Int. J. Anal. Appl. (2022), 20:50 Interval Valued Intuitionistic Fuzzy β-Filters on β-Algebras Kaliyaperumal Palanivel1, Prakasam Muralikrishna2, Perumal Hemavathi3, Ronnason Chinram4, Pattarawan Singavananda5,∗ 1Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, India 2PG and Research Department of Mathematics, Muthurangam Government Arts College (Autonomus), Vellore-632002, India 3Saveetha School of Engineering, SIMATS, Thandalam-602025, India 4Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand 5Program in Mathematics, Faculty of Science and Technology, Songkhla Rajabhat University, Songkhla 90000, Thailand ∗Corresponding author: pattarawan.pe@skru.ac.th Abstract. This study establishes the concept of interval valued intuitionistic fuzzy (InVInF) β-filters on β-algebras and a few of its related properties are investigated. Some compelling results of interval valued fuzzy β-filters have been examined. Further, the notions of products and strong β-filters are also introduced. In addition that, the level set and homomorphism of interval valued intuitionistic fuzzy β-filters are too discussed. Furthermore, we enacted that the intersection between two interval valued intuitionistic fuzzy β−filters is again an interval valued intuitionistic fuzzy β-filter. 1. Introduction In 2002, Neggers et al. [12] proposed the idea of a β-algebra which is an algebraic structure with two operations. The concepts of fuzzy positive implicative and fuzzy associative filters of lattice implication algebras have been initiated in [13,14]. Further, the authors in [13,14] have demonstrated that every fuzzy associative filter is a fuzzy associative filter and that every fuzzy positive implicative Received: Aug. 17, 2022. 2010 Mathematics Subject Classification. 08A72, 03E72. Key words and phrases. fuzzy β-filter; intuitionistic fuzzy β-filter; β-algebra; interval valued intuitionistic fuzzy set. https://doi.org/10.28924/2291-8639-20-2022-50 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-50 2 Int. J. Anal. Appl. (2022), 20:50 filter is a fuzzy implicative filter. The equivalent conditions for both fuzzy positive implicative filters and fuzzy associative filters were also provided. Xu et al. [20] established the thought of intuitionistic fuzzy implicative filters in lattice implication algebras. Jun et al. [8] proposed the concept of fuzzy BCI- subalgebras with interval valued membership functions. In 2011, Ghorbani [4] proposed the notion of intuitioistic fuzzy filters of residuated lattices. They have illustrated that a residual lattice’s collection of all intuitionistic fuzzy filters is a complete lattice and identified its distributive sublattices. Zadeh [21] developed an interval valued fuzzy set was used to extend a fuzzy set (ie. a fuzzy set with an interval valued membership function). An i-v fuzzy set is an interval valued fuzzy set that can be used in various algebraic structures. Biswas et al. [2] created fuzzy subgroups with interval membership values in 1994. Hoo [9] applied the concepts of filters and ideals in BCI-algebras in 1991. In 2015, Hemavathi et al. [5–7] discussed interval valued fuzzy β-subalgebras and also applied the concept to intuitionistic fuzzy sets. Jun et al. [15] introduced foldness of bipolar fuzzy sets and its application in BCK/BCI- algebras. Takallo et al. [19] discussed the concept of multipolar fuzzy p-ideals of BCI-algebras. The concept of multipolar intuitionistic fuzzy hyper BCK-ideals in hyper BCK-algebras has been developed by Seo et al. [16]. Borzooei et al. [3] focused on multipolar intuitionistic fuzzy B-algebras. A new perception of cubic multi-polar structures on BCK/BCI-algebras was approached by Al-Masarwah et al. [1]. In [10], the authors invented a mathematical model for nonlinear optimization which attempts membership functions to address the uncertainties. Muhiuddin et al. [11] applied the theory of linear Diophantine fuzzy set into BCK/BCI-algebras. Sujatha et al. [17, 18] introduced fuzzy filters on β- algebras and also developed the concept of intuitiointic fuzzy filters on β-algebras. With all of this in mind, this paper establishes the idea of interval valued fuzzy β-filters on β-algebras and demonstrate few of its intriguing aspects. 2. Preliminares This section outlines some of the most important definitions and examples relevant to the study. Definition 2.1. A β-algebra is a non-empty set Γ with two binary operations + and − and a constant 0 fulfills the following axioms: (1) %δ − 0 = %δ (2) (0 −%δ) + %δ = 0 (3) (%δ − ỳδ) − z̀δ = %δ − (z̀δ + ỳδ) for all %δ, ỳδ, z̀δ ∈ Γ. Definition 2.2. Let f be a mapping from a β- algebra Γ to a β- algebra Υ , then f is referred as homomorphism, if (1) f (%δ + ỳδ) = f (%) + f (ỳδ) (2) f (%δ − ỳδ) = f (%δ) − f (ỳδ) Int. J. Anal. Appl. (2022), 20:50 3 for all %δ, ỳδ ∈ Γ. Definition 2.3. A β-subalgebra Ξ on a β-algebra Γ is referred as β-filter, if (1) %δ 4 ỳδ = %δ + (%δ + ỳδ) ∈ Ξ (2) %δ 5 ỳδ = %δ − (%δ − ỳδ) ∈ Ξ for all %δ, ỳδ ∈ Ξ. Definition 2.4. A β-subalgebra Ξ on a β-algebra Γ is referred as fuzzy β-filter, if (1) �Ξ(%δ 4 ỳδ) ≥ min{�Ξ(%δ),�Ξ(%δ + ỳδ)} and �Ξ(%δ 5 ỳδ) ≥ min{�Ξ(%δ),�Ξ(%δ − ỳδ)} (2) �Ξ(ỳδ) ≥ �Ξ(%δ) if %δ ≤ ỳδ for all %δ, ỳδ ∈ Ξ. 3. Interval Valued Fuzzy β-Filters The concept of an Interval valued fuzzy (InVF) β-filter on a β-subalgebra is introduced in this section. Definition 3.1. An InVF β-subalgebra Ξ on a β-algebra Γ is referred as an InVF fuzzy β-filter, it satisfies (1) �Ξ(%δ 4 ỳδ) ≥ rmin{�Ξ(%δ),�Ξ(%δ + ỳδ)} and �Ξ(%δ 5 ỳδ) ≥ rmin{�Ξ(%δ),�Ξ(%δ − ỳδ)} (2) �Ξ(ỳδ) ≥ �Ξ(%δ) if %δ ≤ ỳδ for all %δ, ỳδ ∈ Ξ. Example 3.1. Consider a β-algebra Γ = {0,γ1,γ2,γ3} with two binary operations + and − and a constant 0 defined on Γ with the Cayley’s table: + 0 γ1 γ2 γ3 0 0 0 0 0 γ1 γ1 γ1 γ1 γ1 γ2 0 0 γ2 γ3 γ3 γ3 γ3 γ3 γ3 − 0 γ1 γ2 γ3 0 0 0 0 0 γ1 γ1 γ1 γ1 γ1 γ2 γ2 γ2 γ2 γ2 γ3 γ3 γ3 γ3 γ3 Then (Γ, +,−, 0) is a β-algebra. Thus Ξ = {γ1,γ3} is a β-filter on Γ. We have Ξ is an InVF β-subalgebra, with interval membership function �Ξ(%δ) =  [0.3, 0.5] : %δ = γ1 [0.4, 0.6] : %δ = γ3 . Then it is observed that, Ξ is an InVF β-filter on Γ. 4 Int. J. Anal. Appl. (2022), 20:50 Example 3.2. From the Example 3.1, Ξ is an InVF β-subalgebra, define by the membership function �Ξ(%δ) =   [0.2, 0.6] : %δ = γ1 [0.1, 0.4] : %δ = γ2 [0.3, 0.5] : %δ = γ3 . Then it is observed that, Ξ is not an InVF β-filter on Γ because �Ξ(γ2) ≥ �Ξ(γ1) ⇒ [0.1, 0.4] � [0.2, 0.6]. Lemma 3.1. If Ξ1 and Ξ2 are two InVF β-filters on Γ, then so is Ξ1 ∩ Ξ2. Proof. For %δ, ỳδ ∈ Γ, �Ξ1∩Ξ2 (%δ 4 ỳδ) = rmin{�Ξ1 (%δ 4 ỳδ),�Ξ2 (%δ 4 ỳδ)} ≥ rmin{rmin{�Ξ1 (%δ),�Ξ1 (%δ + ỳδ)}, rmin{�Ξ2 (%δ),�Ξ2 (%δ + ỳδ)}} ≥ rmin{rmin{�Ξ1 (%δ),�Ξ2 (%δ)}, rmin{�Ξ1 (%δ + ỳδ),�Ξ2 (%δ + ỳδ)}} = rmin{�Ξ1∩Ξ2 (%δ),�Ξ1∩Ξ2 (%δ + ỳδ)}. Similarly, �Ξ1∩Ξ2 (%δ 5 ỳδ) ≥ rmin{�Ξ1∩Ξ2 (%δ),�Ξ1∩Ξ2 (%δ − ỳδ)}. Hence Ξ1 ∩ Ξ2 is an InVF β-filter of Γ. � Theorem 3.1. Every β-filter in InVF is also a β-subalgebra in InVF. Proof. This proof is self-evident, as it follows clearly from the definition of the InVF β-filter. Every InVF β-subalgebra, on the other hand, does not have to be an InVF β-filter. � Theorem 3.2. For an InVF β-filter � of Γ, we have �Ξ(%δ4 ỳδ) ≥ �Ξ(%δ) and �Ξ(%δ5 ỳδ) ≥ �Ξ(%δ) where %δ ≤ ỳδ. Proof. Assume that �Ξ is an InVF β-filter of Γ. Let %δ, ỳδ ∈ Γ. Then �Ξ(%δ 4 ỳδ) = �Ξ(%δ + (%δ + ỳδ)) ≥ rmin{�Ξ(%δ),�Ξ(%δ + ỳδ)} = rmin{�Ξ(%δ), rmin{�Ξ(%δ),�Ξ(ỳδ)}} (because InVF β-filter is an InVF β-subalgebra) = rmin{�Ξ(%δ),�Ξ(%δ)} (because %δ ≤ ỳδ ⇒ �Ξ(ỳδ) ≥ �Ξ(%δ)) = �Ξ(%δ). Similarly, �Ξ(%δ 5 ỳδ) ≥ �Ξ(%δ). � Int. J. Anal. Appl. (2022), 20:50 5 Definition 3.2. Consider an InVF β-filter �Ξ of a β-subalgebra Γ. For [s1,s2] ∈ D[0, 1], the set �Ξ = {%δ ∈ Γ : �Ξ(%)δ ≥ [s1,s2]} is referred to as a level set of an InVF β-filter �Ξ of Γ. Theorem 3.3. An InVF subset Ξ of a β-algebra Γ is an InVF β-filter if and only if for any t ∈ D[0, 1] the t−InVF level subset Ξt = {%δ ∈ Γ : Ξ(%δ) ≥ t} is either a β-filter or Ξt 6= ∅. Proof. For an InVF level subset of Ξ in Γ, Ξt 6= ∅. Then %δ, ỳδ ∈ Ξt, Ξ(%δ) ≥ t. Now, Ξ(%δ 4 ỳ) = Ξ(%δ + (%δ + ỳδ)) ≥ rmin{Ξ(%δ), Ξ(%δ + ỳδ)} = rmin{Ξ(%δ), rmin{Ξ(%δ), Ξ(ỳδ)}} = rmin{t,rmin{t,t}} = t. This implies that %δ 4 ỳδ ∈ Ξt. Similarly, %δ 5 ỳδ ∈ Ξt. Then Ξt is a β-filter of Γ. Suppose that Ξt is a β-filter of Γ, on the other hand. For %δ, ỳδ ∈ Γ,%δ 4 ỳδ and %δ 5 ỳδ ∈ Ξt, this implies that Ξ(%δ4 ỳδ) ≥ t and Ξ(%δ5 ỳδ) ≥ t Ξ(%δ4 ỳδ) = Ξ(%δ + (%δ + ỳδ)) ≥ t = rmin{Ξ(%δ), Ξ(%δ + ỳδ)}. Similarly, Ξ(%δ 5 ỳδ) ≥ t. This proved Ξ is an InVF β-filter. � Theorem 3.4. Consider an onto β-algebra homomorphism f from Γ to Υ . If Ξ2 is a InVF β-filter of Y , hence its inverse image f−1(Ξ2) is again an InVF β−filter on Γ. Proof. Let Ξ2 be an InVF β-filter of Y . For any %δ, ỳδ ∈ Γ, f−1(�Ξ2 (%δ 4 ỳδ)) = f −1(�Ξ2 (%δ + (%δ + ỳδ))) = �Ξ2 (f (%δ + (%δ + ỳδ))) = �Ξ2 (f (%δ)δ + f (%δ + ỳδ)) ≥ rmin{�Ξ2 (f (%δ)),�Ξ2 (f (%δ + ỳδ))} = rmin{f−1(�Ξ2 (%δ)), f −1(�Ξ2 (%δ + ỳδ))}. Similarly, f−1(�Ξ2 (%δ 5 ỳδ)) ≥ rmin{f −1(�Ξ2 (%δ)), f −1(�Ξ2 (%δ − ỳδ))}. Let %δ, ỳδ ∈ Γ, so that %δ ≥ ỳδ. Subsequently, Ξ2 is an InVF β-filter, �Ξ2 (f (ỳδ)) ≥ �Ξ2 (f (%δ)) = f −1(�B(%δ)) such that f−1(�Ξ2 (ỳδ)) ≥ f −1(�Ξ2 (%δ)). � 3.1. Products on InVF β-Filters on β-Algebras. The basic concepts and examples of product on InVF β-filters settings are covered in this section. Theorem 3.5. An InVF β-filter is the Cartesian product of any two InVF β-filters. 6 Int. J. Anal. Appl. (2022), 20:50 Proof. Take %δ = (%δ1,%δ2) & ỳδ = (ỳδ1, ỳδ2) ∈ Γ ×Y & σ = (�Ξ1 × �Ξ2 ). So σΞ1×Ξ2 (%δ 4 ỳδ) = �Ξ1 ((%δ1,%δ2) 4 (ỳδ1, ỳδ2)) = (�Ξ1 × �Ξ2 ){((%δ1,%δ2) + ((%δ1,%δ2) + (ỳδ1, ỳδ2)))} ≥ rmin{�Ξ1 (%δ1 + (%δ1 + ỳδ1)),�Ξ2 (%δ2 + (%δ2 + ỳδ2))} = rmin{rmin{�Ξ1 (%δ1),�Ξ1 (%δ1 + ỳδ1)}, rmin{�Ξ2 (%δ2),�Ξ2 (%δ2 + ỳδ2)}} = rmin{rmin{�Ξ1 (%δ1),�Ξ2 (%δ2)}, rmin{�Ξ1 (%δ1 + ỳδ1),�Ξ2 (%δ2 + ỳδ2)}} = rmin{σΞ1×Ξ2 (%δ1,%δ2),σΞ1×Ξ2 ((%δ1,%δ2) + (ỳδ1, ỳδ2))} = rmin{σΞ1×Ξ2 (%δ),σΞ1×Ξ2 (%δ + ỳδ)}. Similarly, σΞ1×Ξ2 (%δ5 ỳδ) ≥ rmin{σΞ1×Ξ2 (%δ),σΞ1×Ξ2 (%δ− ỳδ)}. This proved that Ξ1 × Ξ2 is also an InVF β-filter. � Theorem 3.6. Let Γ and Υ be two β-algebras. Let Ξ1t and Ξ2s be InVF β-filters on Γ × Y . Then(Ξ1t × Ξ2s) is also a β-filter, if t ≥ s. Proof. Take %δ = (%δ1,%δ2) & ỳδ = (ỳδ1, ỳδ2) ∈ Γ × Y & σ = (�Ξ1 × �Ξ2 ) if t ≥ s. Using above theorem, σ(Ξ1t × Ξ2s)(%δ 4 ỳδ) ≥ s. Similarly, σ(Ξ1t × Ξ2s)(%δ 5 ỳδ) ≥ s. � 3.2. InVF Strong β-Filters. Beginning with a description and some examples, this section introduces the notion of an InVF strong β-filter on a β-subalgebra. Definition 3.3. An InVF β-subalgebra Ξ of a β-algebra is referred as an InVF strong β-filter, if (1) �Ξ(%δ 4 ỳδ) = �Ξ(%δ 5 ỳδ) (2) �Ξ(ỳδ) ≥ �Ξ(%δ) if %δ ≤ ỳδ for all %δ, ỳδ ∈ Ξ. Example 3.3. For a β-algebra Γ = {0,η1,η2,η3} be a with two binary operations + and − constant 0 and defined on Γ, we have a Cayley’s table + 0 η1 η1 η3 0 0 0 0 0 η1 η1 η1 η1 η1 η1 0 0 η1 η3 η3 η3 η3 η3 η3 − 0 η1 η1 η3 0 0 0 0 0 η1 η1 η1 η1 η1 η1 η1 η1 η1 η1 η3 η3 η3 η3 η3 Then (Γ, +,−, 0) is a β-algebra. Thus Ξ = {η1,η3} is a β-filter on Γ. Defining the membership function for an InVF β-subalgebra Ξ as �Ξ(%δ) =  [0.2, 0.5] : %δ = η1 [0.3, 0.6] : %δ = η3 . Int. J. Anal. Appl. (2022), 20:50 7 Then it is observed that, Ξ is an InVF strong β-filter on Γ. Theorem 3.7. Every InVF strong β-filter is also an InVF β-subalgebra. It is not necessary for the converse part of the theorem to be correct. Example 3.4. For a β-algebra Γ = {0,η1,η2,η3} with two binary operations + and − and constant 0 defined on Γ, we have a Cayley’s table + 0 η1 η1 η3 0 0 0 0 0 η1 η1 η1 η3 0 η1 η1 0 η1 η3 η3 η3 η1 η3 η3 − 0 η1 η1 η3 0 0 0 0 0 η1 η1 η1 η1 η1 η1 η1 η1 η1 η1 η3 η3 η3 η3 η3 Then (Γ, +,−, 0) is a β-algebra. Then Ξ = {η1,η3} is β-filter on Γ. Ξ is an InVF β-subalgebra, define by the membership function �Ξ(%δ) =  [0.4, 0.6] : %δ = η3 [0.3, 0.5] : %δ = η1 . Then it is observed that, Ξ is not an InVF strong β-filter on Γ since �Ξ(η1 4η3) 6= �Ξ(η1 5η3). Theorem 3.8. If σ is an InVF strong β-filter of Γ, then �Ξ(%δ 4 ỳδ) ≥ �Ξ(ỳδ) where ỳδ ≤ %δ. Theorem 3.9. Consider be an onto β-algebra homomorphism f from Γ to Υ . If Ξ2 is a InVF strong β-filter of Υ ,then its inverse image f−1(Ξ2) is again an InVF strong β-filter on Γ. 4. InVInF β-Filters on β-Algebras The concept of Interval valued intuitionstic fuzzy (InVInF) β-filters on a β-subalgebra is introduced in this section, which starts with the definition. Definition 4.1. An InVInF β-subalgebra of a β-algebra Γ is called as an InVInF β-filter on Γ, if (1) �Ξ(%δ 4 ỳδ) ≥ rmin{�Ξ(%δ),�Ξ(%δ + ỳδ)} and φΞ(%δ 4 ỳδ) ≤ rmax{φΞ(%δ),φΞ(%δ + ỳδ)} (2) �Ξ(%δ 5 ỳδ) ≥ rmin{�Ξ(%δ),�Ξ(%δ − ỳδ)} and φΞ(%δ 5 ỳδ) ≤ rmax{φΞ(%δ),φΞ(%δ − ỳδ)} (3) �Ξ(ỳ) ≥ �Ξ(%δ) and φΞ(ỳδ) ≤ φΞ(%δ) if %δ ≤ ỳδ for all %δ, ỳδ ∈ Γ. 8 Int. J. Anal. Appl. (2022), 20:50 Example 4.1. Consider a β-algebra Γ = {0,ρ1,ρ2,ρ3} with two binary operations + and − and a constant 0 defined on Γ with the Cayley’s table: + 0 ρ1 ρ2 ρ3 0 0 0 0 0 ρ1 ρ1 0 ρ3 ρ1 ρ2 ρ2 0 ρ2 ρ3 ρ3 ρ3 ρ1 ρ3 ρ3 − 0 ρ1 ρ2 ρ3 0 0 0 0 0 ρ1 ρ1 ρ1 ρ1 ρ1 ρ2 ρ2 ρ2 ρ2 ρ2 ρ3 ρ3 ρ3 ρ3 ρ3 Now, Ξ = {ρ2,ρ3} is a β-filter on Γ. Defining the membership and non membership function of an InVInF β-subalgebra Ξ as �Ξ(%δ) =  [0.4, 0.6] : %δ = 0,ρ3 [0.3, 0.5] : %δ = ρ1,ρ2 . and φΞ(%δ) =  [0.3, 0.4] : %δ = 0,ρ3 [0.4, 0.6] : %δ = ρ1,ρ2 . Therefore, Ξ is an InVInF β-filter on Γ. Example 4.2. Consider a β-algebra Γ = {0,ρ1,ρ2,ρ3} with two binary operations + and − and a constant 0 defined on Γ with the Cayley’s table: + 0 ρ1 ρ2 ρ3 0 0 0 0 0 ρ1 ρ1 ρ1 ρ1 ρ1 ρ2 0 ρ1 ρ2 ρ3 ρ3 ρ3 ρ1 ρ2 ρ3 − 0 ρ1 ρ2 ρ3 0 0 0 0 0 ρ1 ρ1 ρ1 ρ1 ρ1 ρ2 ρ2 ρ2 ρ2 ρ2 ρ3 ρ3 ρ3 ρ3 ρ3 Now, Ξ = {ρ1,ρ2,ρ3} is a β-filter on Γ. Defining the membership and non membership function of an InVInF β-subalgebra Ξ as �Ξ(%δ) =   [0.2, 0.6] : %δ = 0 [0.4, 0.5] : %δ = ρ1,ρ3 [0.3, 0.7] : %δ = ρ2 and φΞ(%δ) =   [0.1, 0.6] : %δ = 0 [0.2, 0.7] : %δ = ρ1 [0.4, 0.5] : %δ = ρ2,ρ3 . This shows that, Ξ is not an InVInF β-filter on Γ because �Ξ(ρ3) ≥ �Ξ(ρ2) ⇒ [0.4, 0.5] � [0.3, 0.7]. Int. J. Anal. Appl. (2022), 20:50 9 Lemma 4.1. If Ξ1 and Ξ2 be any two InVInF β-filters on Γ, then Ξ1 ∩ Ξ2 is also an InVInF β-filter of Γ. Proof. For %δ, ỳδ ∈ Γ σΞ1∩Ξ2 (%δ 4 ỳδ) = rmin{�Ξ1 (%δ 4 ỳδ),�Ξ2 (%δ 4 ỳδ)} ≥ rmin{rmin{�Ξ1 (%δ),�Ξ1 (%δ + ỳδ)}, rmin{�Ξ2 (%δ),�Ξ2 (%δ + ỳδ)}} ≥ rmin{rmin{�Ξ1 (%δ),�Ξ2 (%δ)}, rmin{�Ξ1 (%δ + ỳδ),�Ξ2 (%δ + ỳδ)}} = rmin{�Ξ1∩Ξ2 (%δ),�Ξ1∩Ξ2 (%δ + ỳδ)}. Also, φΞ1∩Ξ2 (%δ 4 ỳδ) ≤ rmax{φΞ1∩Ξ2 (%δ),φΞ1∩Ξ2 (%δ + ỳδ)}. Hence, Ξ1 ∩ Ξ2 is also an InVInF β-filter of Γ. � Lemma 4.2. Every InVInF β-filter is again an InVInF β-subalgebra. Proof. The definition of the InVInF β-filter leads to this proof. � In general, the converse of the preceding lemma does not seems to be true, as shown by the following example (i.e. Every InVInF β-subalgebra need not be an InVInF β-filter). Example 4.3. Let Γ = {0,ω1,ω2,ω3} be a β-algebra with constant 0 and the Cayley’s table: + 0 ω1 ω2 ω3 0 0 0 0 0 ω1 ω1 ω1 ω1 ω1 ω2 ω1 ω1 ω2 0 ω3 ω3 ω3 ω1 ω1 − 0 ω1 ω2 ω3 0 0 0 0 0 ω1 ω1 ω1 ω1 ω1 ω2 ω2 ω2 ω2 ω2 ω3 ω3 ω3 ω3 ω3 Now, Ξ = {0,ω3} is a β-filter on Γ. Defining the membership and non membership function of an InVInF β-subalgebra Ξ as �Ξ(%δ) =  [0.3, 0.5] : %δ = 0,ω2 [0.2, 0.4] : %δ = ω1,ω3 . and φΞ(%δ) =  [0.3, 0.5] : %δ = 0,ω2 [0.4, 0.6] : %δ = ω1,ω3 . However Ξ is not an InVInF β-filter on Γ because �Ξ(ω3) ≥ �Ξ(ω1) ⇒ [0.2, 0.4] � [0.3, 0.5]. Theorem 4.1. If Ξ is an InVInF β-filter of Γ, then �Ξ(%δ4 ỳδ) ≥ �Ξ(%δ) and φΞ(%δ5 ỳδ) ≤ φΞ(%δ) where %δ ≤ ỳδ. 10 Int. J. Anal. Appl. (2022), 20:50 Proof. Assume that Ξ is an InVInF β-filter of Γ. Let %δ, ỳδ ∈ Γ. Then �Ξ(%δ 4 ỳδ) = �Ξ(%δ + (%δ + ỳδ)) ≥ rmin{�Ξ(%δ),�Ξ(%δ + ỳδ)} = rmin{�Ξ(%δ), rmin{�Ξ(%δ),�Ξ(ỳδ)}} = rmin{�Ξ(%δ),�Ξ(%δ)}∵ %δ ≤ ỳδ ⇒ �Ξ(ỳδ) ≤ �Ξ(%δ) = �Ξ(%δ). Similarly, φΞ(%δ 5 ỳδ) = φΞ(%δ − (%δ − ỳδ)) ≤ rmax{φΞ(%δ),φΞ(%δ − ỳδ)} = rmax{φΞ(%δ), rmax{φΞ(%δ),φΞ(ỳδ)}} = rmax{φΞ(%δ),φΞ(%δ)}∵ %δ ≤ ỳδ ⇒ φΞ(ỳδ) ≤ φΞ(%δ) = φΞ(%δ). � Definition 4.2. Let Ξ be an InVInF β-filter of a β-subalgebra Γ. For s,t ∈ D[0, 1], the set Ξs,t = {%δ ∈ Γ : �Ξ(%δ) ≥ s & φΞ(%δ) ≤ t} is referred as a level set of InVInF β-filter Ξ of Γ. Theorem 4.2. An InVInF subset Ξ of a β-algebra Γ is an InVInF β-filter if and only if for any s,t ∈ D[0, 1] the Ξs t−InVInF level subset Ξs t = {%δ ∈ Γ : �Ξ(%δ) ≥ s & φΞ(%δ) ≤ t} is either a β-filter or Ξs,t 6= ∅. Proof. Consider an InVInF level subset of Ξ in Γ, Ξs,t 6= ∅. For any %δ, ỳδ ∈ Ξs,t , �Ξ(%δ) ≥ s & �Ξ(%δ) ≥ s. Now �Ξ(%δ 4 ỳ) = �Ξ(%δ + (%δ + ỳδ)) ≥ rmin{�Ξ(%),�Ξ(%δ + ỳδ)} = rmin{�Ξ(%δ), rmin{�Ξ(%δ),�Ξ(ỳδ)}} = rmin{s,rmin{s,s}} = s. This implies that %δ 4 ỳδ ∈ Ξs t. Similarly, �Ξ(%δ 5 ỳδ) = rmin{�Ξ(%δ),�Ξ(%δ − ỳδ)}. Analogously, Int. J. Anal. Appl. (2022), 20:50 11 φΞ(%δ 4 ỳ) = φΞ(%δ + (%δ + ỳδ)) ≤ rmax{φΞ(%δ),φΞ(%δ + ỳδ)} = rmax{φΞ(%δ), rmax{φΞ(%δ),φΞ(ỳδ)}} = rmax{t,rmax{t,t}} = t. Similarly, φΞ(%δ5 ỳδ). Then φΞ(%δ4 ỳδ) ∈ Ξs t & φΞ(%δ5 ỳ) ∈ Ξs t. So, (%δ4 ỳδ) ∈ Ξs t & (%δ5 ỳδ) ∈ Ξs t. Hence Ξs t is a β-filter of Γ. On the other hand, assume that Ξs t is a β-filter of Γ. For all %δ, ỳδ ∈ X,%δ4ỳδ and %δ5ỳδ ∈ Ξs t. Thus �Ξ(%δ4 ỳδ) ≥ s and Ξ(%δ5 ỳδ) ≥ s. Take s = rmin{�Ξ(%δ),�Ξ(%δ + ỳδ)} for any %δ, ỳδ ∈ X. We have �Ξ(%δ4 ỳδ) = �Ξ(%δ + (%δ + ỳδ)) ≥ s = rmin{�Ξ(%δ),�Ξ(%δ + ỳδ)}. Similarly, �Ξ(%δ5 ỳδ). Analogously, for the non membership function. This proves Ξ is an InVInF β-filter. � Theorem 4.3. Consider an onto β-algebra homomorphism f from Γ to Υ . If Ξ2 is an InVInF β-filter of Υ , then its inverse image f−1(Ξ2) is also an InVInF β-filter on Γ. Proof. Suppose that Ξ2 is an InVInF β-filter of Υ. For any %δ, ỳδ ∈ Γ, f−1(�Ξ2 (%δ 4 ỳδ)) = f −1(�Ξ2 (%δ + (%δ + ỳδ))) = �Ξ2 (f (%δ + (%δ + ỳδ))) = �Ξ2 (f (%δ) + f (%δ + ỳδ)) ≥ rmin{�Ξ2 (f (%δ)),�Ξ2 (f (%δ + ỳδ))} = rmin{f−1(�Ξ2 (%δ)), f −1(�Ξ2 (%δ + ỳδ))}. Also, f−1(�Ξ2 (%δ 5 ỳδ)) ≥ rmin{f −1(�Ξ2 (%δ)), f −1(�Ξ2 (%δ − ỳδ))}. Analogously, f−1(φΞ2 (%δ 4 ỳδ)) = f −1(φΞ2 (%δ + (%δ + ỳδ))) = φΞ2 (f (%δ + (%δ + ỳδ))) = φΞ2 (f (%δ) + f (%δ + ỳδ)) ≤ rmax{φΞ2 (f (%δ)),φΞ2 (f (%δ + ỳδ))} = rmax{f−1(φΞ2 (%δ)), f −1(φΞ2 (%δ + ỳδ))}. Similarly, f−1(φΞ2 (%δ 5 ỳδ)) ≤ rmax{f −1(φΞ2 (%δ)), f −1(φΞ2 (%δ − ỳδ))}. Let %δ, ỳδ ∈ Γ, so that %δ ≥ ỳδ. Consequently, Ξ2 is an InVInF β-filter, �Ξ2 (f (ỳδ)) ≥ �Ξ2 (f (%δ)) = f −1(�Ξ2 (%δ)) such that f−1(�Ξ2 (ỳδ) ≥ f −1(�Ξ2 (%δ)) and φΞ2 (f (ỳδ)) ≤ φΞ2 (f (%δ)) = f −1(φΞ2 (%δ)) such that f −1(φΞ2 (ỳδ) ≥ f−1(φΞ2 (%δ)). This shows that f −1 is an InVInF β-filter on Γ. � 12 Int. J. Anal. Appl. (2022), 20:50 5. 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Sci. 8 (1975), 199-249. https://doi.org/10.1016/0020-0255(75)90036-5. https://doi.org/10.3390/math8081373 https://doi.org/10.3390/math7111094 https://doi.org/10.1016/0020-0255(75)90036-5 1. Introduction 2. Preliminares 3. Interval Valued Fuzzy -Filters 3.1. Products on InVF -Filters on -Algebras 3.2. InVF Strong -Filters 4. InVInF -Filters on -Algebras 5. Conclusion References