EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 3, 2021, 737-745 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fuzzy Translation and Fuzzy multiplication in BRK-algebras Halimah Alshehri Department Computer Science and Engineering,Faculty Applied Studies and Community Service, King Saud University, Riyadh, Saudi Arabia Abstract. In this paper, the concepts of fuzzy translation and fuzzy multiplication on a BRK- algebra are introduced. We investigated fuzzy translation and fuzzy multiplication (BRK-subalgebras & BRK-ideals) in BRK-algebras and discussed related properties. Finally, we presented the nation fuzzy magnified-αβ-translation on BRK-algebra X. 2020 Mathematics Subject Classifications: 08A72, 03E72, 03B52 Key Words and Phrases: BRK-algebra, fuzzy translation BRK-subalgebras, fuzzy multipli- cation BRK-subalgebras, fuzzy translation BRK-ideals, fuzzy multiplication BRK-ideals, fuzzy magnified-αβ-translation 1. Introduction The fundamental concept of fuzzy set, popularized by Zadeh [10], was used to gener- alize several basic concepts of algebra. Fuzzy sets are extremely useful to deal with the many problems in applied mathematics, control engineering, information sciences, expert systems etc. Although there are several generalizations of fuzzy sets, none of them address the issues of members with membership degree 0 who have opposing qualities. Lee [7] han- dled this problem by introducing the concept of bipolar fuzzy (BF) sets. A BF set is a pair of fuzzy sets, namely a membership and a non-membership function, which represent pos- itive and negative aspects of the given information. Imai and Iseki investigated two classes of abstract algebras: BCI-algebras and BCK-algebras [5]. Recently, Bandaru [1] investi- gated BRK-algebra which is a generalization of BCK/BCI/BCH/Q/QS/BM-algebras. In [2,3], Elgendy introduced fuzzy BRK-ideal of BRK-algebra and cubic BRK-ideal of BRK- algebra. Some properties of n-dimensional fuzzy subalgebra in BRK-algebras investigated by Zulfiqar [11]. Fuzzy translations and fuzzy multiplications of BCK/BCI-algebras pre- sented in [8]. The contents of the current paper are structured as follows: In Sect. 2, we presented some basic definitions and preliminaries. In Sect. 3, we investigated fuzzy DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.3971 Email address: haalshehri@ksu.edu.sa (H. Alshehri) http://www.ejpam.com 737 © 2021 EJPAM All rights reserved. H. Alshehri / Eur. J. Pure Appl. Math, 14 (3) (2021), 737-745 738 translation and fuzzy multiplication of BRK-subalgebras and discussed related properties. In Sect. 4, we introduced fuzzy translation and fuzzy multiplication of BRK-ideals and discussed related results. In Sect. 5, we defined concept of Fuzzy magnified-αβ-translation of BRK-algebras. At last, some conclusions and future work were presented. 2. Preliminaries Some elementary aspects that are important for this paper are included in this section. Definition 1 (1). A BRK-algebra is a non-empty set X with a constant 0 and a binary operation “∗”satisfyingthefollowingconditions : (BRK1) x ∗0 = x, (BRK2) (x ∗ y) ∗ x = 0 ∗ y , for all x, y ∈ X. A partial ordered relation ≤ canbedefinedbyx≤ y if and only if x ∗ y = 0. Throughout this paper, X denotes BRK-algebra. Definition 2 (1). If (X, ∗, 0) is a BRK-algebra, the following conditions hold: (BRK3) x ∗ x = 0, (BRK4) (x ∗ y) = 0 implies 0 ∗ x = 0 ∗ y for all x, y ∈ X, (BRK5) 0 ∗ (x ∗ y) = (0 ∗ x) ∗ (0 ∗ y) for all x, y ∈ X, . Definition 3 (1). A subset S of a BRK-algebra X is said to be BRK-subalgebra of X, if x, y ∈ S, implies x ∗ y ∈ S. Definition 4 (1). A non-empty subset I of a BRK-algebra X is said to be a BRK-ideal of X if it satisfies: (I1) 0 ∈ I, (I2) 0 ∗ (x ∗ y) ∈ I and 0 ∗ y ∈ I imply 0 ∗ x ∈ I, for all x, y ∈ X. Definition 5 (10). A fuzzy subset µ in a non-empty set X is a function µ : X −→ [0, 1]. Definition 6 (2). A Fuzzy Subset µ in a BRK-algebra X is said to be a Fuzzy BRK -subalgebra of X if µ(x ∗ y) ≥ min{µ(x), µ(y)}∀x, y ∈ X. Definition 7 (2). Let (X, ∗, 0) be a BRK-algebra. A fuzzy set µ in X is called a fuzzy BRK-ideal of X if it satisfies: (FI1) µ(0) ≥ µ(x), (FI2) µ(0 ∗ x) ≥ min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)},∀x, y ∈ X. 3. Fuzzy Translation and Fuzzy Multiplication of BRK-subalgebras This section deals with the notion of Fuzzy translation and Fuzzy multiplication on BRK-algebras. In what follows, X denotes a BRK -algebra, and for any fuzzy set µ of X, we denote T = 1− sup{µ(x)|x ∈ X} unless otherwise specified. We start with, H. Alshehri / Eur. J. Pure Appl. Math, 14 (3) (2021), 737-745 739 Definition 8. Let µ be a fuzzy subset of X and α ∈ [0, 1]. A mapping µTα : X −→ [0, 1] is said to be a fuzzy α− translationofµ if it satisfies: µTα (x) = µ(x) + α,∀x ∈ X. Definition 9. Let µ be a fuzzy subset of X and α ∈ [0, 1]. A mapping µMα : X −→ [0, 1] is said to be a fuzzy α- multiplication of µ if it satisfies: µMα (x) = α.µ(x) ,∀x ∈ X. Definition 10. A fuzzy α-translation set µTα(x) of µ is called fuzzy α-translation BRK- subalgebra of X if it satisfies following condition: µTα(x∗y) ≥ min{µTα(x), µTα(y)}, Similarly, wesaidthatµMα (x) is fuzzy α-multiplication BRK-subalgebra of X if it satisfies: µMα (x∗y) ≥ min{µMα (x), µMα (y)}. Example 1. Consider a set X = {0, a, b, c}. We define “∗”onXasthefollowingtable : ∗ 0 a b c 0 0 a 0 a a a 0 a 0 b b a 0 a c c b c 0 Define a fuzzy subset µ of X by µ(0) = µ(a) = 0.6 and µ(b) = µ(c) = 0.1, routine cal- culation gives that µ is fuzzy BRK-subalgebra of X. Here T = 1 − sup{(x) : x ∈ X} = 1− 0.6 = 0.4. Choose α = 0.2 ∈ [0, T ] and β = 0.3 ∈ [0, 1]. Then the mapping µT0 .2(x): X −→ [0, 1] defined by µT0 .2(x)= { 0.6 + 0.2 = 0.8 ;x = 0, a 0.1 + 0.2 = 0.3 ;x = b, c µT0 .2(x)=µ(x) + 0.2 , ∀x ∈ X, isafuzzy0.2−translation. µM0 .3(x): X −→ [0, 1] defined by µM0 .3(x)= { (0.3)(0.6) = 0.18 ;x = 0, a (0.3)(0.1) = 0.3 ;x = b, c µM0 .3(x)=(0.3)µ(x) , ∀x ∈ X, isafuzzy0.3−multiplication. Theorem 1. For any fuzzy BRK-subalgebra µ of X and α ∈ [0, T ] , the fuzzy α − translation µTα(x) of µ is a fuzzy BRK-subalgebra of X. Proof. Let x, y ∈ X and α ∈ [0, T ]. Then µ(x ∗ y) ≥ min{µ(x), µ(y)}. Now, H. Alshehri / Eur. J. Pure Appl. Math, 14 (3) (2021), 737-745 740 µMα (x ∗ y)= µ(x) + α ≥ min{µ(x), µ(y)}+ α =min{µ(x) + α, µ(y) + α} =min{µMα (x), µMα (y)} This completes the proof. The converse of the above theorem is valid. Theorem 2. For any fuzzy subset µ of X and α ∈ [0, T ] , if the fuzzy α-translation µTα(x) of µ is a fuzzy BRK-subalgebra of X then so is µ. Proof. Let x, y ∈ X. Assume that µTα(x) of µ is a fuzzy BRK-subalgebra of X for α ∈ [0, 1] . Then µ(x ∗ y) + α = µTα(x ∗ y) ≥ min{µTα(x), µTα(y)} = min{µ(x) + α, µ(y) + α} = min{µ(x), µ(y)}+ α. Hence, µ(x ∗ y) ≥ min{µ(x), µ(y)}. Therefore, µ is a fuzzy BRK-subalgebra of X. Theorem 3. For any fuzzy BRK-subalgebra µ of X and α ∈ [0, 1], the fuzzy α-multiplication µMα (x) of µ is a fuzzy BRK-subalgebra of X. Proof. Let x, y ∈ X and α ∈ [0, 1]. Then µ(x ∗ y) ≥ min{µ(x), µ(y)}. Now, µMα (x ∗ y) = α.µ(x ∗ y) ≥ α.min{µ(x), µ(y)} = min{α.µ(x), α.µ(y)} = min{µM (x), µM (y)}. This completes the proof. The following is the converse of the above theorem. Theorem 4. For any fuzzy subset µ of X and α ∈ [0, T ] ,if the fuzzy α- multiplication µMα (x) of µ is a fuzzy BRK-subalgebra of X then so is µ. Proof. Let x, y ∈ X. Assume that µMα (x) of µ is a fuzzy BRK-subalgebra of X for α ∈ [0, 1]. Then α.µ(x ∗ y) = µMα (x ∗ y) ≥ min{µM (x), µM (y)} = min{α.µ(x), α.µ(y)} = α.min{µ(x), µ(y)} Hence, µ(x ∗ y) ≥ min{µ(x), µ(y)} Therefore, µ is a fuzzy BRK-subalgebra of X. 4. Fuzzy Translation and Fuzzy Multiplication of BRK-ideals Definition 11. A fuzzy α-translation set µTα(x) of µ is called fuzzy α-translation BRK- ideal of X if it satisfies following condition: (FTI1) µTα(0) ≥ µTα(x), H. Alshehri / Eur. J. Pure Appl. Math, 14 (3) (2021), 737-745 741 (FTI2) µTα(0 ∗ x) ≥ min{µTα(0 ∗ (x ∗ y)), µTα(0 ∗ y)}, ∀x, y ∈ X. Similarly, we said that µTα(x) is fuzzy α-multiplication BRK-ideal of X if it satisfies: (FMI1) µMα (0) ≥ µMα (x), (FMI2) µMα (0 ∗ x) ≥ min{µMα (0 ∗ (x ∗ y)), µMα (0 ∗ y)},∀x, y ∈ X. Theorem 5. For any fuzzy BRK-ideal µ of X and α ∈ [0, T ] , the fuzzy α-translation µTα(x) of µ is a fuzzy BRK-ideal of X. Proof. Let x, y ∈ X and α ∈ [0, T ]. Then µ(0 ∗ x) ≥ min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)} Now, µTα(0 ∗ x) = µ(0 ∗ x) + α ≥ min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)}+ α = min{µ(0 ∗ (x ∗ y)) + α, µ(0 ∗ y) + α} = min{µMα (0 ∗ (x ∗ y)), µMα (0 ∗ y)} Hence, µTα(x) is a fuzzy BRK-ideal of X. The following is the converse of the above theorem. Theorem 6. For any fuzzy subset µ of X and α ∈ [0, T ] , if the fuzzy α-translation µTα(x) of µ is a fuzzy BRK-ideal of X then so is µ. Proof. Let x, y ∈ X. Assume that µTα(x) of µ for α ∈ [0, 1]. Then µ(0 ∗ x) + α = µTα(0 ∗ x) ≥ min{µMα (0 ∗ (x ∗ y)), µMα (0 ∗ y)} = min{µ(0 ∗ (x ∗ y)) + α, µ(0 ∗ y) + α} = min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)}+ α Hence, µ(0 ∗ x) ≥ min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)}. Therefore, µ is a fuzzy BRK-ideal of X. Theorem 7. For any fuzzy BRK-ideal µ of X and α ∈ [0, T ] ,the fuzzy α-multiplication µMα (x) of µ is a fuzzy BRK-ideal of X. Proof. Let x, y ∈ X and α ∈ [0, T ]. Then µ(0 ∗ x) ≥ min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)} H. Alshehri / Eur. J. Pure Appl. Math, 14 (3) (2021), 737-745 742 Now, µMα (0 ∗ x) = α.µ(0 ∗ x) ≥ α.min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)} = min{α.µ(0 ∗ (x ∗ y)), α.µ(0 ∗ y)} = min{µMα (0 ∗ (x ∗ y)), µMα (0 ∗ y)} Hence, µMα (x) is a fuzzy BRK-ideal of X. The following is the converse of the above theorem. Theorem 8. For any fuzzy subset µ of X and α ∈ [0, T ] , if the fuzzy α-multiplication µMα (x) of µ is a fuzzy BRK-ideal of X then so is µ. Proof. Let x, y ∈ X. Suppose that µMα (x) of µ for α ∈ [0, 1]. Then α.µ(0 ∗ x) = µMα (0 ∗ x) ≥ min{µMα (0 ∗ (x ∗ y)), µMα (0 ∗ y)} = min{α.µ(0 ∗ (x ∗ y)), α.µ(0 ∗ y)} = α.min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)} Hence, µ(0 ∗ x) ≥ min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)}. Therefore, µ is a fuzzy BRK-ideal of X. Lemma 1. Let I be a fuzzy α-translation (or fuzzy α-multiplication) BRK-ideal of BRK- algebra X. If x ≤ y holds in X, then µTα(0 ∗ x) = µTα(0 ∗ y) (or µMα (0 ∗ x) = µMα (0 ∗ y)). Proof. Assume that x ≤ y holds in X. Then x ∗ y = 0. By (BRK4) implies µTα(0 ∗ x) = µTα(0 ∗ y) Lemma 2. Let I be a fuzzy α-translation (or fuzzy α-multiplication) BRK-ideal of BRK- algebra X. If x ∗ y ≤ x holds in X, then µTα(0 ∗ y) ≥ µTα(0 ∗ x) (or µMα (0 ∗ y) ≥ µMα (0 ∗ x)). Proof. Assume that x ∗ y ≤ x holds in X. Then (x ∗ y) ∗ x = 0. By (BRK2) µTα(0 ∗ y) = min{µTα(0 ∗ (x ∗ y)), µTα(0 ∗ x)} Also, µTα(0 ∗ (x ∗ y)) ≥ min{µTα((0 ∗ (x ∗ y)) ∗ x), µTα(0 ∗ x)} = min{µTα(0), µTα(x)} = {µTα(0 ∗ x)} Hence, µTα(0 ∗ y) ≥ µTα(0 ∗ x). H. Alshehri / Eur. J. Pure Appl. Math, 14 (3) (2021), 737-745 743 5. Fuzzy magnified-αβ-translation of BRK-algebras Definition 12. Let µ be a fuzzy subset of X, α ∈ [0, T ],where T = 1− sup{µ(x) : x ∈ X} and β ∈ [0, 1]. A mapping µMT (αβ) :X −→ [0, 1] ,is said to be a fuzzy magnified-αβ- translation of µ if it satisfies: µMT (αβ) = α.x+ β. Example 2. Consider the BRK-algebra X = 0, a, b, c in example 3.4. Define a fuzzy Subset µ of X by µ(x)= { 0.6 ;x = 0, a 0.1 ;x = b, c Then µ is fuzzy BRK-subalgebra of X. Here, T = 1 − sup{µ(x) : x ∈ X} = 1 − 0.6 = 0.4.Choose α = 0.5 ∈ [0, T ] and β = 0.2 ∈ [0, 1]. Then the mapping µMT (0.5)(0.2) :X −→ [0, 1] defined by µMT (0.5)(0.2) = { (0.5)(0.6) + 0.2 = 0.5 ;x = 0, a (0.5)(0.1) + 0.2 = 0.7 ;x = b, c which satisfiesµMT (0.5)(0.2) =αµ(x) + β;∀x ∈ X is fuzzy magnified-(0.5)(0.2)-translation. Theorem 9. Let µ be a fuzzy subset of X, α ∈ [0, T ] and β ∈ [0, 1]. A mapping µMT (αβ) :X −→ [0, 1] is said to be a fuzzy magnified-αβ-translation of µ. Then µ is fuzzy BRK-subalgebra of X if and only if µMT (αβ) is fuzzy subalgebra of X. Proof. Let µ be a fuzzy subset of X, α ∈ [0, T ] and β ∈ [0, 1]. A mapping µMT (αβ) :X −→ [0, 1] is said to be a fuzzy magnified-αβ-translation of µ. Assume µ is fuzzy BRK-subalgebra of X. Then µ(x ∗ y) ≥ min{µ(x), µ(y)} Now, µMT (αβ) (x ∗ y)=α.µ(x ∗ y) + β ≥ α.min{µ(x), µ(y)}+ β = min{α.µ(x) + β, α.µ(y) + β} = min{µMT (αβ) (x),µMT (αβ) (y)} Hence, µMT (αβ) is fuzzy subalgebra of X. Conversely assume,µMT (αβ) is fuzzy subalgebra of X. Then, α.µ(x ∗ y) + β=µMT (αβ) (x ∗ y)≥ min{µMT (αβ) (x), µMT (αβ) (y)} = min{α.µ(x) + β, α.µ(y) + β} = α.min{µ(x), µ(y)}+ β REFERENCES 744 Hence, µ(x ∗ y) ≥ min{µ(x), µ(y)}, so µ is fuzzy BRK-subalgebra of X. Theorem 10. Let µ be a fuzzy subset of X, α ∈ [0, T ] and β ∈ [0, 1]. A mapping µMT (αβ) :X −→ [0, 1] is said to be a fuzzy magnified-αβ-translation of µ. Then µ is fuzzy BRK-ideal of X if and only if µMT (αβ) is fuzzy ideal of X. Proof. Let µ be a fuzzy subset of X, α ∈ [0, T ] and β ∈ [0, 1]. A mapping µMT (αβ) :X −→ [0, 1] is said to be a fuzzy magnified-αβ-translation of µ. Assume µ is fuzzy BRK-ideal of X. Then µ(0 ∗ x) ≥ min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)} Now, µMT (αβ) (0 ∗ x) =α.µ(0 ∗ x) + β ≥ α.min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)}+ β = min{α.µ(0 ∗ (x ∗ y)) + β, µ(0 ∗ y) + β} = min{µMT (αβ) (0 ∗ (x ∗ y)),µMT (αβ) (0 ∗ y)} Hence, µMT (αβ) is fuzzy BRK-ideal of X. Conversely, suppose that µMT (αβ) is fuzzy BRK-ideal of X. Then, α.µ(0 ∗ x)= µMT (αβ) (0 ∗ x)≥ min{µMT (αβ) (0 ∗ (x ∗ y)),µMT (αβ) (0 ∗ y)} =min{(α.µ(0 ∗ (x ∗ y)) + β), (α.µ(0 ∗ y) + β)} =α.min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)}+β So, µ(0 ∗ x) ≥ min{µ(0 ∗ (x ∗ y)), µ(0 ∗ y)}. Hence,µ is fuzzy BRK-ideal of X. 6. Conclusion In this article, we investigated the notion of fuzzy translation and fuzzy multiplication BRK-subalgebras and discussed related properties. We introduced fuzzy translation and fuzzy multiplication BRK-ideals and discussed related results. Also, we defined Fuzzy magnified-αβ-translation of BRK-algebras. As an extension of above results, one could study anti fuzzy translation and fuzzy multiplication on BRK-algebras in other algebraic structures. Method implementations to fix similar concerns in machine learning, decision- making, knowledge science, cognitive science, smart decision-making, etc. References [1] R.K.Bandaru, On BRK-algebras. Int. J. Math. Mathematical Sci. , 2012. REFERENCES 745 [2] O. R. Elgendy, Fuzzy BRK-ideal of BRK-algebra., JP J. Algebra Number Theory Appl. , 36, 231–240, 2015. [3] O. R. Elgendy, Cubic BRK-ideal of BRK-algebra., Ann. Fuzzy Math. Inf., 11, 1–9, 2016. [4] Q.P. Hu and X. Li , On BCH-algebras., Math. Sem. Notes Kobe Univ. , 2, 1983. [5] Y. Imai and K. Is´eki, On axiom systems of propositional calculi, XIV. , Proc. Jpn. Acad., 42, 19–22 1966. [6] H. Khizar, L. Xiao and C. Bing, Bipolar Fuzzy BRK-ideals in BRK-algebras., Conference: International workshop on Mathematics and Decision Science, 3-15 2018. [7] K.M. Lee, Bipolar-valued fuzzy sets and their operations., Proceedings of the International Conference on Intelligent Technologies,Bangkok, Thailand, 307–312, 2000. [8] K.J.Lee, Y. B. Jun and M. I. Doh, Fuzzy translations and fuzzy multiplications of BCK/BCI -algebras., Commun. Korean Math. Soc., 24, 353–360, 2009. [9] J. Neggers, S.S. Ahn and H.S.Kim, On Q-algebras., Int. J. Math. Sci., 27, (12), 749-757, 2001. [10] L.A.Zadeh, Fuzzy sets., Inform. and Control, 8, 338–353, 1965. [11] M.Zulfiqar, Some properties of n-dimensional fuzzy subalgebra in BRK-algebras., Analele stiintifice ale Universitatii Ovidius Constanta, 2015.