F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.158546.1 Research Article Articles Fuzzy Pseudo-ideal in  Pseudo-TM  Alegbra

[version 1; peer review: awaiting peer review]

 Melaku Alemayehu Girum Conceptualization https://orcid.org/0000-0002-4462-4021 a 1 Alaba Berhanu Assaye Methodology 1 Bitew Bekalu Tarekegn Methodology 1 Derseh Beza Lamesgin Writing – Original Draft Preparation https://orcid.org/0000-0002-7726-5248 2 Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia Mathematics, Debre Markos University College of Natural and Computational Science, Debre Markos, Amhara, Ethiopia alemayehugirum21@gmail.com

No competing interests were disclosed.

 4 12 2024 2024 13 1476 15 11 2024 Copyright: © 2024 Melaku AG et al. 2024 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Background

Fuzzy sets and ideals play a significant role in the study of algebraic structures, particularly in the context of pseudo-TM algebras, which are non-commutative generalizations of MV-algebras. However, the concept of fuzzy pseudo-ideals within these algebras has not been extensively explored. This paper introduces fuzzy pseudo-ideals in pseudo-TM algebras and investigates their key properties, contributing to the broader understanding of fuzzy algebraic structures.

 Methods

We define fuzzy pseudo-ideals in the framework of pseudo-TM algebras and investigate their properties using level sets. The paper employs techniques from algebraic logic and set theory to characterize fuzzy pseudo-ideals and their interactions with homomorphism’s and Cartesian products. Several theorems are developed to establish the closure of fuzzy pseudo-ideals under intersection, and their relationships with homomorphism’s and Cartesian products are explored.

 Results

The main findings include a comprehensive characterization of fuzzy pseudo-ideals in pseudo-TM algebras, both in terms of their algebraic structure and their behavior under intersections. We also show how these fuzzy pseudo-ideals interact with homomorphism’s and Cartesian products. Concrete examples are provided to illustrate the theoretical results, demonstrating the applicability of the concepts to real-world algebraic problems.

 Conclusions

This research enhances the theoretical understanding of fuzzy sets and ideals within pseudo-TM algebras, offering new insights into their properties and interrelationships. The results pave the way for future work, particularly in extending the concepts to fuzzy pseudo-strong ideals and fuzzy pseudo-TM ideals. These extensions could further advance the study of fuzzy algebraic structures, contributing to the broader field of algebraic logic and fuzzy set theory.

 Pseudo-TM algebra Pseudo-ideals fuzzy pseudo-ideals level pseudo-ideal Cartesian product of fuzzy-pseudo-ideal. The author(s) declared that no grants were involved in supporting this work. Introduction

Y. Imai and K. Iseki in 1966 3 introduced an algebraic structures called BCK-algebra and BCI- algebra as generalization of BCK-algebra. BCK-algebras have several connections with other areas of investigation, such as: implicative commutative semigroups, Wajsberg algebras and MV-algebras. MV-algebras are an algebraic counterpart of the Lukasiewicz infinite many valued propositional logic. The study of pseudo-MV algebras was initiated by Georgescu and Iorgulescu in 2001, 1 and independently by Rachunek in 2002 9 (there they are called generalized MV algebras or, in short, GMV-algebras) as a non-commutative generalization of MV-algebras which were introduced by Chang in 1958. 2 Georgescu and Iorgulescu in 2001 introduced the concept of pseudo-BCK algebra as an extension of BCK algebra. 1 Y. B. Jun, et al in 1998 gave the notion of a BH-algebra. 5 Also, Y. B. Jun, et al presented the notion of a pseudo BH-algebra in 2015. 1 H.H. Abbass and A.H. Nouri in 1998 studied some types of ideals of pseudo BH-algebra. 1 Most authors considered the beginning of a fuzzy logic when L. A. Zadeh in 1965 defined a subset in a fuzzy sets. 4 Xi. O in 1991 studied the BCK-algebra in a fuzzy sense. 7 Since then, the researchers have extensively studied the fuzzification of BCK/BCI/BH- algebras. A fuzzy closed ideals depending on an element in BH-algebra defined by H. H. Abbass and H. M. A. Saeedin in 2011. 6 H.H. Abbass and H.A. Dahham presented a fuzzy completely closed ideal of BH-algebra in 2012 1 . EunMiKim, Sun shin Ahn introduced the notion of fuzzy strong ideals of a BH-algebra with degrees in the interval (0,1] in 2013. 11 Abbass, Husein Hadi and Mutasher, Abbas Abdullah in 2018 1 defines a fuzzify pseudo ideal and pseudo enlarged ideal in apseudo BH-algebra. Nouri, Adel Hashem and Sabbar, Mahdi Ghareep studied some types of ideals of pseudo TM-algebra. In this paper, we applied the idea of fuzzy set to pseudo-ideals in pseudo-TM algebra. We introduced the notion of fuzzy pseudo- ideals of a pseudo-TM algebra and investigated their properties. Additionally, we characterized the fuzzy pseudo-ideals of pseudo-TM algebra by their level sets. Finally, we characterized the idea of fuzzy pseudo-ideals in terms of homomorphism and Cartesian product.

 Preliminaries

In this section, we recall some basic definitions and results that are used in the study of this paper.

3 A Pseudo- BCK algebra is a structure ( X; · , ⋆, 0), where · and ⋆ are binary operations on X and 0 is an element of X verifying, for all x, y, z ∈ X,

(i) ( x · y) ⋆ ( x · z) ⋆ ( z · y) = 0 , ( x ⋆ y) · ( x ⋆ z) · ( z ⋆ y) = 0, (ii) x · ( x ⋆ y) · y = 0 , x ⋆ ( x · y) ⋆ y = 0,

(iii) x · x = 0 = x ⋆ x,

(iv) 0 · x = 0 = 0 ⋆ x,

(v) x ≤ y, y ≤ x =⇒ x = y, and

(vi) x ≤ y ⇐⇒ x · y = 0 ⇐⇒ x ⋆ y = 0.

3 A pseudo-BCI algebras is a structure ( X; · , ⋆, 0), where · and ⋆ are binary operations on X and 0 is an element of X satisfying for all x, y, z ∈ X,

(i) ( x · y) ⋆ ( x · z) ⋆ ( z · y) = 0 , ( x ⋆ y) · ( x ⋆ z) · ( z ⋆ y) = 0, (ii) x · ( x ⋆ y) · y = 0 , x ⋆ ( x · y) ⋆ y = 0,

(iii) x · x = 0 = x ⋆ x,

(iv) x ≤ y and y ≤ x = ⇒ x = y, and (v) x ≤ y ⇐⇒ x · y = 0 ⇐⇒ x ⋆ y = 0.

6 A pseudo T M -algebra is an algebra ( X; ⊙ , ⋆, 0) of type (2 , 2 , 0) satisfying the following axioms:

x ⊙ 0 = x ⋆ 0 = 0, for all x ∈ X

( x ⊙ y) ⋆ ( x ⊙ z) = z ⊙ y and ( x ⋆ y) ⊙ ( x ⋆ z) = z ⋆ y. for all x, y, z ∈ X

In a pseudo-Tm algebra X, define a binary operation ≤ by x ≤ y if and only if x ⊙ y = 0 and x ⋆ y = 0.

8 A non-empty subset S of a TM algebra X algebra is called a TM - subalgebra or subalgebra of X if x ∗ y ∈ S for all x, y ∈ S.

8 Let I be a non-empty subset of a TM algebra X. Then I is called an ideal of X if it satisfies: for all x , y ∈ X.

0 ∈ I.

x ⋆ y ∈ I and y ∈ I imply x ∈ I.

6 A non-empty subset I of pseudo TM -algebra X is called a pseudo ideal of X if:

0 ∈ I,

x ⊙ y, x ⋆ y ∈ I and y ∈ I ⇒ x ∈ I,for all x, y, X.

6 A non-empty subset I of a pseudo TM-algebra X is called a pseudo strong of X

0 ∈ I,

( x ⊙ y) ⋆ z, y ∈ I ⇒ x ⊙ y

( x ⋆ y) ⊙ z, y ∈ I ⇒ x ⋆ y, for all x, y, z ∈ X.

6 Let X be a pseudo TM-algebra, then I is called a pseudo TM ideal of X if:

0 ∈ I,

( x ⊙ ( x ⋆ ( x ⋆ y))) ⊙ z, y ∈ I ⇒ x ⊙ y,

( x ⋆ ( x ⊙ ( x ⊙ y))) ⋆ z, y ∈ I ⇒ x ⋆ y, for all x, y, z ∈ X.

4 Let X be a non-empty set. A fuzzy set in X is a function from X into the closed interval [0 , 1] of the real number.

7 Let X and Y be non-empty sets. Let f: X → Y be a function, and let A, B be fuzzy sets in X and Y respectively. The image of A under f, denoted by f ( A), is a fuzzy set in Y defined by f ( A ) ( y ) = { sup { A ( x ) | x ∈ f − 1 ( y ) } if f − 1 ( y ) ≠ ∅ , 0 if f − 1 ( y ) = ∅ .

The inverse image of B under f, denoted by f −1( B), is a fuzzy set in X defined by f −1( B)( x) = B( f ( x)) , ∀ x ∈ X. The union and intersection of set A and

4 Let A and B be two fuzzy sets in X, then:

( A ∩ B)( x) = min{ A( x) , B( x)}, for all x ∈ X.

( A ∪ B)( x) = max{ A( x) , B( x)}, for all x ∈ X.

( A × B)( x) = min{ A( x) , B( x)},where A ∩ B,A ∪ B and A × B are fuzzy sets in X,

In general, if { A i, I ∈ I} is a family of fuzzy sets in X, then:

T i ∈ I A i ( x) = inf { A i ( x) , i ∈ I}, for all x ∈ X and

S i ∈ I A i ( x) = sup { A i ( x) , i ∈ I}, for all x ∈ X. Which are also fuzzy sets in X.

4 Let A be a fuzzy set in X, for all α ∈ [0 , 1]. The set A α = { x ∈ X: A( x) ≥ α} is called a level subset of A.

4 For any t ∈ [0 , 1] and a fuzzy set μ in a nonempty set of X, the sets:

U ( μ, t) = { x: μ( x) ≥ t}, it’s said to be an upper t-level cut of X.

L( μ, t) = { x, μ( x) ≤ t}, it’s said to be a lower t-level cut of X.

10 A fuzzy subset μ of a TM algebra X is called a fuzzy ideal of X, if

μ(0) ≥ A( x).

μ( x) ≥ min { μ ( x ⋆ y), μ( y)} for all x , y ∈ X.

Fuzzy pseudo ideal in pseudo-TM algebra

In this section, we studied the basic definition of a fuzzy pseudo ideal in a pseudo-TM algebra. Also, we discussed the basic properties of fuzzy pseudo ideals in a pseudo TM algebra are investigated and the relationship between fuzzy pseudo ideal of a pseudo- TM algebra and their level set of a pseudo-TM algebra. Let X and Y denote a pseudo-TM algebras unless otherwise specified throughout this and the following section.

A fuzzy subset φ: X → [0 , 1] is a fuzzy Pseudo ideal of X if it satisfies:

φ(0) ≥ φ( x),for all x ∈ X

φ( x) ≥ min{ φ( x ⊙ y) , φ( x ⋆ y) , φ( y)},for all x, y ∈ X.

Let X = {0 , 1 , 2 , 3 , 4 , 5 , 6} be a set with two binary operations ⊙ and ⋆ which are given by table.
⊙ 0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0
2 2 2 0 0 0 0 0
3 3 2 2 0 0 0 0
4 4 0 0 0 0 0 0
5 5 0 0 0 0 0 0
6 6 0 0 0 0 0 0
⋆ 0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0
2 2 2 0 0 0 0 0
3 3 3 3 0 0 0 0
4 4 0 0 0 0 0 0
5 5 0 0 0 0 0 0
6 6 0 0 0 0 0 0

Hence, ( X, ⊙ , ⋆, 0) is a pseudo TM-algebra. Let φ and ω be a fuzzy subsets of X defined by φ ( x ) = { 0.6 if x = 0 , 0.3 if x = 1 , 2 , 3 , 4 , 5 , 6 and ω ( x ) = { 0.7 if x = 0 , 0.5 if x = 1 , 2 , 3 , 4 , 5 , 6

Clearly, φ and ω are a fuzzy pseudo-TM subalgebra of X. Hence, clearly, φ and ω are a fuzzy pseudo-ideal of X.

Consider a pseudo-TM algebra X in example (3.2). Let φ: X → [0 , 1] be a fuzzy set in X defined by φ ( x ) = { 1 if x = 0 , 0.4 if x = 2 , 3 , 4 , 5 , 6 0.1 if x = 1

Hence, φ is a fuzzy pseudo-ideal of X.

If φ and ω are fuzzy pseudo-ideal of a pseudo-TM algebra X . Then φ ∩ ω is also fuzzy pseudo-ideal of X.

Let x, y, ∈ X. Then ( φ ∩ ω)(0) = min{ φ(0) , ω(0)} ≥ min{ φ( x) , ω( x)} = ( φ ∩ ω)( x). These implies that ( φ ∩ ω)(0) ≥ ( φ ∩ ω)( x). Again, ( φ ∩ ω ) ( x ) = min { φ ( x ) , ω ( x ) } ≥ min { min { φ ( x ⊙ y ) , φ ( x ⋆ y ) , φ ( y ) } , min { ω ( x ⊙ y ) , ω ( x ⋆ y ) , φ ( y ) } } = min { min { φ ( x ⊙ y ) , ω ( x ⊙ y ) ) } , min { φ ( x ⋆ y ) , ω ( x ⋆ y ) ) } , min { φ ( y ) , ω ( y ) } } = min { ( φ ∩ ω ) ( x ⊙ y ) , ( φ ∩ ω ) ( x ⋆ y ) , ( φ ∩ ω ) ( y ) } . ⇒ ( φ ∩ ω ) ( x ) ≥ min { ( φ ∩ ω ) ( x ⊙ y ) , ( φ ∩ ω ) ( x ⋆ y ) , ( φ ∩ ω ) ( y ) } .

Hence, φ ∩ ω is a fuzzy pseudo-ideal of a pseudo-TM algebra X.

Let { φ i | i ∈ I} be a family of a fuzzy pseudo ideal of a pseudo-TM algebra X.

Then ∩ i∈ I φ i is a fuzzy pseudo-ideal of X.

Let { φ i | i ∈ I} be a family of fuzzy pseudo-ideal of X.

Let x ∈ X. Then ∩ i∈ I φ i (0) = inf { φ i (0) | i ∈ I} ≥ φ i(x). Since φ i (0) ≥ φ i (x) ∀ i ∈ I.

Let x , y ∈ X. Then, we have ∩ i∈ I φ i (x) = inf { φ i (x) | i ∈ I} ≥ inf { φ i ( x ⊙ y), φ i ( x ⋆ y) , φ i(y)}.

Because φ i ( x ) = inf { φ i ( x ) | i ∈ I } ≥ inf { φ i ( x ⊙ y ) , φ i ( x ⋆ y ) , φ i ( y ) } , ∀ i ∈ I ⇒ ∩ i ∈ I φ i ≥ inf { φ i ( x ⊙ y ) , φ i ( x ⋆ y ) , φ i ( y ) } .

Therefore, ∩ i∈ I φ i is a fuzzy pseudo-ideal of X.

Assume that X is a pseudo-TM algebra and let φ be a fuzzy subset of X. Then φ is a fuzzy pseudo-ideal of X if and only if the set upper level U ( φ, α 1) is a pseudo-ideal of X or empty of X , ∀ α 1 ∈ [0 , 1].

Let φ be a fuzzy pseudo-ideal of a pseudo-TM algebra X & U ( φ, α 1) 6= ∅ for every α 1 ∈ [0,1]. Obviously 0 ∈ U ( φ, α 1) since φ(0) ≥ α 1. Assume x, y ∈ X such that ( x ⊙ y) , ( x ⋆ y) and y ∈ U ( φ, α 1), then φ ( x ⊙ y) ≥ α 1 , φ ( x ⋆ y) and φ( y) ≥ α 1.

It follows that φ ( x) ≥ min { φ (( x ⊙ y)), φ (( x ⋆ y)), φ( y)} ≥ α 1. It implies that x ∈ U ( φ, α 1).

Therefore, U ( φ, α 1) is a pseudo-ideal of X.

Conversely, for every α 1 ∈ [0 , 1] and U ( φ, α 1) a pseudo-ideal, ∀ x ∈ X.

Let φ( x) = α 1 then x ∈ U ( φ, α 1) &U ( φ, α 1) 6 = ∅. Since U ( φ, α 1) is a pseudo-ideal of X which implies that 0 ∈ U ( φ, α 1).

Hence φ ( 0 ) ≥ α 1 = φ ( x ) ∀ x ∈ X .

Suppose that for every x, y ∈ X such that φ ( x) < min { φ ( x ⊙ y), φ ( x ⋆ y), φ ( y)}.

Assume that α 2 = 1 2 ( φ ( x ) + min { φ ( x ⊙ y ) , φ ( x ⋆ y ) , φ ( y ) } ) then φ ( x ) < α 2 < min { φ ( x ⊙ y ) , φ ( x ⋆ y ) , φ ( y ) } .

Hence x ∉ U ( φ , α 2 ) and φ ( x ⊙ y ) , φ ( x ⋆ y ) , φ ( y ) ∈ U ( φ , α 2 ) . This is impossible.

Therefore, φ is a fuzzy pseudo-ideal of a pseudo-TM algebra X.

If φ is a fuzzy pseudo-ideal of a pseudo-TM algebra, then the set J ≔ { x ∈ X | φ ( x ) = φ ( 0 ) } is a pseudo -ideal of X.

Since φ is a fuzzy pseudo-ideal of a pseudo-TM algebra. So 0 ∈ J.

Let x, y ∈ X be such that x ⊙ y, x ⋆ y ∈ J and y ∈ J. Then φ( x ⊙ y) = φ(0) = φ( x ⋆ y) and φ( y) = φ(0) which implies that φ( x) ≥ min{ φ( x ⊙ y) , φ( x ⋆ y) , φ( y)} = μ(0). But φ is a fuzzy pseudo -ideal of a pseudo-TM algebra. Hence φ(0) ≥ φ( x). It follows that φ( x) = μ(0).

Therefore, x ∈ J, and thus J is a pseudo-ideal of X.

If φ is a fuzzy pseudo-ideal of a pseudo-TM-algebra X, then the set J:=

{ x ∈ X | φ( x) ≥ φ( b)} is a pseudo -ideal of X for any b ∈ X.

Suppose that φ is a fuzzy pseudo-ideal of X. We need to show that J is a pseudo-ideal of X.

Let x, y ∈ X such that x ⊙ y ∈ J, x ⋆ y ∈ J and y ∈ J. Then φ( x ⊙ y) ≥ φ( b) , φ( x ⋆ y) ≥ φ( b) and φ( y) ≥ φ( b) which implies that φ( x) ≥ min{ φ( x ⊙ y) , φ( x ⋆ y) , φ( y)} ≥ φ( b). It follows that φ( x) ≥ μ( b).

Hence, x ∈ J, and thus J is a pseudo-ideal of X.

Let φ be any fuzzy subset of X . Then the fuzzy subset φ ⋆ of X defined by φ ⋆ = sup{ t ∈ [0 , 1] | x ∈ U ( φ: t)} for all x ∈ X. Then φ ⋆ is the least fuzzy pseudo-ideal of X that contains φ and U ( φ: t) is the least pseudo-ideal contains U ( φ: t).

Suppose that φ is a fuzzy subset of X and φ ⋆ = sup{ t ∈ [0 , 1] | x ∈ U ( φ: t) }.

We need to prove that φ ⋆ is a fuzzy pseudo-ideal of X. Let k ∈ Im ( μ ⋆ ). For any ǫ > 0 such that s = k − ǫ.

If x ∈ U ( φ ⋆ ; K), then φ ⋆ ( x) ≥ k.

It implies that sup{ t ∈ [0 , 1] | x ∈ U ( φ; t)} ≥ k > k − ǫ = s. ⇒ s ≤ k .

Hence there exists t 1 ∈ { t ∈ [0 , 1] | x ∈ U ( φ; t)} such that t 1 > s. It follows that U ( φ; t 1) ⊆ U ( φ; s) and so x ∈ U ( φ; t 1) ⊆ U ( φ; s) ⇒ x ∈ U ( φ ; s ) . ⇒ U ( φ ⋆ ; k ) ⊆ U ( φ ; s ) .

Similarly, if x ∈ h U ( φ; s)), then s ∈ { t ∈ [0 , 1] | x ∈ U ( φ; t)} ⇒ s ≤ Sup { t ∈ [ 0 , 1 ] | x ∈ U ( φ ; t ) } = φ ⋆ ( x ) ⇒ s ≤ φ ⋆ ( x ) ⇒ U ( φ ⋆ ; K ) = U ( φ ; s ) .

Hence, U ( φ ⋆ ; K) is a pseudo-ideal of X.

Therefore, φ ⋆ is a fuzzy pseudo-ideal of X.

Again we need to show that φ ⋆ contains φ.

Suppose that for any x ∈ X. Let k ∈ { t ∈ [0 , 1] | x ∈ U ( φ; t)}.

Then x ∈ U ( φ; k) and that x ∈ U ( φ; t).

So, k ∈ { t ∈ [0 , 1] | x ∈ U ( φ; t)}.

This implies that t ∈ [0 , 1] | x ∈ U ( φ; t) ⊆ t ∈ [0 , 1] | x ∈ U ( φ; t). ⇒ φ ( x ) = Sup { t ∈ [ 0 , 1 ] | x ∈ U ( φ ; t ) } ≤ Sup { t ∈ [ 0 , 1 ] | x ∈ U ( φ ; t ) } = φ ⋆ ≤ φ ⋆ ⇒ φ ⊆ φ ⋆ .

Finally, let ω be a fuzzy pseudo-ideal of X containing φ. For any x ∈ X.

If φ ⋆ = 0 implies φ ⋆ ( x) ≤ ω( x).

Assume that φ ⋆ ( x) = k 6= 0. Then x ∈ U ( φ ⋆ ; k) = U ( φ; s). ⇒ x ∈ U ( φ ; s ) ⇒ ω ( x ) ≥ φ ( x ) ≥ s = k − ǫ ⇒ ω ( x ) ≥ s = φ ⋆ ( x ) .

It follows that φ ⋆ is contained in ω.

Every fuzzy pseudo-ideal of a pseudo-TM algebra X is order reversing.

x ≤ y,then x ⊙ y = x ⋆ y = 0 ∈ I where I is Pseudo-ideal of X and,

So, φ ( x ) ≥ min { φ ( x ⊙ y ) , φ ( x ⋆ y ) , φ ( y ) } = min { φ ( 0 ) , φ ( 0 ) , φ ( y ) } = min { φ ( 0 ) , φ ( y ) } = φ ( y )

Thus, φ( x) ≥ φ( y). This shows that φ is order reversing.

Homomorphism of fuzzy pseudo-ideal of pseudo-TM algebra

In this section, we give several characterizations for fuzzy pseudo-ideals and fuzzy homomorphisms of a pseudo-TM algebra.

Let ( X; ⊙ , ⋆, 0) and ( Y; ⊙ , ⋆, 0) be two pseudo-TM algebras. A mapping f: X → Y is called a pseudo TM homomorphism if f ( x⊙ y) = f ( x)⊙ f ( y) and f ( x⋆y) = f ( x) ⋆f ( y) for all x, y ∈ X.

Let ( X, ⊙ , ⋆, 0) and ( Y, ⊙ , ⋆, 0) be a pseudo-TM algebras and let f: X → Y be a homomorphism and φ be a fuzzy pseudo-ideal of Y . Then f −1( φ) is a fuzzy pseudo-ideal of X.

Let x ∈ X. Then f ( x) ∈ Y and φ is a fuzzy pseudo-ideal of Y, we have:

f −1( φ)(0) = φ( f (0)) ≥ φ( f ( x)). Thus we get f −1( φ)(0) ≥ φ( f ( x)) for any x ∈ X. That is f −1( φ) satisfies (i).

Let x, y ∈ X. Since ( φ) is a fuzzy pseudo-ideal of Y,we have: f − 1 ( φ ( x ) ) = φ ( f ( x ) ) ≥ min { φ ( f ( x ) ⊙ f ( y ) ) , φ ( f ( x ) ⋆ f ( y ) ) , φ ( f ( y ) ) } = min { φ ( f ( x ⊙ y ) ) , φ ( f ( x ⋆ y ) ) , φ ( f ( y ) ) } = min f − 1 ( φ ) ( x ⊙ y ) , f − 1 ( φ ) ( x ⋆ y ) , f − 1 ( φ ) ( y ) ⇒ f − 1 ( φ ( x ) ) ≥ min f − 1 ( φ ) ( x ⊙ y ) , f − 1 ( φ ) ( x ⋆ y ) , f − 1 ( φ ) ( y )

That is f −1( φ) satisfies (i).

Therefore, f −1( φ( x)) is a fuzzy pseudo-ideal of X.

Let ( X, ⊙ , ⋆, 0) and ( Y, ⊙ , ⋆, 0) be a pseudo-TM algebras and let f: X → Y be a homomorphism and φ be a fuzzy pseudo-ideal of X . If φ is constant on Kerf = f −1(0) , then f −1( f )( φ) = φ.

Let x ∈ X and f ( x) = y. Hence f − 1 ( f ) ( φ ) ( x ) = ( f ) ( φ ) f ( x ) = ( f ) ( φ ) ( y ) = Sup φ ( b ) : b ∈ f − 1 ( y )

For all b ∈ f −1( y) , we have f ( x) = f ( b). Hence, f ( b ⊙ x) = f ( b ⋆ x) = 0

⇒ b ⊙ x, b ⋆ x ∈ Kerf. It follows that φ( b ⊙ x) = φ(0) = φ( b ⋆ x). Then φ( b) ≥ min { φ( b ⊙ x) , φ( b ⋆ x) , φ( x)} = min { φ(0) , φ( x)} = φ( x)

We get, φ( b) ≥ φ( x). Since by (3.8) φ( x) ≥ φ( b). Hence, φ ( x ) = φ ( b ) .

Thus, f −1( f )( φ) ( x) = Sup φ( b): b ∈ f −1( y) = φ( x).

Therefore, f −1( f )( φ) = φ.

Let ( X, ⊙ , ⋆, 0) and ( Y, ⊙ , ⋆, 0) be a pseudo-TM algebras and let f: X → Y be an epimorphism and φ be a fuzzy pseudo-ideal of X such that φ is constant on Kerf. Then f ( φ) is a fuzzy pseudo-ideal of Y, provided that the sup property holds.

Let 0 ∈ Y, then there exist 0 ∈ X such that f (0) = 0. Now, f ( φ)(0) = sup φ( k) | k ∈ f −1(0) = φ(0) ≥ φ( x) , ∀ x ∈ X. Let y ∈ Y. Since f is an epimorphism, we have f −1( y) = ∅ and φ(0) ≥ φ( k) | k ∈ f −1( y). Which implies that f ( φ)(0) ≥ sup φ( k) | k ∈ f −1( y) = f ( φ)( y) , ∀ y ∈ Y. Hence, f ( φ)(0) ≥ f ( φ)( y) , ∀ y ∈ Y.

Thus f ( φ) satisfies (i) (1).

Assume that ( f ( φ)) ( a) < min {( f ( φ))( a ⊙ b) , ( f ( φ))( a ⋆ b) , ( f ( φ))( b)} for some a, b ∈ Y. Since f is an epimorphism there exists x, y ∈ X such that f ( x) = a and f ( x) = b.

Hence, ( f ( φ)) ( f ( x)) < min {( f ( φ))( f ( x ⊙ y)) , ( f ( φ))( f ( x ⋆ y)) , ( f ( φ))( f ( y))}.

Then f −1( f ( φ))( x) < min f −1( f ( φ))( x ⊙ y) , f −1( f ( φ))( x ⋆ y) , f −1( f ( φ))( y).

But φ is constant on Kerf. Then by (4.3) , we get φ( x) < min { φ( x ⊙ y) , φ( x ⋆ y) , φ( y)} which is a contradiction with the fact that φ is a fuzzy pseudo-ideal of X.

Therefore, f ( φ) is a fuzzy pseudo-ideal of Y.

Cartesian product of fuzzy pseudo-ideal of pseudo-TM algebra

In this section, we introduced the notion of fuzzy pseudo-ideals in pseudo-TM algebra in terms of Cartesian product of any two pseudo-TM algebra.

Let X and Y be pseudo TM-algebras, and let φ be a fuzzy pseudo ideals of X and ω be a fuzzy pseudo ideals of Y. Then the Cartesian product of φ and ω denoted as φ × ω is defined by ( φ × ω)(( x 1 , x 2) ⊙ ( y 1 , y 2)) = min { φ( x 1 ⊙ y 1) , ω( x 2 ⊙ y 2)} and ( φ × ω)(( x 1 , x 2) ⋆ ( y 1 , y 2)) = min { φ( x 1 ⋆ y 1) , ω( x 2 ⋆ y 2)} for all ( x 1 , x 2) and ( y 1 , y 2) in X × Y.

Let ( X, ⊙ , ⋆, 0) and ( Y, ⊙ , ⋆, 0) be a pseudo-TM algebras and let φ and ω be any two fuzzy pseudo-ideal of X and Y respectively. Then φ × ω is a fuzzy pseudo-ideals of X × Y.

Suppose that φ and ω be a pseudo-TM algebras of X and Y respectively.

(a) Let ( x, y) ∈ X × Y. Then ( φ × ω) (0 , 0) = min { φ(0) , ω(0)} ≥ min { φ( x) , ω( y)}.

Hence,( φ × ω) (0 , 0) ≥ ( φ × ω) ( x, y).

(b) Let ( x 1 , x 2) , ( y 1 , y 2) ∈ X × Y. Then ( φ × ω ) ( x 1 , x 2 ) = min { φ ( x 1 ) , ω ( x 2 ) } ≥ min { min { φ ( x 1 ⊙ y 1 ) , φ ( x 1 ⋆ y 1 ) , φ ( y 1 ) } , min { ω ( x 2 ⊙ y 2 ) , ω ( x 2 ⋆ y 2 ) , ω ( y 2 ) } } . = min { min { φ ( x 1 ⊙ y 1 ) , ω ( x 2 ⊙ y 2 ) } , min { φ ( x 1 ⋆ y 1 ) , ω ( x 2 ⋆ y 2 ) } , min { φ ( y 1 ) , ω ( y 2 ) } } . = min { ( φ × ω ) ( x 1 , x 2 ) ⊙ ( y 1 , y 2 ) , ( φ × ω ) ( x 1 , x 2 ) ⋆ ( y 1 , y 2 ) , ( φ × ω ) ( y 1 , y 2 ) }

Hence, ( φ × ω) ( x 1 , x 2) ≥ min {( φ × ω) ( x 1 , x 2) ⊙ ( y 1 , y 2) , ( φ × ω) ( x 1 , x 2) ⋆ ( y 1 , y 2) , ( φ × ω) ( y 1 , y 2)}.

By (a) and (b) above, we have that φ × ω is a fuzzy pseudo-ideal of X.

Let ( X, ⊙ , ⋆, 0) and ( Y, ⊙ , ⋆, 0) be a pseudo-TM algebras. If φ × ω be a fuzzy pseudo-ideal of X × Y for any fuzzy subset of φ and ω of a pseudo-algebra of X and Y respec- tively, then either φ is a fuzzy pseudo-ideal of X or ω be a fuzzy pseudo-ideal of Y.

Suppose that φ × ω is a fuzzy pseudo-ideal of X × Y such that φ and ω be a fuzzy subset of X and Y respectively. Then φ × ω (0 , 0) = min { φ(0) , ω(0)} ≥ min { φ( x) , ω( y)} for all ( x, y) ∈ X × Y.

Suppose that φ( x) > φ(0) and ω( y) > ω(0) for some ( x, y) ∈ X × Y.

Now, ( φ × ω)( x, y) = min { φ( x) , ω( y)} > min { φ(0) , ω(0)} = ( φ × ω)(0 , 0) which is a contradiction.

Thus, φ(0) ≥ φ( x) or ω(0) ≥ ω( x) for all ( x, y) ∈ X × Y.

Let ( x 1 , x 2) , ( y 1 , y 2) ∈ X × Y. Then ( φ × ω ) ( x 1 , x 2 ) = min { φ ( x 1 ) , ω ( x 2 ) } ≥ min { ( φ × ω ) ( x 1 , x 2 ) ⊙ ( y 1 , y 2 ) , ( φ × ω ) ( x 1 , x 2 ) ⋆ ( y 1 , y 2 ) , ( φ × ω ) ( y 1 , y 2 ) } = min { min { φ ( x 1 ⊙ y 1 ) , φ ( x 1 ⋆ y 1 ) , φ ( y 1 ) } , min { ω ( x 2 ⊙ y 2 ) , ω ( x 2 ⋆ y 2 ) , ω ( y 2 ) } } . = min { min { φ ( x 1 ⊙ y 1 ) , ω ( x 2 ⊙ y 2 ) } , min { φ ( x 1 ⋆ y 1 ) , ω ( x 2 ⋆ y 2 ) } , min { φ ( y 1 ) , ω ( y 2 ) } } . ⇒ φ ( x 1 ) ≥ min { φ ( x 1 ⊙ y 1 ) , φ ( x 1 ⋆ y 1 ) , φ ( y 1 ) } or ω ( x 2 ) ≥ min { ω ( x 2 ⊙ y 2 ) , ω ( x 2 ⋆ y 2 ) , ω ( y 2 ) }

Hence, either φ( x 1) ≥ min { φ( x 1 ⊙ y 1) , φ( x 1 ⋆ y 1) , φ( y 1)}

or ω( x 2) ≥ min { ω( x 2 ⊙ y 2) , ω( x 2 ⋆ y 2) , ω( y 2)}.

Let ( X, ⊙ , ⋆, 0) and ( Y, ⊙ , ⋆, 0) be a pseudo-TM algebras and let φ and ω be any two fuzzy subset of X and Y respectively. Then φ × ω is a fuzzy pseudo-ideals of X × Y. For t ∈ [0 , 1] the set U ( φ × ω, t) = {( x, y) ∈ X × Y | ( φ × ω)( x, y) ≥ t} is called the upper t-level set of φ × ω.

Let ( X; ⊙ , ⋆, 0) and ( Y; ⊙ , ⋆, 0) be a pseudo-TM algebras and let φ and ω be any two fuzzy pseudo-ideal of X and Y respectively. Then φ × ω is a fuzzy pseudo-ideals of X × Y if and only if the non-empty upper t− level set U ( φ × ω, t) is a pseudo-ideal of X × Y for any t ∈ [0 , 1].

Let φ × ω be a fuzzy pseudo-ideals of X × Y such that U ( φ × ω, t) 6= ∅ for all t ∈ [0 , 1]. Then there exists ( x, y) ∈ U ( φ × ω, t). Thus ( φ × ω)( x, y) ≥ t. Since φ × ω is a fuzzy pseudo- ideal of X × Y. We have ( φ × ω) (0 , 0) ≥ ( φ × ω) ( x, y) for all ( x, y) ∈ X × Y. ⇒ ( φ × ω ) ( 0 , 0 ) ≥ ( φ × ω ) ( x , y ) ≥ t ⇒ ( φ × ω ) ( 0 , 0 ) ≥ t ⇒ ( 0 , 0 ) ∈ U ( φ × ω , t )

Suppose that ( x 1 , y 1 ) , ( x 2 , y 2 ) ∈ U ( φ × ω , t ) such that ( x 1 , y 1 ) ⊙ ( x 2 , y 2 ) , ( x 1 , y 1 ) ⋆ ( x 2 , y 2 ) , ( x 2 , y 2 ) ∈ U ( φ × ω , t ) ⇒ ( φ × ω ) ( ( x 1 , y 1 ) ⊙ ( x 2 , y 2 ) ) ≥ t , ( φ × ω ) ( ( x 1 , y 1 ) ⋆ ( x 2 , y 2 ) ) ≥ t and ( φ × ω ) ( ( x 2 , y 2 ) ≥ t .

Since φ × ω is a fuzzy pseudo-ideal of X × Y. We have ( φ × ω ) ( x 1 , y 1 ) ≥ min { ( φ × ω ) ( ( x 1 , y 1 ) ⊙ ( x 2 , y 2 ) ) , ( φ × ω ) ( ( x 1 , y 1 ) ⋆ ( x 2 , y 2 ) ) , ( φ × ω ) ( x 2 , y 2 ) } ≥ min { t , t , t } = t ⇒ ( φ × ω ) ( x 1 , y 1 ) ≥ t ⇒ ( x 1 , y 1 ) ∈ U ( φ × ω , t ) Hence, U ( φ × ω, t) is a pseudo ideal of X × Y.

Conversely, assume that the set U ( φ × ω, t) be a pseudo-ideal of X × Y for each t ∈ [0 , 1].

Let ( x, y) ∈ X × Y such that ( φ × ω) = t. Then ( x, y) ∈ U ( φ × ω, t). Since ( φ × ω) is a fuzzy pseudo-ideal of X × Y, then (0 , 0) ∈ U ( φ × ω, t).

Hence, ( φ × ω)(0 , 0) ≥ t = ( φ × ω)( x, y) for all ( x, y) ∈ X × Y.

Assume that

t 1 = {( φ × ω)( x, x) + min { ( φ × ω) (( x, x) ⊙ ( x 1 , y 1)), ( φ × ω) (( x, x) ⋆ ( x 1 , y 1)), ( φ × ω) ( x 1 , y 1)}}.

Then ( φ× ω)( x, x) < t 1 < min {( φ × ω) (( x, x) ⊙ ( x 1 , y 1)), ( φ × ω) (( x, x) ⋆ ( x 1 , y 1)), ( φ × ω) ( x 1 , y 1)}.

It follows that ( x, x) 6= U ( φ × ω, t 1). Also, ( φ × ω) (( x, x) ⊙ ( x 1 , y 1)), ( φ × ω) (( x, x) ⋆ ( x 1 , y 1)) ∈ U ( φ × ω, t 1) and ( φ × ω) ( x 1 , y 1) ∈ U ( φ × ω, t 1) which is a contradiction. Since U ( φ × ω, t 1) is a pseudo-ideal of X × Y. Therefore, φ × ω is a fuzzy pseudo-ideal of X × Y.

Let ( X; ⊙ , ⋆, 0) and ( Y; ⊙ , ⋆, 0) be a pseudo-TM algebras and let φ and ω be any two fuzzy pseudo-ideal of X and Y respectively such that φ × ω is a fuzzy pseudo-ideals of X × Y. Then:

Either φ(0) ≥ φ( x) or ω(0) ≥ ω( x) for all x ∈ X.

If φ(0) ≥ φ( x) for all x ∈ X ¯, then either ω(0) ≥ φ( x) or ω(0) ≥ ω( x)

If ω(0) ≥ φ( x) for all x ∈ X, then either φ(0) ≥ φ(x) or φ(0) ≥ ω(x).

Conclusion

In this paper, we applied the idea of fuzzy set to pseudo-ideals in pseudo-TM algebra. We introduced the notion of fuzzy pseudo-ideals of a pseudo-TM algebra and investigated their properties. Additionally, we characterized the fuzzy pseudo-ideals of pseudo-TM algebra by their level sets. Finally, we characterized the idea of fuzzy pseudo-ideals in terms of homo- morphism and Cartesian product. In our future works, we will extend the concept of fuzzy pseudo-TM algebra to the structure of fuzzy pseudo-strong ideals and fuzzy pseudo-TM ideals of a pseudo-TM algebra. The study reveals several key properties of fuzzy pseudo-ideals, including their behavior under certain algebraic operations. We successfully characterize fuzzy pseudo-ideals in terms of their level sets, providing a clearer understanding of their structure. Furthermore, we present a comprehensive characterization of fuzzy pseudo-ideals using homomorphisms and Cartesian products, offering new insights into their algebraic properties.

This work contributes to the theory of pseudo-TM algebras by introducing fuzzy pseudo-ideals and demonstrating their mathematical significance. The paper suggests future work on extending the concepts to fuzzy pseudo-strong ideals and fuzzy pseudo-TM ideals, further advancing the study of these algebraic structures.

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