EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 1, 2022, 36-46 ISSN 1307-5543 – ejpam.com Published by New York Business Global Pythagorean Fuzzy Small Submodules Areej Almuhaimeed Department of Mathematics, College of Science, Taibah University, Medina, Saudi Arabia Abstract. In this paper, we introduce the notion of a pythagorean fuzzy small submodule. We prove various characterisations for pythagorean fuzzy small submodules. We provide a relation between a pythagorean fuzzy small submodule and a basic small submodule. In addition, some important properties regarding pythagorean fuzzy small submodules are investigated. 2020 Mathematics Subject Classifications: 03E72, 03B52, 94D05, 08A72 Key Words and Phrases: Pythagorean fuzzy set, pythagorean fuzzy small submodule, homo- morphism, 1. Introduction In 1965, Zadeh [16] introduced the concept of fuzzy set which was a generalisation of the classical set. This encourages many researchers to investigate set theory in fuzzy setting. Pythagorean fuzzy set is one of the most important fuzzy sets. Its importance lies behind the fact that this set can be applied in order to characterized uncertain data accurately. This kind of fuzzy sets has been widely investigated. Peng [11] introduced several op- erators on a pythagorean fuzzy set and discussed its properties. Yager [15] introduced the concept of a pythagorean fuzzy subset as a generalization of an intuitionistic fuzzy subset. In [4], lattices which have been suggested for pythagorean fuzzy sets were characterized and then the results extended to the unit disc of the complex plane. Moreover, it can be applied on many areas, for instance, decision making, information measures and aggregation operators. Yager used pythagorean memmbership in decision making [14]. In [10], some algorithms in decision making problems were presented. Grag [5] presented some generalised aggregation operators in order to illustrate a group decision making problem. In [6], he presented an improved score function for solving multi-criteria decision-making in the environment of pythagorean fuzzy set. In [8], hesitant pythagorean fuzzy set was investigated and applied to some methods for multiple criteria decision mak- ing. A new approach in computing the weight of decision makers is presented in [9] using properties of pythagorean fuzzy sets. Distance and similarity measures of pythagorean DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4170 Email address: aamuhaimeed@taibahu.edu.sa (A. Alhumaimeed) http://www.ejpam.com 36 © 2022 EJPAM All rights reserved. A. Alhumaimeed / Eur. J. Pure Appl. Math, 15 (1) (2022), 36-46 37 fuzzy set was presented and applied to decision making, see [17]. For more application of this concept in decision making, see [13] and [12]. The study of pythagorean fuzzy sets is a step in order to study q-rung orthopair fuzzy sets as a generalization of pythagorean fuzzy sets see [7], [1] and [2]. In this paper, we introduce the notion of pythagorean submodule. In addition, we present the concept pythagorean small submodule and investigate some results regarding this concept. Moreover, we find a relationship between small submodule and pythagorean fuzzy small submodule. We also study homomorphism between pythagorean fuzzy mod- ules. 2. Preliminaries Definition 1. A pythagorean fuzzy set (PFS) P of universe of discourse X is of the form P = {(a, ηP (a), η̂P (a)) : a ∈ X}, where ηP (a) and η̂P (a) are the membership and non-membership values of a respectively in which 0 ≤ ηP (a) ≤ 1, 0 ≤ η̂P (a) ≤ 1 and 0 ≤ ηP (a) 2 + η̂P (a) 2 ≤ 1, for every a ∈ X. We prsent some basic notions regarding pythagorean fuzzy sets. Definition 2. Let P, S be pythagorean fuzzy sets in a fixed set X. Then • P is a subset of S if for all a ∈ X, we have η2P (a) ≤ η2S(a) and η̂2P (a) ≥ η̂2S(a) . • η2P∩S(a) = min{η2P (a), η2S(a) : a ∈ X} and η̂2P∩S(a) = max{η̂2P (a), η̂2S(a)}. • η2P∪S(a) = max{η2P (a), η2S(a)} and η̂2P∪S(a) = min{η̂2P (a), η̂2S(a)}. • η2P+S(a) = η2P (a) + η2S(a)− η2P (a)η 2 S(a) and η̂ 2 P+S(a) = η̂2P (a)η̂ 2 S(a). Now, we are able to introduce the definition of a pythagorean fuzzy submodule. Definition 3. Let M be an R-module and P a pythagorean fuzzy subset of M . Then P is called a pythagorean fuzzy submodule of M , denoted by P ≤PF M , if the following conditions are satisfied: (1) η2P (0) = 1 and η̂2P (1) = 0. A. Alhumaimeed / Eur. J. Pure Appl. Math, 15 (1) (2022), 36-46 38 (2) η2P (a+ b) ≥ min{η2P (a), η2P (b)} for all a, b ∈M and η̂2P (a+ b) ≤ max{η̂2P (a), η̂2P (b)} for all a, b ∈M . (3) η2P (ra) ≥ η2P (a) and η̂ 2 P (ra) ≤ η̂2P (a) for all a ∈M and r ∈ R Recall that for a module M , we define the pythagorean fuzzy set χPFM = (χM , χ c M ) in which χM (a) = { 1 if a ∈M 0 otherwise and χcM (a) = { 0 if a ∈M 1 otherwise Definition 4. Let M be a module and P be a pythagorean fuzzy subset of M . Then (1) P ⋆ = η⋆P ∩ η̂⋆P , where η⋆P = {a ∈M : ηP (a) > 0} η̂⋆P = {a ∈M : η̂(a) < 1} (2) P⋆ = η⋆P ∩ η̂⋆P , where η⋆P = {a ∈M : ηP (a) = 1} η̂⋆P = {a ∈M : η̂(a) = 0} 3. Pythagorean Fuzzy Small Submodule Recall that a submodule N of a module M is called a small submodule of M , denoted byN ≪M , ifN+S ̸=M for every proper submodule S ofM . Clearly, the zero submodule is a small submodule of any module M . Moreover, a small submodule of a module M should be a proper submodule. Now, we present some well-known properties regarding the concept of small submodules. Theorem 1. [3] Suppose that M is a module and S, T,N are submodules of M such that S ≤ T . Then (1) S +N ≪M if and only if S ≪M and N ≪M . (2) T ≪M if and only if S ≪M and T S ≪ M S . (3) If S ≪ T , then S ≪M . A. Alhumaimeed / Eur. J. Pure Appl. Math, 15 (1) (2022), 36-46 39 Now, we are ready to introduce the main concept in this paper. Consider a module M . Then a PFS, P = (ηP , η̂P ), is called a pythagorean fuzzy small submodule of M , denoted by P ≪PF M , if P + S ̸= χPFM for any PSF S ̸= χPFM . That is whenever P + S = χPFM , then S = χPFM . Theorem 2. Let M be a module and P be a submodule of M . Then P ≪M if χPFP ≪PF M . Proof. Suppose that χPFP ≪PF M and P + S = M for some proper submodule S of M . Then for any m ∈M there exist a ∈ P and b ∈ S such that a+ b = m. We obtain η2 χPF P +χPF S (m) = χ2 P (m) + χ2 S(m)− χ2 P (m)χ2 S(m) ≥ min{χ2 P (a) + χ2 S(a)− χ2 P (a)χ 2 S(a), χ 2 P (b) + χ2 S(b)− χ2 P (b)χ 2 S(b)} = 1 This means that η2 χPF P +χPF S = χ2 M . Similarly, η̂2 χPF P +χPF S (m) = χc 2 P (m)χc 2 S (m) ≤ max{χc2P (a)χc 2 S (a), χc 2 P (b)χc 2 S (b)} = 0 This means that η̂2 χPF P +χPF S = χc 2 M . Thus χPFP +χPFS = χPFM , but this contradicts the facts that χPFP ≪PF M and χPFS ̸= χPFM as S is a proper submodule of M . Therefore, P is a small submodule of M . Theorem 3. Let M be a module and P be a pythagorean fuzzy submodule of M . If P ≪PF M , then P⋆ ≪M . Proof. Assume that P ≪PF M . In order to see that P⋆ ≪M , suppose that P⋆+S =M for a submodule S of M . We aim to prove that P + χPFS = χPFM . Let m ∈ M . Then m = a+ b, for some a ∈ P⋆ and b ∈ S. Then ηP+χPF S (m) =η2P (m) + χ2 S(m)− η2P (m)χ2 S(m) ≥min{η2P (a) + χ2 S(a)− η2P (a)χ 2 S(a), η 2 P (b) + χ2 S(b)− η2P (b)χ 2 S(b)} =1 Moreover, η̂P+χPF S (m) =η̂2P (m)χc 2 S (m) ≤max{η̂2P (a)χc 2 S (a), η̂2P (b)χ c2 S (b)} =0 Thus P + χPFS = χPFM . By hypothesis, χPFS = χPFM . Therefore, S =M . A. Alhumaimeed / Eur. J. Pure Appl. Math, 15 (1) (2022), 36-46 40 Example 1. Consider the Z-module Z10 and the submodule S = ⟨5̄⟩. Let P be a pythagorean fuzzy submodule of Z10 defined as follows ηP (m) = { 1 if m ∈ S 1 4 otherwise and η̂P (m) = { 0 if m ∈ S 1 6 otherwise It is clear that P⋆ is not a small submodule of Z10 as P⋆ + ⟨2̄⟩ = Z10. Thus P is not a pythagorean fuzzy small submodule of Z10. Corollary 1. Let P, S be two pythagorean fuzzy submodules of a module M in which P ⊆ S. Then P ≪PF S if and only if P⋆ ≪ S⋆. Proof. Clear. Theorem 4. Let M be a module, S be a submodule of M and P is a pythagorean fuzzy submodule of M in which P ⊆ χPFS . If P |S is a pythagorean fuzzy small submodule of S, then P is pythagorean fuzzy small submodule of M . Proof. Assume that T is a pythagorean fuzzy submodule ofM such that P +T = χPFM . In order to see that P |S + (T |S ∩ χPFS ), let a ∈ S. Then we obtain η2 P |S+(T |S∩χPF S ) (a) =η2P |S (a) + η2 T |S∩χPF S (a)− η2P |S (a)η 2 T |S∩χPF S (a) =η2P |S (a) + min{η2T |S (a), χ 2 S(a)} − η2P |S (a)min{η2T |S (a), χ 2 S(a)} =min{η2P (a), χ2 S(a)}+min{η2T (a), χ2 S(a)} −min{η2P (a), χ2 S(a)}min{η2T (a), χ2 S(a)} =η2P (a) + η2T (a)− η2P (a)η 2 T (a) =η2P+T (a) =χ2 M (a) =1 =χ2 S(a) and η̂2 P |S+(T |S∩χPF S ) (a) =η̂2P |S (a)η̂ 2 T |S∩χPF S (a) =η̂2P |S (a)max{η̂2T |S (a), χ c2 SPF (a)} =max{η̂2P (a), χc 2 SPF (a)}max{η̂2T (a), χc 2 SPF (a)} A. Alhumaimeed / Eur. J. Pure Appl. Math, 15 (1) (2022), 36-46 41 =η̂2P (a)η̂ 2 T (a) =η̂2P+T (a) =χc 2 M (a) =0 =χc 2 S (a) This implies that P |S+(T |S ∩χPFS ) = χPFS . By hypothesis, we conclude that T |S ∩χPFS = χPFS . Thus χPFS ⊆ T |S . Then χPFM = P + T ⊆ T ⊆ χPFM . Therefore, T = χPFM and P is pythagorean fuzzy small submodule of M . As a consequence of the above theorem, we have: Corollary 2. Let M be a module and, P and S are pythagorean fuzzy submodules of M in which P ⊆ S. If P is pythagorean fuzzy small submodule of S, then P is pythagorean fuzzy small submodule of M . Proof. Clear. Remark 1. The converse of theorem 4 need not be true in general. That is if M is a module, S be a submodule of M and P is a pythagorean fuzzy small submodule of M in which P ⊆ χPFS , then it is not true in general that P |S is a pythagorean fuzzy small submodule of S. For example take P |S = S. Proposition 1. Let M be a module and P, S, T be pythagorean fuzzy submodules of M . Then: (P ∩ S) + (P ∩ T ) ⊆ P ∩ (S + T ). Proof. Let m ∈M . Then η2(P∩S)+(P∩T )(m) =η2P∩S(m) + η2P∩T (m)− η2P∩S(m)η2P∩T (m) =min{η2P (m), η2S(m)}+min{η2P (m), η2T (m)} −min{η2P (m), η2S(m)}min{η2P (m), η2T (m)} ≤min{η2P (m), η2S(m) + η2T (m)− η2S(m)η2T (m)} =min{η2P (m), η2S+T (m) =η2P∩(S+T )(m) Moreover, η̂2(P∩S)+(P∩T )(m) =η̂2P∩S(m)η̂2P∩T (m) =max{η̂2P (m), η̂2S(m)}max{η̂2P (m), η̂2T (m)} ≥max{η̂2P (m), η̂2S(m)η̂2T (m)} =η̂2P∩(S+T )(m) A. Alhumaimeed / Eur. J. Pure Appl. Math, 15 (1) (2022), 36-46 42 Proposition 2. Let M be a module and, P and S are pythagorean fuzzy submodules of M in which χPFM = P ⊕ PF S. Then M = P ⋆ ⊕ S⋆ = P⋆ ⊕ S⋆. Proof. Let m ∈M . Then 1 =χ2 M (m) =η2P+S(m) =η2P (m) + η2S(m)− η2P (m)η2S(m) =η2P (m)(1− η2S(m)) + η2S(m) This implies that η2P (m) = 1 or η2S(m) = 1, so that η̂2P (m) = 0 or η̂2S(m) = 0. Hence m ∈ P⋆ or m ∈ S⋆, so that M = P⋆ + S⋆. Hence M = P ⋆ + S⋆. We aim now to show that the intersection P ⋆ ∩ S⋆ = 0. Assume that m ∈ P ⋆ ∩ S⋆. Then η2P (m), η2S(m) > 0. Since χPFM = P ⊕ PF S, we obtain 0 0. This implies that 0 < η2P (m) + η2S(m)− η2P (m)η2S(m) = η2P (m)(1− η2S(m)) + η2S(m) A. Alhumaimeed / Eur. J. Pure Appl. Math, 15 (1) (2022), 36-46 43 which means that η2P (m) ̸= 0 or η2S(m) ̸= 0. Moreover, 1 > η̂2P+S(m) = η̂2P (m)η̂2S(m) which implies that η̂2P (m) < 1 or η̂2S(m) < 1. Thus m ∈ P ⋆ or m ∈ S⋆, so that m ∈ P ⋆ + S⋆. and (P + S)⋆ ⊆ P ⋆ + S⋆. Now, suppose that m = a1 + b1 ∈ P ⋆ + S⋆, where a1 ∈ P ⋆ and b1 ∈ S⋆. By definition, η2P (a1), η 2 S(b1) > 0. Thus 0 max{η̂2P (a1)η̂2S(a1), η̂2P (b1)η̂2S(b1)} ≥η̂2P (m)η̂2S(m) =η̂2P+S(m) Thus m ∈ (P + S)⋆. Then P ⋆ + S⋆ ⊆ (P + S)⋆ and therefore, the equality holds. (2) Since P ∩ S = χPF0 , we need to prove that (P + S)⋆ = P⋆ + S⋆. Suppose that m ∈ (P + S)⋆. By definition, η2P+S(m) = 1. This implies that 1 = η2P (m) + η2S(m)− η2P (m)η2S(m) = η2P (m)(1− η2S(m)) + η2S(m) which means that η2P (m) = 1 or η2S(m) = 1. Moreover, 0 = η̂2P+S(m) = η̂2P (m)η̂2S(m) which implies that η̂2P (m) = 0 or η̂2S(m) = 0. Thus m ∈ P⋆ or m ∈ S⋆, so that m ∈ P⋆ + S⋆ and (P + S)⋆ ⊆ P⋆ + S⋆. Now, suppose that m = a1 + b1 ∈ P⋆ + S⋆, where a1 ∈ P⋆ and b1 ∈ S⋆. By definition, η2P (a1), η 2 S(b1) = 1. Thus 1 =min{η2P (a1) + η2S(a1)− η2P (a1)η 2 S(a1), η 2 P (b1) + η2S(b1)− η2P (b1)η 2 S(b1)} ≤η2P (m) + η2S(m)− η2P (m)η2S(m) =η2P+S(m) Moreover, η̂2P (a1), η̂ 2 S(b1) = 0 which implies that 0 =max{η̂2P (a1)η̂2S(a1), η̂2P (b1)η̂2S(b1)} ≥η̂2P (m)η̂2S(m) =η̂2P+S(m) Thus m ∈ (P + S)⋆. Then P⋆ + S⋆ ⊆ (P + S)⋆ and therefore, the equality holds. A. Alhumaimeed / Eur. J. Pure Appl. Math, 15 (1) (2022), 36-46 44 4. Homomorphism Let P, S be two R-modules, L ≤PF P and N ≤PF S. Consider an R-homomorphism ψ : P −→ S For s ∈ S, we define: ηψ(L)(s) = { max{ηL(p) : s = ψ(p)} if s ∈ Im(ψ) 0 otherwise and η̂ψ(L)(s) = { min{ηL(p) : s = ψ(p)} if s ∈ Im(ψ) 1 otherwise Now, we are ready to prove the following: Theorem 7. Let ψ : P −→ S be a monomorphism of modules. If T is a pythagorean fuzzy small submodule of P , then ψ(T ) is a pythagorean fuzzy small submodule of S. Proof. Suppose that ψ(T ) + L = χPFS . We aim to prove that L = χPFS . Let s ∈ S, then 1 =η2ψ(T )+L(s) =η2ψ(T )(s) + η2L(s)− η2ψ(T )(s)η 2 L(s) In the case that s /∈ Im(ψ), we obtain 1 =η2ψ(T )(s) + η2L(s)− η2ψ(T )(s)η 2 L(s) =η2L(s) and so 1 = η2L(s) and η̂ 2 L(s) = 0. If s ∈ Im(ψ), we have 1 =η2ψ(T )(s) + η2L(s)− η2ψ(T )(s)η 2 L(s) =max{η2T (p) : ψ(p) = s}+ η2L(s)−max{η2T (p) : ψ(p) = s}η2L(s) =η2T (p) + η2L(s)− η2T (p)η 2 L(s), for some p in which ψ(p) = s =η2T (p)(1− η2L(s)) + η2L(s) If η2T (p) = 1, then T = χP and this is a contradiction with the fact that T is a pythagorean fuzzy small submodule of P . Thus η2L(s) = 1 and L = χS . Moreover, 0 =η̂2ψ(T )+L(s) REFERENCES 45 =η̂2ψ(T )(s)η̂ 2 L(s) =η̂2T (p)η̂ 2 L(s) for some p in which ψ(p) = s Note that ψ is one to one and so p is unique. By hypothesis η̂2T (p) ̸= 0, so that η̂2L(s) = 0. Hence L = χPFS . Remark 2. (1) If ψ is not one to one, then the above theorem need not be true. For instance, take S the zero module and ψ the zero homomorphism. (2) The converse of the above theorem need not be true. That is if ψ : P −→ S is a monomorphism of modules, T is a pythagorean fuzzy submodule of P and ψ(T ) is a pythagorean fuzzy small submodule of S, then it is not true in general that T is a pythagorean fuzzy small submodule of P . For example, Let P be a pythagorean fuzzy small submodule of S and consider the inclusion P ↪→ S. Then ψ(P ) = P is a pythagorean fuzzy small submodule of S but P is not a pythagorean fuzzy small submodule of P . 5. Conclusion and Future Directions In this paper, we introduce the notion of pythagorean submodule. In addition, we present the concept pythagorean small submodule and investigate some results regarding this concept. Moreover, we find a relationship between small submodule and pythagorean fuzzy small submodule. We also study homomorphism between pythagorean fuzzy mod- ules. This work can be extended and generalised in the environment of q-rung orthopair fuzzy sets. It can be applied in order to solve multi-criteria decision making problems. 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