Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra

Alachew Amaneh Mechderso; Berhanu Assaye Alaba; Tilahun Mekonnen Munie; Teferi Getachew Alemayhu

doi:10.12688/f1000research.157173.1

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Research Article

Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra

[version 1; peer review: 2 approved, 1 approved with reservations]

Alachew Amaneh Mechderso ¹, Berhanu Assaye Alaba¹, Tilahun Mekonnen Munie¹, Teferi Getachew Alemayhu²

PUBLISHED 18 Nov 2024

Author details Author details

¹ Department of Mathematics, Bahir Dar University College of Science, Bahir Dar, Amhara, 6000, Ethiopia
² Department of Mathematics, Debre Berhan University, Debre Birhan, Amhara, Ethiopia

Alachew Amaneh Mechderso
Roles: Conceptualization, Writing – Original Draft Preparation, Writing – Review & Editing

Berhanu Assaye Alaba
Roles: Conceptualization

Tilahun Mekonnen Munie
Roles: Conceptualization

Teferi Getachew Alemayhu
Roles: Conceptualization

OPEN PEER REVIEW

REVIEWER STATUS

Abstract

In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of two bipolar fuzzy pseudo-UP ideals is also a bipolar fuzzy pseudo-UP ideal, while the union of two such ideals does not always result in a bipolar fuzzy pseudo-UP ideal. Additionally, we discuss the concepts of bipolar fuzzy pseudo-UP ideals under homomorphism and explore several related properties. The homomorphic image and inverse image of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra are also examined in detail. Furthermore, we study the notion of a bipolar fuzzy pseudo-UP ideal under the Cartesian product of pseudo-UP algebra. The Cartesian product of any two bipolar fuzzy pseudo-UP ideals is also the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra, and then some related results are obtained. MSC: 03G25, 06D30.

Keywords

Pseudo-UP algebra, pseudo-UP ideal, Fuzzy pseudo-UP ideal, bipolar Fuzzy pseudo- UP ideal

Corresponding author: Alachew Amaneh Mechderso

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2024 Mechderso AA et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Mechderso AA, Alaba BA, Munie TM and Alemayhu TG. Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations]. F1000Research 2024, 13:1386 (https://doi.org/10.12688/f1000research.157173.1) First published: 18 Nov 2024, 13:1386 (https://doi.org/10.12688/f1000research.157173.1) Latest published: 18 Nov 2024, 13:1386 (https://doi.org/10.12688/f1000research.157173.1)

Introduction

Iseki¹ introduced BCK algebras. It is known that the class of BCK algebras is a proper subclass of the class of BCI-algebras. In 2020, Romano² introduced the concept of pseudo-UP ideals and pseudo-UP filters and derived basic properties. In 1965, Zadeh³ popularized the concept of fuzzy sets. Since then, the concepts of fuzzy sets have been extensively used in many branches of mathematics. The fuzzifications of algebraic structures were initiated by Rosenfeld,⁴ and he introduced the notion of fuzzy subgroups. In 2023, Mechderso⁵ investigated the concept of fuzzy pseudo-UP ideals of pseudo-UP algebra. Wechler⁶ studied how fuzzy algebraic structures plaid vital roles in mathematics with wide applications in many other branches such as computer sciences, theoretical physics, information sciences, control engineering, topological spaces and coding theory. For a fuzzy set, its degree of membership expresses its degree of containment of elements in it. Sometimes, the degree of membership also means the degree of satisfaction of elements to some property or constraint corresponding to a fuzzy set.⁷ Keeping in view this notion, the membership degree 0 is in general assigned to those elements of the set that do not satisfy some property. In the usual study of fuzzy set representation, the elements with membership degree 0 are usually regarded as having the same characteristic.

Keeping in view these facts, Lee⁸ proposed an extension of fuzzy sets named bipolar fuzzy sets. Lee⁹ used the notion of bipolar fuzzy set and worked on bipolar valued fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI algebra. In recent times, Alaba¹⁰ studied the notion of intuitionist fuzzy PMS ideals under homomorphism and Cartesian product and investigated several related properties.

A wide variety of human decision-making is based on double-sided or bipolar judgmental thinking on a positive side and a negative side, for instance, cooperation and competition, friendship and hostility, common interests and conflict interests, effect and side effect, likelihood and unlikelihood, feed forward and feedback. The notion of bipolar fuzzy sets (YinYang bipolar fuzzy sets) was introduced by Zhang¹¹ as a generaztion of fuzzy set.

Consider a bipolar fuzzy set, as follows:

Frog’s prey = {(mosquito, 1, 0), (dragon fly, 0.4, 0), (turtle, 0, 0), (snake, 0, −1)}.

We can see that membership degree 0 and non-membership degree 0 of turtle mean that frog never hunts turtle and turtle never hunts frog. While membership degree 0 and non-membership degree –1 of snake mean that frog never hunts snake but snake always hunts frog. Here, the implicit counter-property is “predator of frog,” which created the difference between fuzzy set and bipolar fuzzy set of frog’s prey.

Motivated by this, we introduced the notion of a bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra and proved some results. We present several results related to the bipolar fuzzy pseudo-UP ideal in pseudo-UP algebras. We investigate the properties of the homomorphic image and preimage of bipolar fuzzy pseudo-UP ideals and demonstrate that both the homomorphic image and preimage of a bipolar fuzzy pseudo-UP ideal are themselves bipolar fuzzy pseudo-UP ideals. Furthermore, the Cartesian product of bipolar fuzzy pseudo-UP ideals of pseudo-UP algebras is introduced, and several properties are investigated.

Preliminaries

Definition 2.1.

[Ref. 12] A pseudo-UP algebra is algebra (X, ·, _*, 0) of type (2, 2, 0) which satisfies the following axioms: for any x, y, and z ∈ X

1) $(y \cdot z) \cdot ((x \cdot y) * (x \cdot z)) = 0 and (y^{*} z) * ((x^{*} y) \cdot (x^{*} z)) = 0$ ,
2) $x \cdot y = 0 = y \cdot x and x^{*} y = 0 = y^{*} x \Rightarrow x = y$ ,
3) $(y \cdot 0) * x = x and (y^{*} 0) \cdot x = x$ ,
4) x ≤ y if and only if x · y = 0 and x ≤ y if and only if x * y = 0.

Proposition 2.2.

[Ref. 12] In a pseudo-UP algebra X the following holds: for any x, y ∈ X

1) $x \leq y \cdot x$ and
2) $x \leq y^{*} x$ .

Definition 2.3.

[Ref. 2] A pseudo-UP ideal of X is a nonempty subset J of a pseudo-UP algebra X

That has the following: for each x, y, z ∈ X.

1) $0 \in J$ ,
2) $x \cdot (y^{*} z) \in J and y \in J \Rightarrow x \cdot z \in J$ and
3) $x^{*} (y \cdot z) \in J and y \in J \Rightarrow x^{*} z \in J$ .

Lemma 2.4.

[Ref. 2] In a pseudo-UP algebra X the following holds, for each x ∈ X,

1) x · 0 = 0 and x * 0 = 0,
2) 0 · x = x and 0 * x = x and
3) x · x = 0 and x * x = 0.

Definition 2.5.

[Ref. 5] A fuzzy subset λ of a pseudo-UP algebras of X is called fuzzy pseudo-UP ideal of X if and only if it fulfill the following axioms: for any x, y, z ∈ X

1) $λ (0) \geq λ (x)$ ,
2) $λ (x \cdot y) \geq min {λ (x \cdot (y^{*} z)), λ (y)}$ and
3) $λ (x^{*} y) \geq min {λ (x^{*} (y \cdot z)), λ (y)}$ .

Definition 2.6.

[Ref. 8] Let X be the universe of discourse. A bipolar fuzzy set λ in X is an object having the form

λ = {(x, λ^{-} (x), λ^{+} (x)) : x \in X}

Where λ⁻: X→ [−1, 0] and λ⁺: X → [0, 1] are mappings. Form this paper we use the symbol

λ = (X; λ⁻, λ⁺) for the bipolar valued fuzzy set λ = {(x, λ⁻(x), λ⁺(x)): x ∈ X}, and use the notion of bipolar fuzzy sets instead of the notion of bipolar valued fuzzy sets.

Definition 2.7.

[Ref. 13] A bipolar fuzzy set λ = (X; λ⁺, λ⁻) in a set X with the positive membership

λ⁺: X → [0, 1] and negative membership λ⁻: X → [−1, 0] is indicated to have Sup-Inf property, if for any subset T of X, there exists x₀ ∈ T such that $λ^{+} (x_{0}) = sup_{t \in T} λ^{+} (t)$ and $λ^{-} (x_{0}) = inf_{t \in T} λ^{-} (t)$

Lemma 2.8.

Let λ = (X, λ⁺, λ⁻) be bipolar fuzzy set in X. Then the following statements hold for any x, y ∈ X.

1) $1 - max {λ^{+} (x), λ^{+} (y)} = min {1 - λ^{+} (x), 1 - λ^{+} (y)}$ ,
2) $1 - min {λ^{+} (x), λ^{+} (y)} = max {1 - λ^{+} (x), 1 - λ^{+} (y)}$ ,
3) $- 1 - max {λ^{-} (x), λ^{-} (y)} = min {- 1 - λ^{-} (x), - 1 - λ^{-} (y)}$ and
4) $- 1 - min {λ^{-} (x), λ^{-} (y)} = max {- 1 - λ^{-} (x), - 1 - λ^{-} (y)}$ .

Definition 2.9.

[Ref. 14] Let f: X → Y be a homomorphism from a set X onto a set Y and let λ = (X; λ⁻, λ⁺) be a bipolar fuzzy set of X and σ = (Y; σ⁻, σ⁺) be two bipolar fuzzy set of Y, then the homomorphic image f (λ) is f (λ) = (f (λ⁻), f (λ⁺)) defined as for all y ∈ Y.

And

f (λ^{-}) (x) = inf λ^{-} (x) / x \in f^{- 1} (y) if f^{- 1} (y) \neq Ø

f (λ^{+}) (x) = sup λ^{+} (x) / x \in f^{- 1} (y) if f^{- 1} (y) \neq Ø .

The pre-image f ⁻¹ (σ) of σ under f is a bipolar set defined as f ⁻¹ (σ⁻) (x) = σ⁻ (f(x)) and f ⁻¹ (σ⁺) (x)) =σ⁺ (f(x)), for all x ∈ X.

Definition 2.10.

[Ref. 15] Let λ and σ are any two bipolar fuzz set of X and Y respectively. The Cartesian product of λ and σ is defined as λ × σ = (X × Y, λ⁺ × σ⁺, λ⁻ × σ⁻) with λ⁺ × σ⁺ (x, y) = $min {λ^{+} (x), σ^{+} (y)}$ and $λ^{-} \times σ^{-} (x, y) = max {λ^{-} (x), σ^{-} (y)}$ where $λ^{+} \times σ^{+}$ : $X \times Y \to [0, 1]$ and $λ^{-} \times σ^{-}$ : $X \times Y \to [- 1, 0]$ , $\forall (x, y) \in X \times Y$ .

Bipolar fuzzy pseudo-UP ideal

In this section, we introduce the concept of a bipolar fuzzy pseudo-UP ideal in pseudo-UP algebras and examine several important properties associated with bipolar fuzzy pseudo-UP ideals.

Definition 3.1.

A bipolar fuzzy set λ = (X, λ⁺, λ⁻) in X is called bipolar fuzzy pseudo-UP ideal of X if it satisfies the following conditions for all x, y, z ∈ X.

i) λ⁺(0) ≥ λ⁺(x) and λ⁻(0) ≤ λ⁻(x),
ii) λ⁺(x · z) ≥ min {λ⁺(x · (y * z)), λ⁺(y)},
iii) λ⁺(x * z) ≥ min {λ⁺(x * (y · z)), λ⁺(y)},
iv) λ⁻(x · z) ≤ max {λ⁻(x · (y * z)), λ⁻(y)},
v) λ⁻(x * z) ≤ max {λ⁻(x * (y · z)), λ⁻(y)}.

Example 3.2.

Let X = {0, a, b, c} be a set with binary operations “·” and “*” defined by the following cayley Table 1

Clearly, (X, ·, *, 0) is pseudo-UP algebra. We define a bipolar fuzzy pseudo-UP ideal of X as follows.

X	0	a	b	c
λ⁺	0.8	0.6	0.5	0.4
λ⁻	-0.7	-0.5	-0.4	-0.4

Then, λ is a bipolar fuzzy pseudo-UP ideal of X.

Theorem 3.3.

Let λ = (X, λ⁺, λ⁻) be a bipolar fuzzy pseudo-UP ideal of a pseudo-UP algebra X with y ≤ z, for any y, z ∈ X, then λ⁺(y) ≥ λ⁺(z) and λ⁻(y) ≤ λ⁻(z), that is λ⁺ is order reversing and λ⁻ is order preserving.

Proof.

Let λ = (X, λ⁺, λ⁻) be a bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra X such that y ≤ z, for all y, z ∈ X. Then by the binary relation ≤ define in X, we have z · y = 0 and z * y = 0.

\begin{array}{l} Now, λ^{+} (y) = λ^{+} (0 \cdot y), by Lemma 2.4 \\ \geq min {λ^{+} (0 \cdot (z^{*} y)), λ^{+} (z)}, (by Definition 3.1) \\ = min {λ^{+} (z^{*} y), λ^{+} (z)}, (by Lemma 2.4) \\ = min {λ^{+} (0), λ^{+} (z)}, (since y \leq z \Rightarrow z^{*} y = 0) \\ = λ^{+} (z) . \\ \Rightarrow λ^{+} (y) \geq λ^{+} (z) . \end{array}

\begin{array}{l} λ^{+} (y) = λ^{+} (0^{*} y), by Lemma 2.4 \\ \geq min {λ^{+} (0^{*} (z \cdot y)), λ^{+} (z)}, (by Definition 3.1) \\ = min {λ^{+} (z \cdot y), λ^{+} (z)}, (by Lemma 2.4) \\ = min {λ^{+} (0), λ^{+} (z)}, (since y \leq z \Rightarrow z \cdot y = 0) \\ = λ^{+} (z) . \Rightarrow λ^{+} (y) \geq λ^{+} (z), and \end{array}

\begin{array}{l} λ^{-} (y) = λ^{-} (0 \cdot y), (by Lemma 2.4) \\ \leq max {λ^{-} (0 \cdot (z^{*} y)), λ^{-} (z)}, (by Definition 3.1) \\ = max {λ^{-} (z^{*} y), λ^{-} (z)}, (by Lemma 2.4) \\ = max {λ^{-} (0), λ^{-} (z)}, (since y \leq z \Rightarrow z^{*} y = 0) \\ = λ^{-} (z) . \\ \Rightarrow λ^{-} (y) \leq λ^{-} (z) . \end{array}

\begin{array}{l} λ^{-} (y) = λ^{-} (0^{*} y), (by Lemma 2.4) \\ \leq max {λ^{-} (0^{*} (z \cdot y)), λ^{-} (z)}, (by Definition 3.1) \\ = max {λ^{-} (z \cdot y), λ^{-} (z)}, (by Lemma 2.4) \\ = max {λ^{-} (0), λ^{-} (z)}, (since y \leq z \Rightarrow z \cdot y = 0) \\ = λ^{-} (z) . \\ \Rightarrow λ^{-} (y) \leq λ^{-} (z) . \end{array}

Hence, λ⁺(y) ≥ λ⁺ (z) and λ⁻(y) ≤ λ⁻ (z). □

Theorem 3.4.

Every bipolar fuzzy pseudo-UP ideal λ = (X, λ⁺, λ⁻) of X is a bipolar fuzzy pseudo- UP subalgebra of X.

Proof.

Suppose λ is a fuzzy pseudo-UP ideal of X. Then,

\begin{array}{l} λ^{+} (x \cdot y) \geq min {λ^{+} (x \cdot (y^{*} y)), λ^{+} (y)}, (by Definition 3.1) \\ = min {λ^{+} (x \cdot 0), λ^{+} (y)}, (by Lemma 2.4) \\ = min {λ^{+} (0), λ^{+} (y)}, (by Lemma 2.4) \\ \geq min {λ^{+} (x), λ^{+} (y)}, (since λ^{+} (0) \geq λ^{+} (x)) \\ \Rightarrow λ^{+} (x \cdot y) \geq min {λ^{+} (x), λ^{+} (y)} . \end{array}

\begin{array}{l} λ^{+} (x^{*} y) \geq min {λ^{+} (x^{*} (y \cdot y)), λ^{+} (y)}, (by Definition 3.1) \\ = min {λ^{+} (x^{*} 0), λ^{+} (y)}, (by Proposition 2.2) \\ = min {λ^{+} (0), λ^{+} (y)}, (by Lemma 2.4) \\ \geq min {λ^{+} (x), λ^{+} (y)} (since λ^{+} (0) \geq λ^{+} (x)) . \\ \Rightarrow λ^{+} (x^{*} y) \geq min {λ^{+} (x), λ^{+} (y)}, and \end{array}

\begin{array}{l} λ^{-} (x \cdot y) \leq max {λ^{-} (x \cdot (y^{*} y)), λ^{-} (y))}, (by Definition 3.1) \\ = max {λ^{-} (x \cdot 0), λ^{-} (y)}, (by Lemma 2.4) \\ = max {λ^{-} (0), λ^{-} (y)}, (by Lemma 2.4) \\ \leq max {λ^{-} (x), λ^{-} (y)} . \\ \Rightarrow λ^{-} (x \cdot y) \leq max {λ^{-} (x), λ^{-} (y)} . \end{array}

\begin{array}{l} Finally, λ^{-} (x^{*} y) \leq max {λ^{-} (x^{*} (y \geq y)), λ^{-} (y)}, (by Definition 3.1) \\ = max {λ^{-} (x^{*} 0), λ^{-} (y)}, (by Proposition 2.2) \\ = max {λ^{-} (0), λ^{-} (y)}, (by Lemma 2.4) \\ \leq max {λ^{-} (x), λ^{-} (y)} . \\ \Rightarrow λ^{-} (x^{*} y) \leq max {λ^{-} (x), λ^{-} (y)} . \end{array}

Thus, λ = (X, λ⁺, λ⁻) is a bipolar fuzzy pseudo-UP subalgebra of pseudo-UP algebra X. □

Remark 3.5.

The converse is may not be true.

Example 3.6.

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

Then, (X, ·, *, 0) is a pseudo-UP algebra. We define a bipolar fuzzy set λ = (X; λ⁺, λ⁻) in X as follows:

X	0	1	2	3
λ⁺	0.8	0.4	0.2	0.1
λ−	-0.9	-0.5	-0.3	-0.2

Then λ = (X; λ⁺, λ⁻) is a bipolar fuzzy pseudo-UP subalgebra of X, but it is not a bipolar fuzzy pseudo-UP ideal of X. Indeed,

λ⁺(0 · 2) ≥ min {λ⁺(0 · (1 * 2)), λ⁺(1)} = λ⁺(2) ≥ min {λ⁺(0 · 1), λ⁺(1)} implies that 0.2 ≥ 0.4 which is contradict to Definition 3.1. And similarly, λ⁻(0 · 2) ≤ max {λ⁻(0 · (1 * 2)), λ⁻(1)} = λ⁻(2) ≤ max {λ⁻(0 · 1), λ⁻(1)} implies that −0.3 ≤ −0.5 which is contradict to Definition 3.1.

Theorem 3.7.

Let λ = (X; λ⁺, λ⁻) be a bipolar fuzzy pseudo-UP ideal of a pseudo-UP algebra X. If the inequality x ≤ y · z and x ≤ y * z holds in X for all x, y, z ∈ X, the λ⁺(z) ≥ min {λ⁺(x), λ⁺(y)} and λ⁻(z) ≤ max {λ⁻(x), λ⁻(y)}.

Proof.

Let x, y, z ∈ X such that x ≤ y * z and x ≤ y · z. Then by the binary relation “≤” defined in X, we have x · (y * z) = 0 and x * (y · z) = 0. By Definition 3.1, we have

(3.1)

λ^{+} (x \cdot z) \geq min {λ^{+} (x \cdot (y^{*} z)), λ^{+} (y)}

And by Theorem 3.4, we have

(3.2)

λ^{+} (z) = λ^{+} (0^{*} z) \geq min {λ^{+} (0^{*} (x \cdot z)), λ^{+} (x)} = min {λ^{+} (x \cdot z), λ^{+} (x)}

By (3.1) and Definition 3.1

(3.3)

λ^{+} (x \cdot z) \geq min {λ^{+} (x^{*} (y \cdot z)), λ^{+} (y)} = min {λ^{+} (0), λ^{+} (y)}

By (3.2) and (3.3), we have

Similarly,

λ^{+} (z) \geq min {λ^{+} (x \cdot z), λ^{+} (x)} \geq min {λ^{+} (y), λ^{+} (x)} = min {λ^{+} (x), λ^{+} (y)} .

(3.4)

λ^{-} (x \cdot z) \leq max {λ^{-} (x \cdot (y^{*} z)), λ^{-} (y)}

And by Theorem 3.4, we have

(3.5)

λ^{-} (z) = λ^{-} (0^{*} z) \leq max {λ^{-} (0^{*} (x \cdot z)), λ^{-} (x)} = max {λ^{-} (x \cdot z), λ^{-} (x)}

By (3.4) and Definition 3.1

(3.6)

λ^{-} (x \cdot z) \leq max {λ^{-} (x^{*} (y \cdot z)), λ^{-} (y)} = max {λ^{-} (0), λ^{-} (y)}

By (3.5) and (3.6), we have

λ^{-} (z) \leq max {λ^{-} (x \cdot z), λ^{-} (x)} \leq max {λ^{-} (y), λ^{-} (x)} = max {λ^{-} (x), λ^{-} (y)} .

Hence, $λ^{+} (z) \geq min {λ^{+} (x), λ^{+} (y)} and λ^{-} (z) \leq max {λ^{-} (x), λ^{-} (y)} .$ □

Theorem 3.8.

The intersection of any two bipolar fuzzy pseudo-UP ideals of a pseudo-UP algebra X is also a bipolar fuzzy pseudo-UP ideal.

Proof.

Let λ = (X; λ⁺, λ⁻) and η = (X; η⁺, η⁻) be two bipolar fuzzy pseudo-UP ideals of X. Then we claim that λ ∩ η is a bipolar fuzzy pseudo-UP ideal of X. Let x, y, z ∈ X. Then

$λ^{+} \cap η^{+} (0) = min {λ^{+} (0), η^{+} (0)} \geq min {λ^{+} (x), η^{+} (x)} = λ^{+} \cap η^{+} (x) .$

And $λ^{-} \cap η^{-} (0) = max {λ^{-} (0), η^{-} (0)} \leq max {λ^{-} (x), η^{-} (x)} = λ^{-} \cap η^{-} (x) .$

\begin{array}{l} Also, λ^{+} \cap η^{+} (x \cdot z) = min {λ^{+} (x \cdot z), η^{+} (x \cdot z)} \\ \geq min {min {λ^{+} (x \cdot (y^{*} z)), λ^{+} (y)}, min {η^{+} (x \cdot (y^{*} z)), η^{+} (y)}} \\ = min {min {λ^{+} (x \cdot (y^{*} z)), η^{+} (x \cdot (y^{*} z))}, min {λ^{+} (y), η^{+} (y)}} \\ = min {λ^{+} \cap η^{+} (x \cdot (y^{*} z)), λ^{+} \cap η^{+} (y)}, and \end{array}

\begin{array}{l} λ^{+} \cap η^{+} (x^{*} z) = min {λ^{+} (x^{*} z), η^{+} (x^{*} z)} \\ \geq min {min {λ^{+} (x^{*} (y \cdot z)), λ^{+} (y)}, min {η^{+} (x^{*} (y \cdot z)), η^{+} (y)}} \\ = min {min {λ^{+} (x^{*} (y \cdot z)), η^{+} (x^{*} (y \cdot z))}, min {λ^{+} (y), η^{+} (y)}} \\ = min {λ^{+} \cap η^{+} (x^{*} (y \cdot z)), λ^{+} \cap η^{+} (y)} . \end{array}

\begin{array}{l} Additionally, λ^{-} \cap η^{-} (x \cdot z) = max {λ^{-} (x \cdot z), η^{-} (x \cdot z)} \\ \leq max {max {λ^{-} (x \cdot (y^{*} z)), λ^{-} (y)}, max {η^{-} (x \cdot (y^{*} z)), η^{-} (y)}} \\ = max {max {λ^{-} (x \cdot (y^{*} z)), η^{-} (x \cdot (y^{*} z))}, max {λ^{-} (y), η^{-} (y)}} \\ = max {λ^{-} \cap η^{-} (x \cdot (y^{*} z)), λ^{-} \cap η^{-} (y)} . \end{array}

\begin{array}{l} And λ^{-} \cap η^{-} (x^{*} z) = max {λ^{-} (x^{*} z), η^{-} (x^{*} z)} \\ \leq max {max {λ^{-} (x^{*} (y \cdot z)), λ^{-} (y)}, max {η^{-} (x^{*} (y \cdot z)), η^{-} (y)}} \\ = max {max {λ^{-} (x^{*} (y \cdot z)), η^{-} (x^{*} (y \cdot z))}, max {λ^{-} (y), η^{-} (y)}} \\ = max {λ^{-} \cap η^{-} (x^{*} (y \cdot z)), λ^{-} \cap η^{-} (y)} . \end{array}

Hence, λ ∩ η is a bipolar fuzzy pseudo-UP ideal of X. □

Corollary 3.9.

The intersection of any set of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra X is also a bipolar fuzzy pseudo-UP ideal.

Remark 3.10.

The union of two bipolar fuzzy pseudo-UP ideals may not be bipolar fuzzy pseudo-UP ideal.

Example 3.11.

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

Clearly, (X, ·, *, 0) is a pseudo-UP algebra. We define a fuzzy set λ⁺: X → [0, 1] as follows, λ⁺(0) = 1, λ⁺(1) = 0.6, λ⁺(2) = 0.4, λ⁺(3) = 0.3 and we define a fuzzy set λ⁻: X → [−1, 0] as follows λ⁻(0) = −1, λ⁻(1) = −0.5, λ⁻(2) = −0.6, λ⁻(3) = −0.4 and a fuzzy set η⁺: X → [0, 1] define as follows, η⁺(0) = 1, η⁺(1) = 0.4, η⁺(2) = 0.5, η⁺(3) = 0.3. and η⁻: X → [−1, 0] define as follows η⁻(0) = −0.9, η⁻(1) = −0.5, η⁻(2) = −0.3, η⁻(3) = −0.2. Now, (λ⁺ ∪ η⁺)(1 · 3) = max {λ⁺(1 · 3), η⁺(1 · 3)} = max {λ⁺(3), η⁺(3)} = max{0.3, 0.3} = 0.3.

Hence, (λ⁺ ∪ η⁺) (1 · 3) = 0.3=.....................................(*).

\begin{array}{l} (λ^{+} \cup η^{+}) (1 \cdot 3) = max {λ^{+} (1 \cdot 3), η^{+} (1 \cdot 3)} \\ \geq max {min {λ^{+} (1 \cdot (2^{*} 3)), λ^{+} (2)}, min {η^{+} (1 \cdot (2^{*} 3)), η^{+} (2)}} \\ = max {min {λ^{+} (1 \cdot 0), η^{+} (1 \cdot 0)}, min {λ^{+} (2), η^{+} (2)}} \\ = max {min {λ^{+} (0), η^{+} (0)}, min {λ^{+} (2), η^{+} (2)}} \\ = max {min {1, 1}, min {0.4, 0.5}} = max {1, 0.5} = 1 . \end{array}

From (*) we get 0.3 ≥ 1 which is contradict to Definition 3.1.

And $(λ^{-} \cup η^{-}) (1 \cdot 3) = min {λ^{-} (1 \cdot 3), η^{-} (1 \cdot 3)} = min {λ^{-} (3), η^{-} (3)} = min {- 0.4, - 0.2} = - 0.4 (^{*} *)$ .

\begin{array}{l} (λ^{-} \cup η^{-}) (1 \cdot 3) = min {λ^{-} (1 \cdot 3), η^{-} (1 \cdot 3)} \\ \leq min {max {λ^{-} (1 \cdot (2^{*} 3)), λ^{-} (2)}, max {η^{-} (1 \cdot (2^{*} 3)), η^{-} (2)}} \\ = min {max {λ^{-} (1 \cdot 0), η^{-} (1 \cdot 0)}, max {λ^{-} (2), η^{-} (2)}} \\ = min {max {λ^{-} (0), η^{-} (0)}, max {λ^{-} (2), η^{-} (2)}} \\ = min {max {- 1, - 0.9}, max {- 0.6, - 0.3}} = min {- 0.9, - 0.3} = - 0.9 . \end{array}

From (**), we get −0.4 ≤ −0.9 which is contradict to Definition 3.1. This shows that the union of any two bipolar fuzzy pseudo-UP ideal of X may not be a bipolar fuzzy pseudo-UP ideal of X.

Theorem 3.12.

A bipolar fuzzy set λ = (X, λ⁺, λ⁻) in X is a bipolar fuzzy pseudo-UP ideal of X

If and only if the fuzzy subset λ⁺ and (λ−)^c are fuzzy pseudo-UP ideals of X.

Proof.

Let λ = (X, λ⁺, λ⁻) be a bipolar fuzzy pseudo-UP algebra of X. We need to show that the fuzzy subset λ⁺ and (λ−)^c are fuzzy pseudo-UP ideals of X. Clearly, λ⁺ is a fuzzy pseudo-UP ideal of X follows from the fact that (X, λ⁺, λ⁻) is a bipolar fuzzy pseudo-UP ideal of X. Now, it remains to show that (λ−)^c is a fuzzy pseudo-UP ideal of X. Let x, y, z ∈ X, then we have (λ−)^c (0) = −1 − λ⁻ (0) ≥ −1 − λ⁻(x) = (λ⁻)^c (x).

\begin{array}{l} Next, {(λ -)}^{c} (x \cdot z) = - 1 - λ^{-} (x \cdot z) \geq - 1 - max {λ^{-} (x \cdot (y^{*} z)), λ^{-} (y)} \\ = min {- 1 - λ^{-} (x \cdot (y^{*} z)), - 1 - λ^{-} (y)} (by Lemma) 2.8 (3) \\ = min {{(λ -)}^{c} (x \cdot (y^{*} z)), {(λ -)}^{c} (y)} . \end{array}

\begin{array}{l} Moreover, {(λ -)}^{c} (x^{*} z) = - 1 - λ^{-} (x^{*} z) \geq - 1 - max {λ^{-} (x^{*} (y \cdot z)), λ^{-} (y)} \\ = min {- 1 - λ^{-} (x^{*} (y \cdot z)), - 1 - λ^{-} (y)} (by Lemma) 2.8 (3) \\ = min {{(λ -)}^{c} (x^{*} (y \cdot z)), {(λ -)}^{c} (y)} . \end{array}

Hence, (λ−)^c is a fuzzy pseudo-UP ideal of X.

Conversely, assume that λ⁺ and (λ−)^c is fuzzy pseudo-UP ideal of X. Then for every x, y, z ∈ X, we get λ⁺ (0) ≥ λ⁺(x) and (λ−)^c (0) ≥ (λ−)^c(x), by Definition 3.1

Now, (λ−)^c (0) ≥ (λ−)^c (x) implies that −1 − λ⁻ (0) ≥ −1 − λ⁻(x), then λ⁻ (0) ≤ λ⁻(x). Now it is suffices to show that λ⁻(x~z) ≤ max {λ⁻(x·(y*z)), λ⁻(y)} and λ⁻(x*z) ≤ max {λ⁻(x*(y ·z)), λ⁻(y)}, for all x, y, z ∈ X.

\begin{array}{l} - 1 - λ^{-} (x \cdot z) = {(λ -)}^{c} (x \cdot z) \geq min {{(λ -)}^{c} (x \cdot (y^{*} z)), {(λ -)}^{c} (y)} \\ = min {- 1 - λ^{-} (x \cdot (y^{*} z)), - 1 - λ^{-} (y)} \\ = - 1 - max {λ^{-} (x \cdot (y^{*} z)), λ^{-} (y)}, (by Lemma) 2.8 \\ \Rightarrow λ^{-} (x \cdot z) \leq max {λ^{-} (x \cdot (y^{*} z)), λ^{-} (y)} . \end{array}

\begin{array}{l} Finally, - 1 - λ^{-} (x^{*} z) = {(λ^{-})}^{c} (x^{*} z) \geq min {{(λ^{-})}^{c} x^{*} (y \cdot z), {(λ^{-})}^{c} (y)} \\ = min {- 1 - λ^{-} (x^{*} (y \cdot z)), - 1 - λ^{-} (y)} \\ = - 1 - max {λ^{-} (x^{*} (y \cdot z)), λ^{-} (y)}, \\ \Rightarrow λ^{-} (x^{*} z) \leq max {λ^{-} (x^{*} (y \cdot z)), λ^{-} (y)} . (by Lemma) 2.8 \end{array}

Hence, λ = (X, λ⁺, λ⁻) is a bipolar fuzzy pseudo-UP ideal of X. □

Corollary 3.13.

If λ⁺ is a fuzzy pseudo-UP ideal of X, then λ = (X, λ⁺, ((λ⁺)^c) is a bipolar fuzzy pseudo-UP ideal of X.

Proof.

Suppose λ⁺ is a fuzzy pseudo-UP ideal of X. Then we need to show that λ = (X, λ⁺, (λ−)^c) is a bipolar fuzzy pseudo-UP ideal of X. Since λ⁺ is a fuzzy pseudo-UP ideal of X, it follows that λ⁺ (0) ≥ λ⁺(x), λ⁺(x· z) ≥ min {λ⁺(x· (y *z)), λ⁺(y)} and λ⁺(x*z) ≥ min {λ⁺(x*(y · z)), λ⁺(y)}, for all x, y, z ∈ X. Then it is enough to show that (λ⁺)^c (0) ≥ (λ⁺)^c (x), (λ⁺)^c (x · z) ≥ min{(λ⁺)^c (x · (y * z)), (λ⁺)^c (y)} and (λ⁺)^c (x * z) ≥ min{(λ⁺)^c (x * (y · z)), (λ⁺)^c (y)}.

Now, {(λ^{+})}^{c} (0) = 1 - λ^{+} (0) \leq 1 - λ^{+} (x) = {(λ^{+})}^{c} (x) \Rightarrow {(λ^{+})}^{c} (0) \leq {(λ^{+})}^{c} (x) and

\begin{array}{l} {(λ^{+})}^{c} (x \cdot z) = 1 - λ^{+} (x \cdot z) \leq 1 - min {λ^{+} (x \cdot (y^{*} z)), λ^{+} (y)} \\ = max {1 - λ^{+} (x \cdot (y^{*} z)), 1 - λ^{+} (y)} \\ = max {{(λ^{+})}^{c} (x \cdot (y^{*} z)), {(λ^{+})}^{c} (y)} . \\ \Rightarrow {(λ^{+})}^{c} (x \cdot z) \leq max {{(λ^{+})}^{c} (x \cdot (y^{*} z)), {(λ^{+})}^{c} (y)} . \end{array}

\begin{array}{l} Finally {(λ^{+})}^{c} (x^{*} z) = 1 - λ^{+} (x^{*} z) \leq 1 - min {λ^{+} (x^{*} (y . z)), λ^{+} (y)} \\ = max {1 - λ^{+} (x^{*} (y . z)), 1 - λ^{+} (y)} \\ = max {{(λ^{+})}^{c} (x^{*} (y . z)), {(λ^{+})}^{c} (y)} . \\ \Rightarrow {(λ^{+})}^{c} (x \cdot z) \leq max {{(λ^{+})}^{c} (x^{*} (y . z)), {(λ^{+})}^{c} (y)} . \end{array}

Therefore, λ = (X, λ⁺, (λ⁺)^c) is a bipolar fuzzy pseudo-UP ideal of X. □

Corollary 3.14.

If (λ−)^c is a fuzzy pseudo-UP ideal of X, then λ = (X, (λ⁻)^c), λ⁻ is a bipolar fuzzy pseudo-UP ideal of X.

Proof.

Similar to Corollary 3.13.

Using those, corollary the following is obtained.

Theorem 3.15.

A bipolar fuzzy set λ = (X, λ⁺, λ⁻) of X is a bipolar fuzzy pseudo-UP ideal of X if and only if λ = (X, λ⁺, (λ⁺)^c) and ♦λ = (X, (λ⁻)^c, λ⁻) are bipolar fuzzy pseudo-UP ideals of X.

Proof.

Suppose λ = (X, λ⁺, λ⁻) is bipolar fuzzy pseudo-UP ideal of X, then for any x, y, z ∈ X, we have λ⁺(0) ≥ λ⁺(x), λ⁺(x·z) ≥ min {λ⁺(x·(y*z)), λ⁺(y)} and λ⁺(x*z) ≥ min {λ⁺(x*(y·z)), λ⁺(y)}. Next we have to show that (λ⁺)^c satisfies the conditions such that (λ⁺)^c (0) ≤ (λ⁺)^c (x), (λ⁺)^c (x·z) ≤ max{(λ⁺)^c (x · (y * z)), (λ⁺)^c (y)} and (λ⁺)^c (x * z) ≤ max{(λ⁺)^c (x * (y · z)), (λ⁺)^c (y)}.

Now, {(λ^{+})}^{c} (0) = 1 - λ^{+} (0) \leq 1 - λ^{+} (x) = {(λ^{+})}^{c} (x) \Rightarrow {(λ^{+})}^{c} (0) \leq {(λ^{+})}^{c} (x) .

\begin{array}{l} Next, {(λ^{+})}^{c} (x \cdot z) = 1 - λ^{+} (x \cdot z) \leq 1 - min {λ^{+} (x \cdot (y^{*} z)), λ^{+} (y)} \\ = max {1 - λ^{+} (x \cdot (y^{*} z)), 1 - λ^{+} (y)} \\ = max {{(λ^{+})}^{c} (x \cdot (y^{*} z)), {(λ^{+})}^{c} (y)} . \\ \Rightarrow {(λ^{+})}^{c} (x \cdot z) \leq max {{(λ^{+})}^{c} (x \cdot (y^{*} z)), {(λ^{+})}^{c} (y)} . \end{array}

\begin{array}{l} And also {(λ^{+})}^{c} (x^{*} z) = 1 - λ^{+} (x^{*} z) \leq 1 - min {λ^{+} (x^{*} (y . z)), λ^{+} (y)} \\ = max {1 - λ^{+} (x^{*} (y . z)), 1 - λ^{+} (y)} \\ = max {{(λ^{+})}^{c} (x^{*} (y . z)), {(λ^{+})}^{c} (y)} . \\ \Rightarrow {(λ^{+})}^{c} (x^{*} z) \leq max {{(λ^{+})}^{c} (x^{*} (y . z)), {(λ^{+})}^{c} (y)} . \end{array}

Hence, λ is a bipolar fuzzy pseudo-UP ideal of X.

Also, for any x, y, z ∈ X, λ⁻ (0) ≤ λ⁻(x), λ⁻(x · z) ≤ max {λ⁻(x · (y * z)), λ⁻(y)} and λ⁻(x * z) ≥ max {λ⁻(x * (y · z)), λ⁻(y)}. Now we have to show that (λ⁺)^c (0) ≥ (λ⁺)^c (x), (λ⁺)^c (x · z) ≥ min {(λ⁺)^c (x · (y * z)), (λ⁺)^c (y)} and (λ⁺)^c (x * z) ≤ min {(λ⁺)^c (x * (y · z)), (λ⁺)^c (y)}. So for any x, y, z ∈ X, we have (λ^-)^c (0) = −1 − λ⁻(0) ≥ −1 − λ⁻(x) = (λ^-)^c (x) ⇒ (λ⁻)^c (0) ≥ (λ⁻)^c (x).

\begin{array}{l} Next, {(λ^{-})}^{c} ((x \cdot z) = - 1 - λ^{-} (x \cdot z) \geq - 1 - max {λ^{-} (x \cdot (y^{*} z)), λ^{-} (y)} \\ = min {- 1 - λ^{-} (x \cdot (y^{*} z)), - 1 - λ^{-} (y)} \\ = min {{((λ^{-})}^{c} (x \cdot (y^{*} z)), {(λ^{-})}^{c} (y)} . \\ \Rightarrow {(λ^{-})}^{c} (x \cdot z) \geq min {{(λ^{-})}^{c} (x \cdot (y^{*} z)), {(λ^{-})}^{c} (y)} . \end{array}

\begin{array}{l} Finally, {(λ^{-})}^{c} (x^{*} z) = - 1 - λ^{-} (x^{*} z) \geq - 1 - max {λ^{-} (x^{*} (y \cdot z)), λ^{-} (y)} \\ = min {- 1 - λ^{-} (x^{*} (y \cdot z)), - 1 - λ^{-} (y)} \\ = min {{(λ^{-})}^{c} (x^{*} (y \cdot z)), {(λ^{-})}^{c} (y)} . \\ \Rightarrow {(λ^{-})}^{c} (x^{*} z) \geq min {{(λ^{-})}^{c} (x^{*} (y \cdot z)), {(λ^{-})}^{c} (y)} . \end{array}

Hence, ♦λ is a bipolar fuzzy pseudo-UP ideal of X. The proof of the converse follows from Definition 3.1.□

Theorem 3.16.

If λ = (X, λ⁺, λ⁻) is a bipolar fuzzy pseudo-UP ideal of X, then λ^c is also a bipolar fuzzy pseudo-UP ideal of X.

Proof.

Let λ = (X, λ⁺, λ⁻) be a bipolar fuzzy pseudo-UP ideal of X. Then λ^c = (X, (λ⁺)^c (λ⁻)^c), where (λ⁺)^c (x) = 1 − λ⁺(x) and (λ⁻)^c (x) = −1 − λ⁻(x). Now, for any x, y, z ∈ X, we have (λ⁻)^c (0) = −1−λ⁻(0) ≥ −1−λ⁻(x) = (λ⁻(λ⁻)^c (x) ⇒ (λ⁻)^c (0) ≥ (λ⁻)^c (x) and (λ⁺)^c 0) = 1−λ⁺(0) ≤ 1 − λ⁺(x) = (λ⁺)^c (x) ⇒ (λ⁺)^c (0) ≤ (λ⁺)^c (x).

\begin{array}{l} Next, {(λ^{-})}^{c} (x \cdot z) = - 1 - λ^{-} (x \cdot z) \geq - 1 - max {λ^{-} (x \cdot (y^{*} z)), λ^{-} (y)} \\ = min {- 1 - λ^{-} (x \cdot (y^{*} z)), - 1 - λ^{-} (y)} \\ = min {{(λ^{-})}^{c} (x \cdot (y^{*} z)), {(λ^{-})}^{c} (y)} . \\ \Rightarrow {(λ^{-})}^{c} (x \cdot z) \geq min {{(λ^{-})}^{c} (x \cdot (y^{*} z)), {(λ^{-})}^{c} (y)} . \end{array}

\begin{array}{l} Finally, {(λ^{-})}^{c} (x^{*} z) = - 1 - λ^{-} (x^{*} z) \geq - 1 - max {λ^{-} (x^{*} (y \cdot z)), λ^{-} (y)} \\ = min {- 1 - λ^{-} (x^{*} (y \cdot z)), - 1 - λ^{-} (y)} \\ = min {{(λ^{-})}^{c} (x^{*} (y \cdot z)), {(λ^{-})}^{c} (y)} . \\ \Rightarrow {(λ^{-})}^{c} (x^{*} z) \geq min {{(λ^{-})}^{c} (x^{*} (y \cdot z)), {(λ^{-})}^{c} (y)} . \end{array}

\begin{array}{l} {(λ^{-})}^{c} (x \cdot z) = 1 - λ^{+} (x \cdot z) \leq 1 - min {λ^{+} (x \cdot (y^{*} z)), λ^{+} (y)} \\ = max {1 - λ^{+} (x \cdot (y^{*} z)), 1 - λ^{+} (y)} \\ = max {{(λ^{-})}^{c} (x \cdot (y^{*} z)), {(λ^{-})}^{c} (y)} . \\ \Rightarrow {(λ^{+})}^{c} (x \cdot z) \leq max {{(λ^{+})}^{c} (x \cdot (y^{*} z)), {(λ^{+})}^{c} (y)} . \end{array}

\begin{array}{l} And also, {(λ^{+})}^{c} (x^{*} z) = 1 - λ^{+} (x^{*} z) \leq 1 - min {λ^{+} (x^{*} (y \cdot z)), λ^{+} (y)} \\ = max {1 - λ^{+} (x^{*} (y \cdot z)), 1 - λ^{+} (y)} \\ = max {{(λ^{+})}^{c} (x^{*} (y \cdot z)), {(λ^{+})}^{c} (y)} . \\ \Rightarrow {(λ^{+})}^{c} (x^{*} z) \leq max {{(λ^{+})}^{c} (x^{*} (y \cdot z)), {(λ^{+})}^{c} (y)} . \end{array}

Therefore, λ^c is a bipolar fuzzy pseudo-UP ideal of X. □

Table 1. Bipolar fuzzy Pseudo-UP ideal.

·	a	b	c
0	a	b	c
a	0	b	c
b	a	0	c
c	a	0	0
*	a	b	c
0	a	b	c
a	0	b	c
b	a	0	c
c	a	b	0

Table 2. Bipolar fuzzy Pseudo-UP subalgebras is not bipolar fuzzy Pseudo-UP ideal.

·	1	2	3
0	1	2	3
1	0	1	3
2	0	0	3
3	0	0	0
*	1	2	3
0	1	2	3
1	0	1	3
2	0	0	3
3	0	2	0

Table 3. The union of bipolar fuzzy Pseudo-UP ideals is not bipolar fuzzy Pseudo-UP ideal.

·	1	2	3
0	1	2	3
1	0	3	3
2	1	0	0
3	1	3	0
*	1	2	3
0	1	2	3
1	0	3	3
2	1	0	0
3	2	3	0

Homomorphism on bipolar fuzzy pseudo-UP ideal

In this section, we explore bipolar fuzzy pseudo-UP ideals in the context of homomorphism. We examine the homomorphic images and inverse images of bipolar fuzzy pseudo-UP ideals in pseudo-UP algebras, and present some key findings.

Theorem 4.1.

Let f: X→Y is an epimorphism of pseudo-UP algebras. If λ = (X, λ⁺, λ⁻) is a bipolar fuzzy pseudo-UP ideal of X with sup-inf property, then the image f (λ) is a bipolar pseudo-UP ideal of Y.

Proof.

Assume that λ = (X, λ⁺, λ⁻) is a bipolar fuzzy pseudo-UP ideal of X with sup-inf property. Let x, y, z ∈ Y with a ∈ f ⁻¹(x), b ∈ f ⁻¹(y) and c ∈ f ⁻¹ (z).

λ^{+} (a) = sup_{t \in f^{- 1} (x)} λ^{+} (t), λ^{+} (b) = sup_{t \in f^{- 1} (y)} λ^{+} (t) and λ^{+} (c) = sup_{t \in f^{- 1} (z)} λ^{+} (t)

λ^{-} (a) = inf_{t \in f^{- 1} (x)} λ^{-} (t), λ^{-} (b) = inf_{t \in f^{- 1} (y)} λ^{-} (t) and λ^{-} (c) = inf_{t \in f^{- 1} (z)} λ^{-} (t)

Then by Definition 2.7, we have

(f) (λ^{+}) (0) = sup_{t \in f^{- 1} (0)} λ^{+} (t) \geq λ^{+} (0) \geq λ^{+} (a) = sup_{t \in f^{- 1} (x)} λ^{+} (t) = (f) (λ^{+}) (x)

(f) (λ^{-}) (0) = inf_{t \in f^{- 1} (0)} λ^{-} (t) \geq λ^{-} (0) \geq λ^{-} (a) = inf_{t \in f^{- 1} (x)} λ^{-} (t) = (f) (λ^{-}) (x)

\begin{array}{l} (f) (λ^{+}) (x \cdot z) = sup_{t \in f^{- 1} (x \cdot z)} λ^{+} (t) = λ^{+} (a \cdot c) \\ \geq min {λ^{+} (a \cdot (b^{*} c)), λ^{+} (b)} \\ = min {sup_{t \in f^{- 1} (x \cdot (y^{*} z))} λ^{+} (t), sup_{t \in f^{- 1} (y)} λ^{+} (t)} \\ = min {(f) (λ^{+}) (x \cdot (y^{*} z)), (f) (λ^{+}) (y)} . \\ \Rightarrow (f) (λ^{+}) (x \cdot z) \geq min {(f) (λ^{+}) (x \cdot (y^{*} z)), (f) (λ^{+}) (y)} . \end{array}

\begin{array}{l} And also, (f) (λ^{+}) (x^{*} z) = sup_{t \in f^{- 1} (x^{*} z)} λ^{+} (t) = λ^{+} (a^{*} c) \\ \geq min {λ^{+} (a^{*} (b \cdot c)), λ^{+} (b)} \\ = min {sup_{t \in f^{- 1} (x^{*} (y \cdot z))} λ^{+} (t), sup_{t \in f^{- 1} (y)} λ^{+} (t)} \\ = min {(f) (λ^{+}) (x^{*} (y \cdot z)), (f) (λ^{+}) (y)} . \\ \Rightarrow (f) (λ^{+}) (x^{*} z) \geq min {(f) (λ^{+}) (x^{*} (y \cdot z)), (f) (λ^{+}) (y)} . \end{array}

\begin{array}{l} (f) (λ^{-}) (x \cdot z) = inf_{t \in f^{- 1} (x \cdot z)} λ^{-} (t) = λ^{-} (a \cdot c) \\ \leq max {λ^{-} (a \cdot (b^{*} c)), λ^{-} (b)} \\ = max {inf_{t \in f^{- 1} (x \cdot (y^{*} z))} λ^{-} (t), inf_{t \in f^{- 1} (y)} λ^{-} (t)} \\ = max {(f) (λ^{-}) (x \cdot (y^{*} z)), (f) (λ^{-}) (y)} . \\ \Rightarrow (f) (λ^{-}) (x \cdot z) \leq max {(f) (λ^{-}) (x \cdot (y^{*} z)), (f) (λ^{-}) (y)} . \end{array}

Finally, $(f) (λ^{-}) (x^{*} z) \leq max {(f) (λ^{-}) (x^{*} (y \cdot z)), (f) (λ^{-}) (y)} .$

Therefore, f (λ) is a bipolar fuzzy pseudo-UP ideal of Y. □

Theorem 4.2.

Let f be a homomorphism on pseudo-UP algebra X onto Y and η is a bipolar fuzzy pseudo-UP ideal of Y. Then f ⁻¹ (η) is a bipolar fuzzy pseudo-UP ideal of X.

Proof.

Let f be a homomorphism of pseudo-UP algebra. Assume that η is a bipolar pseudo- UP ideal of Y and let x ∈ X. Then f ⁻¹(η⁺)(0) = η⁺(f(0)) ≥ η⁺(f(x)) = f ⁻¹(η⁺)(x) and f ⁻¹(η⁻)(0) = η⁻(f(0)) ≤ η⁻(f(x)) = f ⁻¹(η⁻)(x). Let x, y, z ∈ X. Then

\begin{array}{l} f^{- 1} (η^{+}) (x \cdot z) = η^{+} (f (x \cdot z)) = η^{+} (f (x) \cdot f (z)) \\ \geq min {η^{+} (f (x) \cdot (f (y) * f (z)), η^{+} (f (y))} \\ = min {η^{+} (f (x \cdot (y^{*} z)), η^{+} (f (y))} \\ = min {f^{- 1} (η^{+} (x \cdot (y^{*} z))), f^{- 1} (η^{+} (y))} . \\ \Rightarrow f^{- 1} (η^{+}) (x \cdot z) \geq min {f^{- 1} (η^{+} (x \cdot (y^{*} z))), f^{- 1} (η^{+} (y))} . \end{array}

\begin{array}{l} And f^{- 1} (η^{+}) (x^{*} z) = η^{+} (f (x^{*} z)) = η^{+} (f (x) * f (z)) \\ \geq min {η^{+} (f (x) * (f (y) \cdot f (z)), η^{+} (f (y))} \\ = min {η^{+} (f (x^{*} (y \cdot z)), η^{+} (f (y))} \\ = min {f^{- 1} (η^{+} (x^{*} (y \cdot z))), f^{- 1} (η^{+} (y))} . \end{array}

\begin{array}{l} And f^{- 1} (η^{-}) (x \cdot z) = η^{-} (f (x \cdot z)) = η^{-} (f (x) \cdot f (z)) \\ \leq max {η^{-} (f (x) \cdot (f (y) * f (z)), η^{-} (f (y))} \\ = max {η^{-} (f (x \cdot (y^{*} z)), η^{-} (f (y))} \\ = max {f^{- 1} (η^{-} (x \cdot (y^{*} z))), f^{- 1} (η^{-} (y))} . \end{array}

Again, $f^{- 1} (η^{-}) (x^{*} z) \leq max {f^{- 1} (η^{-} (x^{*} (y \cdot z))), f^{- 1} (η^{-} (y))} .$

Hence, f ⁻¹ (η) is a bipolar fuzzy pseudo-UP ideal of X.

The next theorem shows that the converse of the above Theorem 4.2 is also true if f is an epimorphism of pseudo-UP algebra. □

Theorem 4.3.

Let f: X →Y be an epimorphism of pseudo-UP algebras. Let η be a bipolar fuzzy subset of Y. If f ⁻¹ (η) is a bipolar fuzzy pseudo-UP ideal of X, then η is a bipolar fuzzy pseudo-UP ideal of Y.

Proof.

Assume that f is an epimorphism of pseudo-UP algebra and f ⁻¹ (η) is a bipolar fuzzy pseudo- UP ideal of X. We need to show that η is a bipolar fuzzy pseudo-UP ideal of Y. Since f is an epimorphism of pseudo-UP algebra for any x ∈ Y, there exist a ∈ X such that f (a) = x.

Then $η^{+} (0) = η^{+} (f (0)) = f^{- 1} (η^{+}) (0)) \geq f^{- 1} (η^{+}) (a) = η^{+} (f (a)) = η^{+} (x) .$

$η^{-} (0) = η^{-} (f (0)) = f^{- 1} (η^{-}) (0)) \leq f^{- 1} (η^{-}) (a) = η^{-} (f (a)) = η^{-} (x) .$

Let x, y, z ∈ Y. Then f (a) = x, f (b) = y and f(c) = z, for all a, b, c ∈ X. Thus

\begin{array}{l} η^{+} (x \cdot z) = η^{+} (f (x) \cdot f (z)) = η^{+} (f (a \cdot c)) \\ = f^{- 1} (η^{+}) (a \cdot c) \\ \geq min {f^{- 1} (η^{+}) (a \cdot (b^{*} c)), f^{- 1} (η^{+}) (b)} \\ = min {η^{+} (f (a \cdot (b^{*} c)), η^{+} (f (b))} \\ = min {η^{+} (f (a) \cdot (f (b) * f (c))), η^{+} (f (b))} \\ = min {η^{+} (x \cdot (y^{*} z)), η^{+} (y)} . \end{array}

\begin{array}{l} η^{+} (x^{*} z) = η^{+} (f (x) \cdot f (z)) = η^{+} (f (a^{*} c)) \\ = f^{- 1} (η^{+}) (a^{*} c) \\ \geq min {f^{- 1} (η^{+}) (a^{*} (b \cdot c)), f^{- 1} (η^{+}) (b)} \\ = min {η^{+} (f (a^{*} (b \cdot c)), η^{+} (f (b))} \\ = min {η^{+} (f (a) * (f (b) \cdot f (c))), η^{+} (f (b))} \\ = min {η^{+} (x^{*} (y \cdot z)), η^{+} (y)} . \\ And \end{array}

\begin{array}{l} η^{-} (x \cdot z) = η^{-} (f (x) \cdot f (z)) = η^{-} (f (a \cdot c)) \\ = f^{- 1} (η^{-}) (a \cdot c) \\ \leq max {f^{- 1} (η^{-}) (a \cdot (b^{*} c)), f^{- 1} (η^{-}) (b)} \\ = max {η^{-} (f (a \cdot (b^{*} c)), η^{-} (f (b))} \\ = max {η^{-} (f (a) \cdot (f (b) * f (c))), η^{-} (f (b))} \\ = max {η^{-} (x \cdot (y^{*} z)), η^{-} (y)} . \end{array}

Similarly, $η^{-} (x \cdot z) \leq max {η^{-} (x \cdot (y^{*} z)), η^{-} (y)}$ .

Therefore, η is a bipolar fuzzy pseudo-UP ideal of Y. □

Cartesian products of bipolar fuzzy pseudo-UP ideal

In this section, we discuss the Cartesian product of two bipolar fuzzy pseudo-UP ideal is also again a bipolar fuzzy pseudo-UP ideal and some other results are also investigated.

Theorem 5.1.

Let λ = (X, λ⁺, λ⁻) and σ = (Y, σ⁺, σ⁻) be any two bipolar fuzzy pseudo-UP ideals of X and Y respectively. Then λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y.

Proof.

Assume that λ and σ be any two bipolar fuzzy pseudo-UP ideals of X and Y respectively.

Take (x, y) ∈ X × Y. Then

\begin{array}{l} λ^{+} \times σ^{+} (0, 0) = min {λ^{+} (0), σ^{+} (0)} \\ \geq min {λ^{+} (x), σ^{+} (y)} \\ = λ^{+} \times σ^{+} (x, y) . \end{array}

\begin{array}{l} And λ^{-} \times σ^{-} (0, 0) = max {λ^{-} (0), σ^{-} (0)} \\ \leq max {λ^{-} (x), σ^{-} (y)} \\ = λ^{-} \times σ^{-} (x, y) . \end{array}

Let (x₁, x₂), (y₁, y₂), (z₁, z₂) ∈ X × Y. Put x = (x₁, x₂), y = (y₁, y₂), z = (z₁, z₂). Then,

\begin{array}{l} λ^{+} \times σ^{+} (x \cdot z) = λ^{+} \times σ^{+} ((x_{1}, x_{2}) \cdot (z_{1}, z_{2})) \\ = λ^{+} \times σ^{+} (x_{1} \cdot z_{1}, x_{2} \cdot z_{2}) \\ = min {λ^{+} (x_{1} \cdot z_{1}), σ^{+} (x_{2}, z_{2})} \\ \geq min {min {λ^{+} (x_{1} \cdot (y_{1} * z_{1})), λ^{+} (y_{1})}, min {σ^{+} (x_{2} \cdot (y_{2} * z_{2})), σ^{+} (y_{2})}} \\ = min {min {λ^{+} (x_{1} \cdot (y_{1} * z_{1})), σ^{+} (x_{2} \cdot (y_{2} * z_{2}))}, min {λ^{+} (y_{1}), σ^{+} (y_{2})}} \\ = min {λ^{+} \times σ^{+} ((x_{1} \cdot (y_{1} * z_{1})), (x_{2} \cdot (y_{2} * z_{2})), λ^{+} \times σ^{+} (y_{1}, y_{2})} \\ = min {λ^{+} \times σ^{+} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * ((z_{1}, z_{2}))), λ^{+} \times σ^{+} (y_{1}, y_{2})} \\ = min {λ^{+} \times σ^{+} (x \cdot (y * z)), λ^{+} \times σ^{+} (y)} . \end{array}

And, $λ^{+} \times σ^{+} (x^{*} z) \geq min {λ^{+} \times σ^{+} (x \cdot (y^{*} z)), λ^{+} \times σ^{+} (y)}$ .

And,

\begin{array}{l} λ^{-} \times σ^{-} (x \cdot z) = λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot (z_{1}, z_{2})) \\ = λ^{-} \times σ^{-} (x_{1} \cdot z_{1}, x_{2} \cdot z_{2}) \\ = max {λ^{-} (x_{1} \cdot z_{1}), σ^{-} (x_{2}, z_{2})} \\ \leq max {max {λ^{-} (x_{1} \cdot (y_{1} * z_{1})), λ^{-} (y_{1})}, max {σ^{-} (x_{2} \cdot (y_{2} * z_{2})), σ^{-} (y_{2})}} \\ = max {max {λ^{-} (x_{1} \cdot (y_{1} * z_{1})), σ^{-} (x_{2} \cdot (y_{2} * z_{2}))}, max {λ^{-} (y_{1}), σ^{-} (y_{2})}} \\ = max {λ^{-} \times σ^{-} ((x_{1} \cdot (y_{1} * z_{1})), (x_{2} \cdot (y_{2} * z_{2})), λ^{-} \times σ^{-} (y_{1}, y_{2})} \\ = max {λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * ((z_{1}, z_{2}))), λ^{-} \times σ^{-} (y_{1}, y_{2})} \\ = max {λ^{-} \times σ^{-} (x \cdot (y * z)), λ^{-} \times σ^{-} (y)} . \end{array}

Additionally, $λ^{-} \times σ^{-} (x^{*} z) \leq max {λ^{-} \times σ^{-} (x \cdot (y^{*} z)), λ^{-} \times σ^{-} (y)} .$

Hence, λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. □

Lemma 5.2.

Let λ and σ be two bipolar fuzzy subsets of X and Y. If λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. Then the followings are true.

1) λ⁺(0) ≥ σ⁺(y) and σ⁺(0) ≥ λ⁺(x),
2) λ⁻ (0) ≤ σ⁻(y) and σ⁻ (0) ≤ λ⁻(x).

Proof.

1) Assume σ⁺(y) ≥ λ⁺ (0) and λ⁺(x) ≥ σ⁺ (0), ∀x ∈ X, y ∈ Y

\begin{array}{l} Then, λ^{+} \times σ^{+} (x, y) = min {λ^{+} (x), σ^{+} (y)} \\ \geq min {λ^{+} (0), σ^{+} (0)} \\ = λ^{+} \times σ^{+} (0, 0) which is a contradiction . \end{array}

2) Similarly, $λ^{-} (x) \leq σ^{-} (0) and σ^{-} (y) \leq λ^{-} (y), \forall x \in X, y \in Y$

\begin{array}{l} Then, λ^{-} \times σ^{-} (x, y) = max {λ^{-} (x), σ^{-} (y)} \\ \leq max {λ^{-} (0), σ^{-} (0)} \\ = λ^{-} \times σ^{-} (0, 0) which is a contradiction . \end{array}

Thus proving the result. □

Theorem 5.3.

If λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y for any two bipolar fuzzy subsets λ and σ of X and Y respectively, then either λ is a bipolar fuzzy pseudo-UP ideal of X or σ is a bipolar fuzzy pseudo-UP ideal of Y.

Proof.

Let λ and σ be any two bipolar fuzzy subsets of pseudo-UP algebra X and Y respectively such that λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. Then λ⁺ × σ⁺ (0, 0) ≥ λ⁺ × σ⁺(x, y). Now by Lemma 5.2, let λ⁺ (0) ≥ λ⁺(x) or σ⁺ (0) ≥ σ⁺(y) and λ⁻ (0) ≤ λ⁻(x) or σ⁻ (0) ≤ σ⁻(y).

Let (x₁, x₂), (y₁, y₂), (z₁, z₂) ∈ X × Y. Put x = (x₁, x₂), y = (y₁, y₂), z = (z₁, z₂).

\begin{array}{l} Then λ^{+} \times σ^{+} (x \cdot z) \geq min {λ^{+} \times σ^{+} (x \cdot (y^{*} z)), λ^{+} \times σ^{+} (y)} \\ = min {λ^{+} \times σ^{+} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2})), λ^{+} \times σ^{+} (y_{1}, y_{2})} \\ = min {λ^{+} \times σ^{+} ((x_{1} \cdot ({y_{1}}^{*} z_{1}), (x_{2} \cdot ({y_{2}}^{*} z_{2})) σ^{+} \times σ^{+} (y_{1}, y_{2})} \\ = min {min {λ^{+} ({x_{1}}^{*} ({y_{1}}^{*} z_{1})), σ^{+} (x_{2} \cdot ({y_{2}}^{*} z_{2}))}, min {λ^{+} (y_{1}), σ^{+} (y_{2})}} \end{array}

\begin{array}{l} λ^{+} \times σ^{+} (x_{1} \cdot z_{1}, x_{2} \cdot z_{2}) \geq min {min {λ^{+} ({x_{1}}^{*} ({y_{1}}^{*} z_{1})), σ^{+} (x_{2} \cdot ({y_{2}}^{*} z_{2}))}, min {λ^{+} (y_{1}), σ^{+} (y_{2})}} \\ = min {min {λ^{+} ({x_{1}}^{*} ({y_{1}}^{*} z_{1}), λ^{+} (y_{1})}, min {σ^{+} (x_{2} \cdot ({y_{2}}^{*} z_{2})), σ^{+} (y_{2})}} . \end{array}

⇒ either λ⁺(x₁ · z₁) ≥ min {λ⁺(x₁ · (y₁ *z₁)), λ⁺ (y₁)} or σ⁺(x₂ · z₂) ≥ min {σ⁺(x₂ · (y₂ *z₂)), σ⁺ (y₂)}. And either λ⁺(x₁*z₁) ≥ min {λ⁺(x₁*(y₁·z₁)), λ⁺ (y₁)} or σ⁺(x₂*z₂) ≥ min {σ⁺(x₂*(y₂·z₂)), σ⁺ (y₂)}. Similarly we can show that either λ⁻(x₁ · z₁) ≤ max {λ⁻(x₁ · (y₁ * z₁)), λ⁻(y₁)} or σ⁻(x₂ · z₂) ≤ max {σ⁻(x₂ · (y₂ * z₂)), σ⁻(y₂)} and λ⁻(x₁ * z₁) ≤ max {λ⁻(x₁ * (y₁ · z₁)), λ⁻(y₁)} or σ⁻(x₂ * z₂) ≤ max {σ⁻(x₂ * (y₂ · z₂)), σ⁻(y₂)}. Hence, either λ is a bipolar fuzzy pseudo-UP ideal of X or σ is a bipolar fuzzy pseudo-UP ideal of Y. □

Theorem 5.4.

Let λ and σ be any bipolar fuzzy subsets of X and Y respectively. Then λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y if and only If λ⁺ × σ⁺ and (λ⁻ × σ⁻)^care fuzzy pseudo-UP ideals of X × Y.

Proof.

Let λ and σ be any two bipolar fuzzy subsets of X and Y respectively such that λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. Now we need to show that λ⁺ × σ⁺ and (λ⁻ × σ⁻)^c are fuzzy pseudo-UP ideals of X × Y. Clearly, λ⁺ × σ⁺ is a fuzzy pseudo-UP ideal of X × Y. So it remains to show that (λ⁻ × σ⁻)^c is a fuzzy pseudo-UP ideal of X × Y.

Now, for every (x₁, x₂), (y₁, y₂), (z₁, z₂) ∈ X × Y.

{(λ^{-} \times σ^{-})}^{c} (0, 0) = - 1 - λ^{-} \times σ^{-} (0, 0) \geq - 1 - λ^{-} \times σ^{-} (x_{1}, x_{2}) = {(λ^{-} \times σ^{-})}^{c} (x_{1}, x_{2}) .

\begin{array}{l} {(λ^{-} \times σ^{-})}^{c} ((x_{1}, x_{2}) \cdot ((z_{1}, z_{2})) = - 1 - λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot ((z_{1}, z_{2})) \\ \geq - 1 - max {λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2}))), λ^{-} \times σ^{-} (y_{1}, y_{2})} \\ = min {- 1 - λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2}))), - 1 - λ^{-} \times σ^{-} (y_{1}, y_{2})} \\ = min {{(λ^{-} \times σ^{-})}^{c} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2}))), {(λ^{-} \times σ^{-})}^{c} (y_{1}, y_{2})} . \end{array}

\begin{array}{l} {(λ^{-} \times σ^{-})}^{c} ((x_{1}, x_{2}) * ((z_{1}, z_{2})) = - 1 - λ^{-} \times σ^{-} ((x_{1}, x_{2}) * ((z_{1}, z_{2})) \\ \geq - 1 - max {λ^{-} \times σ^{-} ((x_{1}, x_{2}) * ((y_{1}, y_{2}) \cdot (z_{1}, z_{2}))), λ^{-} \times σ^{-} (y_{1}, y_{2})} \\ = min {- 1 - λ^{-} \times σ^{-} ((x_{1}, x_{2}) * ((y_{1}, y_{2}) \cdot (z_{1}, z_{2}))), - 1 - λ^{-} \times σ^{-} (y_{1}, y_{2})} \\ = min {{(λ^{-} \times σ^{-})}^{c} ((x_{1}, x_{2}) * ((y_{1}, y_{2}) \cdot (z_{1}, z_{2}))), {(λ^{-} \times σ^{-})}^{c} (y_{1}, y_{2})} . \end{array}

Hence, (λ⁻ × σ⁻)^c is a fuzzy pseudo-UP ideal of X × Y.

Conversely, assume that λ⁺ × σ⁺ and (λ⁻ × σ⁻)^c are fuzzy pseudo-UP ideals of X × Y. We need to show that λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. Let (x₁, x₂), (y₁, y₂), (z₁, z₂) ∈ X × Y. Then $λ^{+} \times σ^{+} (0, 0) \geq λ^{+} \times σ^{+} (x_{1}, x_{2}) and λ^{-} \times σ^{-} (0, 0) \leq λ^{-} \times σ^{-} (x_{1}, x_{2})$ .

Now, we have ${(λ^{-} \times σ^{-})}^{c} (0, 0) \geq {(λ^{-} \times σ^{-})}^{c} (x_{1}, x_{2}) \Rightarrow - 1 - λ^{-} \times σ^{-} (0, 0) \geq - 1 - λ^{-} \times σ^{-} (x_{1}, x_{2}) \Rightarrow (λ^{-} \times σ^{-}) (0, 0) \leq (λ^{-} \times σ^{-}) (x_{1}, x_{2})$

$λ^{+} \times σ^{+} ((x_{1}, x_{2}) \cdot (z_{1}, z_{2})) \geq min {λ^{+} \times σ^{+} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2}))), λ^{+} \times σ^{+} (y_{1}, y_{2})} .$

And, $λ^{+} \times σ^{+} ((x_{1}, x_{2}) * (z_{1}, z_{2})) \geq min {λ^{+} \times σ^{+} ((x_{1}, x_{2}) * ((y_{1}, y_{2}) \cdot (z_{1}, z_{2}))), λ^{+} \times σ^{+} (y_{1}, y_{2})}$ .

\begin{array}{l} - 1 - λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot (z_{1}, z_{2})) = {(λ^{-} \times σ^{-})}^{c} ((x_{1}, x_{2}) \cdot (z_{1}, z_{2})) \\ \geq min {{(λ^{-} \times σ^{-})}^{c} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2}))), {(λ^{-} \times σ^{-})}^{c} (y_{1}, y_{2})} \\ = min {- 1 - λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2}))), - 1 - λ^{-} \times σ^{-} (y_{1}, y_{2})} \\ = - 1 - max {λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2}))), λ^{-} \times σ^{-} (y_{1}, y_{2})} \end{array}

\Rightarrow λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot (z_{1}, z_{2})) \leq max {λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2}))), λ^{-} \times σ^{-} (y_{1}, y_{2})} .

Similarly, $λ^{-} \times σ^{-} ((x_{1}, x_{2}) * (z_{1}, z_{2})) \leq max {λ^{-} \times σ^{-} ((x_{1}, x_{2}) * ((y_{1}, y_{2}) \cdot (z_{1}, z_{2}))), λ^{-} \times σ^{-} (y_{1}, y_{2})}$ .

Hence, λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. □

Theorem 5.5.

Let λ and σ be any two bipolar fuzzy subsets of X and Y, respectively. Then λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y if and only if □ (λ × σ) = (X × Y, λ⁺ × σ⁺, (λ⁺ × σ⁺)^c) and ♦ (λ × σ) = (X × Y, (λ⁻ × σ⁻)^c, (λ⁻ × σ⁻) are bipolar fuzzy pseudo-UP ideals of X × Y.

Proof.

Assume that λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. Now we need to show that (λ×σ) and ♦ (λ×σ) are bipolar fuzzy pseudo-UP ideal of X×Y. Then for any (x₁, x₂), (y₁, y₂), (z₁, z₂) ∈ X ×Y, we have λ⁺ ×σ⁺(0, 0) ≥ λ⁺ ×σ⁺(x₁, x₂), λ⁺ ×σ⁺((x₁, x₂) · ((z₁, z₂)) ≥ min {λ⁺ ×σ⁺((x₁, x₂) · ((y₁, y₂)*(z₁, z₂))), λ⁺×σ⁺(y₁, y₂)} and λ⁺×σ⁺((x₁, x₂)*((z₁, z₂)) ≥ min {λ⁺×σ⁺((x₁, x₂)*((y₁, y₂)· (z₁, z₂))), λ⁺ × σ⁺(y₁, y₂)}.

Next, let (x₁, x₂), (y₁, y₂), (z₁, z₂) ∈ X × Y. we have (λ⁺ × σ⁺)^c (0, 0) = 1 − λ⁺ × σ⁺ (0, 0) ≤1 − λ⁺ × σ⁺(x₁, x₂) = (λ⁺ × σ⁺)^c (x₁, x₂).

\begin{array}{l} {(λ^{+} \times σ^{+})}^{c} ((x_{1}, x_{2}) \cdot (z_{1}, z_{2})) = 1 - λ^{+} \times σ^{+} ((x_{1}, x_{2}) \cdot (z_{1}, z_{2})) \\ \leq 1 - min {λ^{+} \times σ^{+} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (x_{1}, z_{2}))), λ^{+} \times σ^{+} (y_{1}, y_{2})} \\ = max {1 - λ^{+} \times σ^{+} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (x_{1}, z_{2}))), 1 - λ^{+} \times σ^{+} (y_{1}, y_{2})} \\ = max {{(λ^{+} \times σ^{+})}^{c} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (x_{1}, z_{2}))), {(λ^{+} \times σ^{+})}^{c} (y_{1}, y_{2})} . \end{array}

\begin{array}{l} {(λ^{+} \times σ^{+})}^{c} ((x_{1}, x_{2}) * (z_{1}, z_{2})) = 1 - λ^{+} \times σ^{+} ((x_{1}, x_{2}) * (z_{1}, z_{2})) \\ \leq 1 - min {λ^{+} \times σ^{+} ((x_{1}, x_{2}) * ((y_{1}, y_{2}) \cdot (x_{1}, z_{2}))), λ^{+} \times σ^{+} (y_{1}, y_{2})} \\ = max {1 - λ^{+} \times σ^{+} ((x_{1}, x_{2}) * ((y_{1}, y_{2}) \cdot (x_{1}, z_{2}))), 1 - λ^{+} \times σ^{+} (y_{1}, y_{2})} \\ = max {{(λ^{+} \times σ^{+})}^{c} ((x_{1}, x_{2}) * ((y_{1}, y_{2}) \cdot (x_{1}, z_{2}))), {(λ^{+} \times σ^{+})}^{c} (y_{1}, y_{2})} . \end{array}

Hence, (λ × σ) is a bipolar fuzzy pseudo-UP ideal of X × Y.

And, ${(λ^{-} \times σ^{-})}^{c} (0, 0) = - 1 - λ^{-} \times σ^{-} (0, 0) \geq - 1 - λ^{-} \times σ^{-} (x_{1}, x_{2}) = {(λ^{-} \times σ^{-})}^{c} (x_{1}, x_{2})$ .

\begin{array}{l} {(λ^{-} \times σ^{-})}^{c} ((x_{1}, x_{2}) \cdot (z_{1}, z_{2})) = - 1 - λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot (z_{1}, z_{2})) \\ \geq - 1 - max {λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2}))), λ^{-} \times σ^{-} (y_{1}, y_{2})} \\ = min {- 1 - λ^{-} \times σ^{-} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2}))), - 1 - λ^{-} \times σ^{-} (y_{1}, y_{2})} \\ = min {{(λ^{-} \times σ^{-})}^{c} ((x_{1}, x_{2}) \cdot ((y_{1}, y_{2}) * (z_{1}, z_{2}))), {(λ^{-} \times σ^{-})}^{c} (y_{1}, y_{2})} . \end{array}

\begin{array}{l} {(λ^{-} \times σ^{-})}^{c} ((x_{1}, x_{2}) * (z_{1}, z_{2})) = - 1 - λ^{-} \times σ^{-} ((x_{1}, x_{2}) * (z_{1}, z_{2})) \\ \geq - 1 - max {λ^{-} \times σ^{-} ((x_{1}, x_{2}) * ((y_{1}, y_{2}) \cdot (z_{1}, z_{2}))), λ^{-} \times σ^{-} (y_{1}, y_{2})} \\ = min {- 1 - λ^{-} \times σ^{-} ((x_{1}, x_{2}) * ((y_{1}, y_{2}) \cdot (z_{1}, z_{2}))), - 1 - λ^{-} \times σ^{-} (y_{1}, y_{2})} \\ = min {{(λ^{-} \times σ^{-})}^{c} ((x_{1}, x_{2}) * ((y_{1}, y_{2}) \cdot (z_{1}, z_{2}))), {(λ^{-} \times σ^{-})}^{c} (y_{1}, y_{2})} . \end{array}

Hence, ♦ (λ × σ) is a bipolar fuzzy pseudo-UP ideal of X × Y.

Conversely, assume that γ(λ × σ) and ♦ (λ × σ) are bipolar fuzzy pseudo-UP ideals of X × Y. It is clear that λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. □

Conclusion

In this paper, we discussed the bipolar fuzzy pseudo-UP ideal of a pseudo-UP algebra and investigated several related properties. We proved that the intersection of two bipolar fuzzy pseudo-UP ideals is bipolar fuzzy pseudo-UP ideal. The union of two bipolar fuzzy pseudo-UP ideals is not always a bipolar fuzzy pseudo-UP ideal. We studied the homomorphic image and the inverse image of the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra and obtained some interesting results. We proved the Cartesian product of any two bipolar fuzzy pseudo-UP ideals in pseudo-UP algebra and investigated related results. We hope that the findings of this study will add new dimensions to the structures of bipolar fuzzy pseudo-UP ideals based on bipolar fuzzy sets and will serve as a foundation for further study into the structures of fuzzy pseudo-UP ideals by using the concept of bipolar fuzzy subset of pseudo-UP algebra.

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¹ Department of Mathematics, Bahir Dar University College of Science, Bahir Dar, Amhara, 6000, Ethiopia
² Department of Mathematics, Debre Berhan University, Debre Birhan, Amhara, Ethiopia

Alachew Amaneh Mechderso
Roles: Conceptualization, Writing – Original Draft Preparation, Writing – Review & Editing

Berhanu Assaye Alaba
Roles: Conceptualization

Tilahun Mekonnen Munie
Roles: Conceptualization

Teferi Getachew Alemayhu
Roles: Conceptualization

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No competing interests were disclosed.

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Mechderso AA, Alaba BA, Munie TM and Alemayhu TG. Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations]. F1000Research 2024, 13:1386 (https://doi.org/10.12688/f1000research.157173.1)

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It is quite interesting to review this article entitled “Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra”
In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of ... Continue reading

It is quite interesting to review this article entitled “Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra”
In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of two bipolar fuzzy pseudo-UP ideals is also a bipolar fuzzy pseudo-UP ideal, while the union of two such ideals does not always result in a bipolar fuzzy pseudo-UP ideal. Additionally, we discuss the concepts of bipolar fuzzy pseudo-UP ideals under homomorphism and explore several related properties. The homomorphic image and inverse image of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra are also examined in detail. Furthermore, we study the notion of a bipolar fuzzy pseudo-UP ideal under the Cartesian product of pseudo-UP algebra. The Cartesian product of any two bipolar fuzzy pseudo-UP ideals is also the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra, and then some related results are obtained. However, the paper has been minor revisions in present form but need the following corrections:

In page 8, 0.3= ………………(*), remove =
In page 9, theorem 3.12 and corollary 3.14, (l-)c instead of (l-)c and l = (X, (l-)c, l-) instead of l = (X, (l-)c), l-
In page 9, corollary 313, l = (X, l+, (l+)c) instead of l = (X, l+, ((l+)c). Throughout the paper check it.
In page 14, proof of the condition 2) σ-(y) ≤ l-(0) instead of σ-(y) ≤ l-(y).
In references, the recent references to add in the related work. [Ref 1[ and [Ref 2].

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

No
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

References

1. Mursaleen M, Balamurugan M, Loganathan K, Nisar K: ()-Bipolar Fuzzy-Ideals of BCK/BCI-Algebras. Journal of Function Spaces. 2021; 2021: 1-8 Publisher Full Text
2. Balamurugan M, Alessa N, Loganathan K, Kumar M: Bipolar Intuitionistic Fuzzy Soft Ideals of BCK/BCI-Algebras and Its Applications in Decision-Making. Mathematics. 2023; 11 (21). Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Fuzzy Algebra

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Reviewer Report 24 Dec 2024

Ubaid Ur Rehman, University of Management and Technology,, Lahore, Punjab, Pakistan

Approved with Reservations

https://doi.org/10.5256/f1000research.172589.r344905

I have read the manuscript thoroughly and decided that the idea is new but the paper in recent shape cannot be accepted. I am adding some specific comments below. So, my decision regarding the current version of the manuscript is ... Continue reading

I have read the manuscript thoroughly and decided that the idea is new but the paper in recent shape cannot be accepted. I am adding some specific comments below. So, my decision regarding the current version of the manuscript is a major revision.
Comments:

The abstract should be improved. The present abstract is not up to the mark.
The motivation of the proposed theory need a lot of improvement. The authors must discuss that what is the requirement of this theory to be developed.
The authors interpreted the concept of bipolar valued algebraic structure, so they must discuss some recent work on fuzzy groups such as Bipolar Complex Fuzzy Subgroups [ref 1], Bipolar Complex Fuzzy Semigroups [ref 2], Analysis of Γ-Semigroups Based on Bipolar Complex Fuzzy Sets [ref 3] , T-Bipolar Soft Groups and Their Fundamental Laws [ref 4], Bipolar complex fuzzy submodules [ref 5] in the introduction.
There must be suitable punctuation mark after every equation.
The theoretical work is okay but what is the application of the proposed work? The authors must add an application of the bipolar valued fuzzy groups to show the practicality of their work.
Improve the language of the whole manuscript with the help of native English speakers, and the structure of each section to ensure the readability of the manuscript.
The conclusion is written like a contribution, but there is no deep analysis of obtained knowledge, no discussion, or wider application.
How the authors can expand their work to other mathematical frameworks such bipolar complex fuzzy set devised by Tahir Mahmood? The author should discuss it in the future direction by citing 2 to 3 articles of bipolar complex fuzzy set for better future work.
Add some merit and limitations of the proposed work.

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Partly

References

1. Yang X, Mahmood T, ur Rehman U: Bipolar Complex Fuzzy Subgroups. Mathematics. 2022; 10 (16). Publisher Full Text
2. Rehman U, Mahmood T, Naeem M: Bipolar complex fuzzy semigroups. AIMS Mathematics. 2023; 8 (2): 3997-4021 Publisher Full Text
3. Mahmood T, ur Rehman U, Albaity M: Analysis of $$Gamma $$-semigroups based on bipolar complex fuzzy sets. Computational and Applied Mathematics. 2023; 42 (6). Publisher Full Text
4. Mahmood T, Hussain K, Ahmmad J, Shahab S, et al.: T-Bipolar soft groups and their fundamental laws. Journal of Intelligent & Fuzzy Systems. 2024; 46 (4): 9479-9490 Publisher Full Text
5. Alsuraiheed T, ur Rehman U, Khan M, Mahmood T: Bipolar complex fuzzy submodules. Physica Scripta. 2024; 99 (6). Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: fuzzy algebra, decision making, algebraic structures

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

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Reviewer Report 18 Dec 2024

Baravan Asaad, University of Zakho, Zakho, Kurdistan, Iraq; Cihan University-Duhok, Duhok, Iraq

Approved

https://doi.org/10.5256/f1000research.172589.r344899

The manuscript presented a new concept of pseudo-UP algebras called bipolar fuzzy pseudo-UP ideal which is combine between bipolar fuzzy sets and pseudo-UP ideal. Some theoretical results are presented via this new concept of pseudo-UP algebras.

I ... Continue reading

The manuscript presented a new concept of pseudo-UP algebras called bipolar fuzzy pseudo-UP ideal which is combine between bipolar fuzzy sets and pseudo-UP ideal. Some theoretical results are presented via this new concept of pseudo-UP algebras.

I have only the following points:
- What is the comparison between the existing concept and the previous concept?

Therefore, I recommend to accept this manuscript for indexing.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Topology, Fuzzy Set Theory, Soft Set Theory, Decision Making.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

CITE

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Baravan Asaad, University of Zakho, Zakho, Iraq; Cihan University-Duhok, Duhok, Iraq
Ubaid Ur Rehman, University of Management and Technology,, Lahore, Pakistan
M Balamurugan, Vel Tech, Chennai, India

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Reviewer Report

8 Views

26 Dec 2024 | for Version 1

M Balamurugan, Vel Tech, Chennai, Tamil Nadu, India

8 Views Cite this report Responses(0)

Approved

It is quite interesting to review this article entitled “Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra”
In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of two bipolar fuzzy pseudo-UP ideals is also a bipolar fuzzy pseudo-UP ideal, while the union of two such ideals does not always result in a bipolar fuzzy pseudo-UP ideal. Additionally, we discuss the concepts of bipolar fuzzy pseudo-UP ideals under homomorphism and explore several related properties. The homomorphic image and inverse image of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra are also examined in detail. Furthermore, we study the notion of a bipolar fuzzy pseudo-UP ideal under the Cartesian product of pseudo-UP algebra. The Cartesian product of any two bipolar fuzzy pseudo-UP ideals is also the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra, and then some related results are obtained. However, the paper has been minor revisions in present form but need the following corrections:

In page 8, 0.3= ………………(*), remove =
In page 9, theorem 3.12 and corollary 3.14, (l-)c instead of (l-)c and l = (X, (l-)c, l-) instead of l = (X, (l-)c), l-
In page 9, corollary 313, l = (X, l+, (l+)c) instead of l = (X, l+, ((l+)c). Throughout the paper check it.
In page 14, proof of the condition 2) σ-(y) ≤ l-(0) instead of σ-(y) ≤ l-(y).
In references, the recent references to add in the related work. [Ref 1[ and [Ref 2].

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

No
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

References

1. Mursaleen M, Balamurugan M, Loganathan K, Nisar K: ()-Bipolar Fuzzy-Ideals of BCK/BCI-Algebras. Journal of Function Spaces. 2021; 2021: 1-8 Publisher Full Text
2. Balamurugan M, Alessa N, Loganathan K, Kumar M: Bipolar Intuitionistic Fuzzy Soft Ideals of BCK/BCI-Algebras and Its Applications in Decision-Making. Mathematics. 2023; 11 (21). Publisher Full Text

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Fuzzy Algebra

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Respond to this report

Responses (0)

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Reviewer Report

13 Views

24 Dec 2024 | for Version 1

Ubaid Ur Rehman, University of Management and Technology,, Lahore, Punjab, Pakistan

13 Views Cite this report Responses(0)

Approved With Reservations

I have read the manuscript thoroughly and decided that the idea is new but the paper in recent shape cannot be accepted. I am adding some specific comments below. So, my decision regarding the current version of the manuscript is a major revision.
Comments:

The abstract should be improved. The present abstract is not up to the mark.
The motivation of the proposed theory need a lot of improvement. The authors must discuss that what is the requirement of this theory to be developed.
The authors interpreted the concept of bipolar valued algebraic structure, so they must discuss some recent work on fuzzy groups such as Bipolar Complex Fuzzy Subgroups [ref 1], Bipolar Complex Fuzzy Semigroups [ref 2], Analysis of Γ-Semigroups Based on Bipolar Complex Fuzzy Sets [ref 3] , T-Bipolar Soft Groups and Their Fundamental Laws [ref 4], Bipolar complex fuzzy submodules [ref 5] in the introduction.
There must be suitable punctuation mark after every equation.
The theoretical work is okay but what is the application of the proposed work? The authors must add an application of the bipolar valued fuzzy groups to show the practicality of their work.
Improve the language of the whole manuscript with the help of native English speakers, and the structure of each section to ensure the readability of the manuscript.
The conclusion is written like a contribution, but there is no deep analysis of obtained knowledge, no discussion, or wider application.
How the authors can expand their work to other mathematical frameworks such bipolar complex fuzzy set devised by Tahir Mahmood? The author should discuss it in the future direction by citing 2 to 3 articles of bipolar complex fuzzy set for better future work.
Add some merit and limitations of the proposed work.

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Partly

References

1. Yang X, Mahmood T, ur Rehman U: Bipolar Complex Fuzzy Subgroups. Mathematics. 2022; 10 (16). Publisher Full Text
2. Rehman U, Mahmood T, Naeem M: Bipolar complex fuzzy semigroups. AIMS Mathematics. 2023; 8 (2): 3997-4021 Publisher Full Text
3. Mahmood T, ur Rehman U, Albaity M: Analysis of $$Gamma $$-semigroups based on bipolar complex fuzzy sets. Computational and Applied Mathematics. 2023; 42 (6). Publisher Full Text
4. Mahmood T, Hussain K, Ahmmad J, Shahab S, et al.: T-Bipolar soft groups and their fundamental laws. Journal of Intelligent & Fuzzy Systems. 2024; 46 (4): 9479-9490 Publisher Full Text
5. Alsuraiheed T, ur Rehman U, Khan M, Mahmood T: Bipolar complex fuzzy submodules. Physica Scripta. 2024; 99 (6). Publisher Full Text

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

fuzzy algebra, decision making, algebraic structures

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Respond to this report

Responses (0)

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Reviewer Report

16 Views

18 Dec 2024 | for Version 1

Baravan Asaad, University of Zakho, Zakho, Kurdistan, Iraq; Cihan University-Duhok, Duhok, Iraq

16 Views Cite this report Responses(0)

Approved

The manuscript presented a new concept of pseudo-UP algebras called bipolar fuzzy pseudo-UP ideal which is combine between bipolar fuzzy sets and pseudo-UP ideal. Some theoretical results are presented via this new concept of pseudo-UP algebras.

I have only the following points:
- What is the comparison between the existing concept and the previous concept?

Therefore, I recommend to accept this manuscript for indexing.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Topology, Fuzzy Set Theory, Soft Set Theory, Decision Making.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Respond to this report

Responses (0)

[1] 1. Iséki K: On axiom systems of propositional calculi. XV. Proc. Jpn. Acad. 1966; 42(3): 217–220. Publisher Full Text

[2] 2. Romano DA: Pseudo-UP ideals and pseudo-UP filters in pseudo- UP algebras. Math. Sci. Appl. E-Notes. 2020; 8(1): 155–158. Publisher Full Text

[3] 3. Zadeh LA: Fuzzy sets. Inf. Control. 1965; 8(3): 338–353. Publisher Full Text

[4] 4. Rosenfeld A: Fuzzy groups. J. Math. Anal. Appl. 1971; 35(3): 512–517. Publisher Full Text

[5] 5. Mechdesro AA, Alaba BA, Munie TM, et al.: Fuzzy pseudo-UP ideal of pseudo-UP algebra. Res. Math. 2023; 10(1): 2234217. Publisher Full Text

[6] 6. Wechler W: The concept of fuzziness in automata and language theory.1978. (No Title).

[7] 7. Zimmermann HJ: Fuzzy set theory and its applications. Springer Science & Business Media; 2011.

[8] 8. Lee KM: Bipolar-valued fuzzy sets and their operations. Proc. Int. Conf. on Intelligent Technologies. Bangkok, Thailan: 2000; pp. 307–312.

[9] 9. Lee KJ: Bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras. Bull. Malaysian Math. Sci. Soc. 2009; 32(3): 361–373.

[10] 10. Alaba BA, Derseh BL, Wondifraw YG: On Intuitionistic Fuzzy PMS-Ideals of a PMS-Algebra under Homomorphism and Cartesian product. Int. J. Comput. Intell. Syst. 2023; 16(1): 68. Publisher Full Text

[11] 11. Zhang WR: Bipolar fuzzy sets and relations. A computational framework for cognitive modeling and multiagent decision analysis. NAFIPS/IFIS/NASA’94. Proceedings of the First International Joint Conference of the North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intelligent. 1994; pp. 305–309.

[12] 12. Romano DA: Pseudo-UP algebras. An introduction. Bull. Int. Math. Virtual Inst. 2020; 10(2): 349–355.

[13] 13. Lele C, Wu C, Weke P, et al.: Fuzzy ideals and weak ideals in BCK-algebras.2001.

[14] 14. Jun YB: Fuzzy d-ideals of d-algebras. J. Fuzzy Math. 2000; 8(1): 123–130.

[15] 15. Sabarinathan S, Kumar DC, Muralikrishna P: Bipolar valued fuzzy α-ideal of BF-algebra. Circuits and Systems. 2016; 07(10): 3054–3062. Publisher Full Text