ κ(x) < s.

Assume that there exist x ∈ X such that t ≤ κ(x) < s.

This contradicts to the assumption that U(κ, t) = U(κ, s).

Conversely, suppose that there is no x ∈ X such that t ≤ κ(x) < s. Then we have to prove that

Now, x ∈ U(κ, t) ⇒ κ(x) ≥ t ⇒ κ(x) > s (since κ(x) does not lie between t and s) ⇒ x ∈ U(κ, s)

From (1) and (2) we get U(κ, t) = U(κ, s). □

Corollary 3.13.

Let X be a finite PUP algebra and κ be a fuzzy PUP subalgebra of X. If Im (κ) = {t₁, …, t _n}, then the family of PUP subalgebra {U (κ, t_i) | 1 ≤ i ≤ n}, constitutes all the upper level PUP subalgebras of κ in X.

Lemma 3.14.

Let X be a PUP algebra and κ be a fuzzy PUP subalgebra of X. If Im(κ) is finite, say {t₁, t₂ …, t _n}, then for any t_i, t_j ∈ Im(κ), U (κ, t_i) = U (κ, t_j) ⇒ t_i = t_j.

Theorem 3.15.

Let κ and ξ be two fuzzy PUP subalgebra of a PUP algebra X with identical family of level PUP Subalgebras. If Im(κ) = {t₁, t₂ …, t_r} and Im(ξ) = {q₁, q₂ …, q_k} where t₁ ≥ t₂ ≥ ··· ≥ t_r and q₁ ≥ q₂≥ ··· ≥ q_k. Then

Corollary 3.16.

Let κ and ξ be two fuzzy PUP subalgebras of a PUP algebra X with identical family of level PUP subalgebras. Then Im(κ) = Im(ξ) ⇒ κ = ξ.

Proof.

Let Im(κ) = Im(ξ) = {q₁, q₂, …, q_r} where q₁ ≥ q₂ ≥ · · · ≥ q_r. By Theorem 3.15, for any x ∈ X there exists q_j such that κ(x) = q_i = ξ(x). Thus κ(x) = ξ(x) for all x ∈ X. This implies κ = ξ. □

Theorem 3.17.

Every PUP ideal of X can be realized as a level PUP ideal of some fuzzy PUP ideal of X.

Proof.

Let J be a PUP ideal of a PUP algebra X and κ be a fuzzy subset in X defined by

Clearly, U(κ, t) = J. Since 0 ∈ J, we have κ(0) = t. Thus κ(0) ≥ κ(x), for all x ∈ X. Now, consider the following cases:

Case i) If y ∈ J, x · (y ٨ z) ∈ J, then x · z ∈ J and x ٨ (y · z) ∈ J then x ٨ z ∈ J. Since J is a PUP ideal of X such that κ(x · z) = κ(y) = κ(x · (y ٨ z)) = t and κ(x ٨ z) = κ(y) = κ(x ٨ (y · z)) = t.

⇒ κ(x · z) = min{κ(x · (y ٨ z)), κ(y)} and κ(x ٨ z) = min{κ(x ٨ (y · z)), κ(y)}.

Case ii) If x · (y ٨ z), x ٨ (y · z) ∈ J, y ∈ J, then κ(x · (y ٨ z)) = t, κ(y) = 0 and κ(x ٨ (y · z)) = t. Then we have κ(x · z) ≥ 0 = min{0, t} = min{κ(x · (y ٨ z)), κ(y)} and κ(x ٨ z) ≥ 0 = min{0, t} = min{κ(x ٨ (y · z)), κ(y)}.

Case iii) If x · (y ٨ z), x ٨ (y · z) ∈ J, y ∈ J, then κ(x · (y ٨ z)) = 0, κ(y) = t and κ(x ٨ (y · z)) = 0.

Then we get similar resualt as in case(ii).

Case iv) If y ∈ J, x · (y ٨ z), x ٨ (y · z) ∈ J, then κ(y) = 0 = κ(x · (y ٨ z)) = 0 and κ(x ٨ (y · z)) = 0.

⇒ κ(x · z) ≥ 0 = min{κ(x · (y ٨ z)), κ(y)} and κ(x ٨ z) ≥ 0 = min{κ(x ٨ (y · z)), κ(y)} , for all x, y, z ∈ X.

Thus κ is a fuzzy PUP ideal of a PUP algebra X. Hence, J is a level PUP ideal of X corresponding to a fuzzy PUP ideal of X. □

Theorem 3.18.

Let {J_i: i = 0, 1, 2, …, n} be any family of a PUP ideal of a PUP algebra X such that J₀ ⊂ J₁ ⊂ J₂ ⊂ … ⊂ J_n = X, then there exist a fuzzy PUP ideal of X.

Theorem 3.19.

Let κ be a fuzzy PUP ideal of X, then the upper level PUP ideals U(κ, s) and U(κ, t), with (s < t) of a fuzzy PUP ideals κ of X are equal if and only if there is no x ∈ X such that s ≤ κ(x) < t.

Proof.

Consider κ is a fuzzy PUP ideal of X and U(κ, s) = U(κ, t), for some s < t. We need to show that there is no x ∈ X such that s ≤ κ(x) < t. Assume that there exist x ∈ x ∈ X such that s ≤ κ(x) < t. This implies that x ∈ U(κ, s) and x ∈ U(κ, t), which contradicts the assumption U(κ, s) = U(κ, t). Hence, there is no x ∈ X such that s ≤ κ(x) < t.

Conversely, assume that there is no x ∈ X such that s ≤ κ(x) < t. We need to prove that U(κ, s) = U(κ, t). Since s < t, we get U(κ, s) ⊆ U(κ, t) (٨).

Let x ∈ U(κ, s) then κ(x) ≥ s. By our assumption there is no x ∈ X such that κ(x) < t. Then κ(x) ≥ t ⇒ x ∈ U(κ, t) because κ(x) does not lie between s and t. Hence U(κ, s) ⊆ U(κ, t)................................ (٨ ٨).

From (٨) and (٨٨), it results in U(κ, s) = U(κ, t). □

Corollary 3.20.

Let κ be a fuzzy PUP ideal of X with finite image. If U(κ, s) = U(κ, t), for some s, t ∈ Im(κ), then s = t.

Theorem 3.21.

Let κ be a fuzzy PUP ideal of a PUP algebra X and let x ∈ X. Then κ(x) = s if and only if x ∈ U(κ, s) but x /∈ U(κ, t), for all t > s.

Definition 3.22.

Let (X, ·_X, ٨_X, 0_X) and (Y, ·_Y, ٨_Y, 0_Y) be PUP algebras. Let f: X → Y be homomorphism on PUP algebra. Then any fuzzy set κ in Y, we define a new fuzzy set κ^f in X by κ^f (x) = κ(f(x)).

Theorem 3.23.

Let f: X → Y be a homomorphism of PUP algebra.

Proof.

1) Let f be a homomorphism of a PUP algebra and let κ be a fuzzy PUP subalgebra of Y. Let x, y ∈ X. Then

Hence, κ^f is a fuzzy PUP subalgebra of X.

2) Assume f is a homomorphism of a PUP algebras and κ is a fuzzy PUP ideal of X, then κ^f (0) = κ(f(0)) = κ(0) ≥ κ(x) = κ(f(x)) = κ^f (x), for all x ∈ X.

Additionally, let x, y, z ∈ X, then

Moreover,

Therefore, κ^f is a fuzzy PUP ideal of X. □

Theorem 3.24.

Let f: X → Y be an epimorphism of PUP algebra. Then κ^f is a fuzzy PUP subalgebra of X if and only if κ is a fuzzy PUP subalgebra of Y .

Definition 3.25.

Let f be a mapping defined on a PUP Algebra X. If κ is a fuzzy set in X, then the fuzzy set ξ in f(X) defined by

for all y ∈ f(X) is called image of κ under f . If ξ is a fuzzy set in f(X), then the fuzzy set κ = ξ ◦f

in X i.e. the fuzzy set defined by κ(x) = ξ(f(x)) for all x ∈ X is called the pre-image of ξ under f .

Theorem 3.26.

Let f: X → Y be an epimorphism of PUP algebras. If ξ is a fuzzy PUP subalgebra of Y and let κ is the pre-image of ξ under f , then κ is a fuzzy PUP subalgebra of X.

Proof.

Suppose ξ is a fuzzy PUP subalgebra of Y and κ is pre-image of ξ under f. For x, y ∈ X, we need to show that κ is a fuzzy PUP subalgebra of X. By Definition 3.25 κ = ξ ◦ f defined as κ(x) = ξ(f(x)).

Hence, κ is a fuzzy PUP subalgebra of X. □

Theorem 3.27.

Let f: X → Y be an onto homomorphism of PUP algebra. If κ is a fuzzy PUP subalgebra of Y, then f ⁻¹(κ^c) = (f ⁻¹(κ))^c.

Proof.

Let κ be a fuzzy PUP subalgebra of Y. Then, for x ∈ X.

□

Definition 3.28.

An ideal J of a PUP algebra X is said to be characteristic if f(J) = J, for all f ∈ Aut(X), where Aut(X) is the set of all automorphisms of X. Similarly, a fuzzy PUP ideal κ of a PUP algebra X is said to be fuzzy characteristic if κ^f (x) = κ(x), for all x ∈ X and f ∈ Aut(X).

Theorem 3.29.

A subset J of X is a PUP ideal if and only if the characteristic function of J is a fuzzy PUP ideal of X.

Proof.

Suppose that κ is a fuzzy characteristic, and let s ∈ Im(κ), f ∈ Aut(X) and x ∈ U(κ, s). Then κ^f $\left(\mathrm{x}\right)=\mathrm{\kappa }\left(\mathrm{x}\right)\text{implies}\mathrm{\kappa }\left(\mathrm{f}\left(\mathrm{x}\right)\right)\ge \mathrm{s},\text{whence}\mathrm{f}\left(\mathrm{x}\right)\in \mathrm{U}(\mathrm{\kappa },\mathrm{s}).$

Thus $\mathrm{f}\left(\mathrm{U}(\mathrm{\kappa },\mathrm{s})\right)\subseteq \mathrm{U}(\mathrm{\kappa },\mathrm{s})(٨).$

Let x ∈ U(κ, s) and y ∈ X such that f(y) = x. Then κ(y) = κ^f (y) = mu(x) ≥ s. Consequnetly, y ∈ U(κ, s). So, x = f(y) ∈ U(κ, s).

Thus, $\mathrm{U}(\mathrm{\kappa },\mathrm{s})\subseteq \mathrm{f}\left(\mathrm{U}(\mathrm{\kappa },\mathrm{s})\right)(٨٨).$

From (٨) and (٨٨) we get f(U(κ, s)) = U(κ, s). Therefore, U(κ, s) is characteristic.

Conversely, suppose thta each level PUP ideal of κ is characteristic, and let x ∈ X, f ∈ Aut(X), κ(x) = s. Then by virtue of Theorem 3.21, x ∈ U(κ, s) and x ∈ U(κ, t), for all t > s. It follows from the assumption that f(x) ∈ f(U(κ, s)) = U(κ, s), so that κ^f (x) = κ(f(x)) ≥ s. Let t = κ^f (x) and assume that t > s. Then, f(x) ∈ U(κ, t) = f(U(κ, t)), which implies from the injectivety of f that x ∈ U(κ, t), a contradiction. Hence, κ^f (x) = κ(f(x)) = s = κ(x) show that κ is a fuzzy characteristic. □

Theorem 3.30.

Let f: X → Y be an epimorphism of a PUP algebra. κ is a fuzzy PUP ideal of X with ker f ⊆ J, then f ⁻¹(f(κ)) = κ. □

Proof.

Let y ∈ Y, then there exist x ∈ X such that f(x) = y. Then f ⁻¹(f(κ))(x) = f(κ)(f(x)) = $\mathrm{\kappa }\left(\mathrm{y}\right)=\underset{x\in f^{-1}}{sup}\left(\mathrm{y}\right)\left\{\mathrm{\kappa }\left(\mathrm{x}\right)\right\}$ . For any z ∈ X, z ∈ f ⁻¹(y) ⇒ f(z) = y ⇒ f(z) = f(x) implies f(z)·f(x) = 0 and f(z) ٨ f(x) = 0. Therefore, f(z · x) = 0 and f(z ٨ x) = 0, since f is homomorphism.

Again we can prove that κ(x) ≥ κ(z) ⇒ κ(z) = κ(x). Thus, ${\mathrm{f}}^{-1}\left(\mathrm{f}\left(\mathrm{\kappa }\right)\right)\left(\mathrm{x}\right)=\underset{z\in f^{-1}\left(y\right)}{sup}$

{κ(z)} = κ(x), for some x ∈ X. Hence, f ⁻¹(f(κ)) = κ. □

Theorem 3.31.

Let f: X → Y be an epimorphism of PUP algebras and α ∈ [0, 1]. If κ^α is a fuzzy PUP ideal of X, then f(κ^α) is a fuzzy PUP ideal of Y .

Theorem 3.32.

Let f: X → Y be a homomorphism of fuzzy PUP algebra X into Y and α ∈ [0, 1]. If κ^α is a fuzzy PUP ideal of Y, then the pre-image f ⁻¹(κ^α) is a fuzzy PUP ideal of X.

Theorem 3.33.

Let f: X → Y be an onto homomorphism of PUP algebras and κ^α is a fuzzy subset in Y . If f ⁻¹(κ^α) is a fuzzy PUP ideal of X, then κ^α is a fuzzy PUP ideal of Y, for α ∈ [0, 1].

Theorem 3.34.

Let f: X → Y be an epimorphism of PUP algebras, for all ν ∈ [0, 1], and we define κ^ν(x) = min{1 − κ(x), 1 − ν} is a fuzzy PUP ideal of X, then f(κ^ν) is a fuzzy PUP ideal of Y .

Proof.

Since f: X → Y is an epimorphism of PUP algebras for any y ∈ Y, there exist x ∈ X such that f(x) = y. First we need to show that (f(κ))^ν = f(κ^ν). Now

Hence, (f(κ))^ν = f(κ^ν).

Then, (f(κ))^ν(0) = f(κ^ν)(x) = f(κ^ν)(f(0)) = f ⁻¹(f(κ^ν))(0) ≥ κ^ν(0) ≥ κ^ν(x), by Theorem 3.31 (٨). Implies κ^ν(x) = f ⁻¹(f(κ^ν))(x) = f(κ^ν)(f(x)) = f(κ^ν)(y). Thus (f(κ))^ν(0) ≥ (f(κ))^ν(y), y ∈Y.

Also, let x, y, z ∈ Y, since f: X → Y is an epimorphisms of PUP algebras, there exist a, b, c ∈ X, such that f(a) = x, f(b) = y, f(c) = z. Then

And also,

Hence, f(κ^ν) is a fuzzy PUP ideal of Y. □

Theorem 3.35.

Let f: X → Y be a homomorphism of fuzzy PUP algebra X into Y and ν ∈ [0, 1].

If κ^ν is a fuzzy PUP ideal of Y, then the pre-image f ⁻¹(κ^ν) is a fuzzy PUP ideal of X.

Proposition 3.36.

Let κ and ξ be fuzzy sets on a PUP algebra X. Then

Lemma 3.37.

Let κ and ξ be any two fuzzy PUP subalgebras of X and Y, respectively. Then (κ × ξ)(0, 0) ≥ (κ × ξ)(x, y), for all (x, y) ∈ X × Y.

Proof.

Let (x, y) ∈ X × Y. Then by definition, it follows that (κ × ξ)(0, 0) = min{κ(0), ξ(0)} ≥ min{κ(x), ξ(y)} = (κ × ξ)(x, y). □

Theorem 3.38.

Let κ₁ and κ₁ be any two fuzzy PUP subalgebras of X₁ and X₂, respectively. Then the cartesian product κ₁ × κ₂ of κ₁ and κ₂ defined as (κ₁ × κ₂)(x₁, x₂) = min{κ₁(x₁), κ₂(x₂)} is also a fuzzy PUP subalgebra of X₁ × X₂.

Proof.

Let X = X₁ × X₂ and κ = κ₁ × κ₂. For x = (x₁, x₂), y = (y₁, y₂) ∈ X, we have

Also,

This shows that, κ₁ × κ₂ is a fuzzy PUP subalgebra of X₁ × X₂. □

Theorem 3.39.

Let κ and ξ be fuzzy subsets of the PUP algebras X and Y, respectively. Suppose 0_X and 0_Y are the constant elements of X and Y, respectively. If κ × ξ is a fuzzy PUP subalgebra of X × Y then at least one of the following two statements holds.

Theorem 3.40.

Let κ be a fuzzy subset of a PUP algebra X. Let ξ be a strong fuzzy relation on X. If ξ is a fuzzy PUP subalgebra of X × X, then κ(0) ≥ κ(x), for all x ∈ X.

Proof.

Since ξ is a fuzzy PUP subalgebra of X × X and it follows that ξ(x, y) = min{κ(x), κ(y)}. By Lemma 3.37 ξ(0, 0) ≥ ξ(x, x). Then we have κ(0) = min{κ(0), κ(0)} = ξ(0, 0) ≥ ξ(x, x) = min{κ(x), κ(x)} = κ(x). Hence, κ(0) ≥ κ(x), for all x ∈ X. □

Theorem 3.41.

Let κ be a fuzzy subset of a PUP algebra X and let ξ be the strongest fuzzy relation on a PUP algebra X. Then κ is a fuzzy PUP subalgebra of X if and only if ξ is a fuzzy PUP subalgebra of X × X.

Theorem 3.42.

If κ and ξ are fuzzy PUP ideals of X and Y, respectively, then κ × ξ is a fuzzy PUP ideal of X × Y .

Theorem 3.43.

Let κ and ξ be two fuzzy subsets of X and Y, resepecively such that κ × ξ is a fuzzy PUP ideal of κ × ξ of X × Y. Then either κ is a fuzzy PUP ideal of X or ξ is a fuzzy PUP ideal of Y.

Theorem 3.44.

Let κ and ξ be fuzzy subsets of X and Y such that κ × ξ is a fuzzy PUP ideal of X × Y. Then

Proof.

Let κ × ξ be a fuzzy PUP ideal of X × Y. Then

Theorem 3.45.

Let κ and Q be fuzzy PUP ideals of X and Y, respectively. Then κ × Q is a fuzzy PUP ideal of X × Y if and only if each non-empty upper t-level set U(κ × Q, t) is a PUP ideal of X × Y, for all t ∈ [0, 1].

Theorem 3.46.

Let κ be a fuzzy set of X and let ξ be the strongest fuzzy relation on X. If ξ , we define ξ(x, y) = min{κ(x), κ(y)} is a fuzzy PUP ideal X × X, then κ(0) ≥ κ(x), for all x, y ∈ X.

Proof.

Suppose ξ is a fuzzy PUP ideal of X × X, it follows that ξ(0, 0) ≥ ξ(x, x), for all x ∈ X. Then we have κ(0) = min{κ(0), κ(0)} = ξ(0, 0) ≥ ξ(x, x) = min{κ(x), κ(x)} = κ(x). Hence, κ(0) ≥ κ(x), for all x ∈ X. □

Theorem 3.47.

Let κ be a fuzzy subset of X and let ξ be the strongest fuzzy relation on X, then κ is a fuzzy PUP ideal of X if and only if ξ is a fuzzy PUP ideal of X × X.