Int. J. Anal. Appl. (2022), 20:41 Picture N-Sets and Applications in Semigroups Anusorn Simuen1, Ronnason Chinram1,∗, Winita Yonthanthum1, Aiyared Iampan2 1Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand 2Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand ∗Corresponding author: ronnason.c@psu.ac.th Abstract. In this paper, we study picture N-structures and apply it to semigroups. Moreover, we define picture N-ideals in semigroups and investigate several properties of these ideals in semigroups. 1. Introduction Fuzzy sets were introduced by Zadeh [8] in 1965 as an extension of the classical notion of sets. Fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the unit interval [0,1]. Next, fuzzy sets were generalized to other concepts. Atanassov [1] generalized fuzzy sets into intuitionistic fuzzy sets in 1986 by considering for each element of the sets is a degree of membership and a degree of non- membership. The notion of the classical sets, fuzzy sets, and intuitionistic fuzzy sets were extended into neutrosophic sets which is the tool for dealing with incomplete, inconsistent, and indeterminate information by Smaradache [6]. In 2009, Jun et al. [4] gave the concept of a negative-valued function and constructed N-structures. Later, Smaradache et al. [7] introduced the notion of neutrosophic N-structures and applied it to semigroups in 2017. Next, Elavarasan et al. [3] introduced neutrosophic N-ideals in semigroups and investigated its several properties. In 2014, the concept of picture fuzzy set was first introduced by Cuong [2] in 2014, which is a generalization of the concept of fuzzy sets and intuitionistic fuzzy sets. This concept is based on adequate in situations when we face human Received: Jul. 4, 2022. 2010 Mathematics Subject Classification. 03E72. Key words and phrases. N-sets; picture N-structures; picture N-ideals. https://doi.org/10.28924/2291-8639-20-2022-41 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-41 2 Int. J. Anal. Appl. (2022), 20:41 opinions involving more answers of types: yes, abstain, no, refusal. Picture fuzzy sets were focused on the degree of positive memberships, the degree of neutral memberships, and the degree of negative memberships. Picture fuzzy sets are generalizations of fuzzy sets and intuitionistic fuzzy sets. The concept of picture fuzzy sets will differ from the concept of neutrosophic sets. The applications of picture fuzzy sets in semigroups were studied by Yiarayong [5] in 2020. In this paper, we study picture N-structures and apply it to semigroup. Moreover, we define picture N-ideals in semigroups and investigate several properties of these ideals in semigroups. 2. Notations In this section, we introduce the concept of picture N-structures of sets. Definition 2.1. A picture N-structure over a set S defined to be the structure: PN := { x TN(x), IN(x),FN(x) | x ∈ S } where TN : S → [−1,0] is called the negative positive membership function, IN : S → [−1,0] is called the negative neutral membership function, and FN : S → [−1,0] is called the negative false membership function with the condition −1 ≤ TN(x)+ IN(x)+ FN(x) ≤ 0 for all x ∈ S. We denote this structure by PN = S (TN,IN,FN) . Definition 2.2. Let PN = S(TN,IN,FN) and PM = S (TM,IM,FM) be picture N-structures over a set S. Then (1) PN is called a picture N-substructure of PM over S, denote by PN ⊆ PM, if (a) TN(x)≥ TM(x), (b) IN(x)≤ IM(x), (c) FN(x)≤ FM(x) for all x ∈ S. (2) The union of PN and PM is defined to be a picture N-structure over S PN∪M =(S;TN∪M, IN∪M,FN∪M) where TN∪M(x)=min{TN(x),TM(x)}, IN∪M(x)=max{IN(x), IM(x)}, FN∪M(x)=max{FN(x),FM(x)} for all x ∈ S. (3) The intersection of PN and PM is defined to be a picture N-structure over S PN∩M =(S;TN∩M, IN∩M,FN∩M) where TN∩M(x)=max{TN(x),TM(x)}, IN∩M(x)=min{IN(x), IM(x)}, Int. J. Anal. Appl. (2022), 20:41 3 FN∩M(x)=min{FN(x),FM(x)} for all x ∈ S. Definition 2.3. For a subset A of a set S, consider the picture N-structure χA(PN)= S (χA(T)N,χA(I)N,χA(F)N) where χA(T)N(x)=  −1 if x ∈ A, 0 otherwise, χA(I)N(x)=  0 if x ∈ A,−1 otherwise, χA(F)N(x)=  0 if x ∈ A,−1 otherwise for all x ∈ S. The picture N-structure χA(PN) is called the characteristic picture N-structure of A over S. Definition 2.4. Let PN be a picture N-structure over S and let α,β,γ ∈ [−1,0] be such that −1≤ α+β +γ ≤ 0. Consider the following sets: TαN = {x ∈ S | TN(x)≤ α}, I β N = {x ∈ S | IN(x)≥ β}, F γ N = {x ∈ S | FN(x)≥ γ}. The set PN(α,β,γ) := {x ∈ S | TN(x)≤ α,IN(x)≥ β,FN(x)≥ γ} is called an (α,β,γ)-level set of PN. Note that PN(α,β,γ)= TαN ∩ I β N ∩Fγ N . 3. Applications of picture N-sets in semigroups The picture N-product of PN and PM is defined to be a picture N-structure over a semigroup S PN �PM := { x TN◦M(x), IN◦M(x),FN◦M(x) | x ∈ S } where TN◦M(x)=  infx=abmax{TN(a),TM(b)} for x = ab for some a,b ∈ S, 0 otherwise, IN◦M(x)=  supx=abmin{IN(a), IM(b)} for x = ab for some a,b ∈ S, 0 otherwise, 4 Int. J. Anal. Appl. (2022), 20:41 FN◦M(x)=  supx=abmin{FN(a),FM(b)} for x = ab for some a,b ∈ S, 0 otherwise. We denote the picture N-product of PN and PM by PN �PM = S(TN◦M,IN◦M,FN◦M). For x ∈ S, the element x (TN◦M,IN◦M,FN◦M) is simply denoted by (PN �PM)(x)= (TN◦M(x), IN◦M(x),FN◦M(x)) for the sake of convenience. Definition 3.1. A picture N-structure PN over a semigroup S is called a picture N-subsemigroup of S if it satisfies: (1) TN(xy)≤max{TN(x),TN(y)}, (2) IN(xy)≥min{IN(x), IN(y)}, (3) FN(xy)≥min{FN(x),FN(y)} for all x,y ∈ S. Definition 3.2. A picture N-structure PN over a semigroup S is called a picture N-left ideal of S if it satisfies: (1) TN(xy)≤ TN(y), (2) IN(xy)≥ IN(y), (3) FN(xy)≥ FN(y) for all x,y ∈ S. Definition 3.3. A picture N-structure PN over a semigroup S is called a picture N-right ideal of S if it satisfies: (1) TN(xy)≤ TN(x), (2) IN(xy)≥ IN(x), (3) FN(xy)≥ FN(x) for all x,y ∈ S. We called PN a picture N-ideal if it is both a picture N-left ideal and a picture N-right ideal of S. Theorem 3.1. Let PN be a picture N-structure over a semigroup S and let α,β,γ ∈ [−1,0] be such that −1 ≤ α + β + γ ≤ 0. If PN is a picture N-left ideal of S, then (α,β,γ)-level set of PN is a picture N-left ideal of S whenever it is nonempty. Proof. Assume that PN(α,β,γ) 6= ∅ for α,β,γ ∈ [−1,0] with −1 ≤ α + β + γ ≤ 0. Let PN be a picture N-left ideal of S, and let x,y ∈ PN(α,β,γ). Then TN(xy)≤ TN(y)≤ α, IN(xy)≥ IN(y)≥ β, and FN(xy) ≥ FN(y) ≥ γ which imply xy ∈ PN(α,β,γ). Hence PN(α,β,γ) is a picture N-left ideal of S. � Int. J. Anal. Appl. (2022), 20:41 5 Theorem 3.2. Let PN be a picture N-structure over a semigroup S and let α,β,γ ∈ [−1,0] be such that −1 ≤ α + β + γ ≤ 0. If PN is a picture N-right ideal of S, then (α,β,γ)-level set of PN is a picture N-right ideal of S whenever it is nonempty. Proof. It is similar to Theorem 3.1. � Theorem 3.3. Let PN be a picture N-structure over a semigroup S and let α,β,γ ∈ [−1,0] be such that −1 ≤ α+β +γ ≤ 0. If PN is a picture N-ideal of S, then (α,β,γ)-level set of PN is a picture N-ideal of S whenever it is nonempty. Proof. It follows from Theorem 3.1 and 3.2. � Theorem 3.4. Let PN be a picture N-structure over a semigroup S and let α,β,γ ∈ [−1,0] be such that −1≤ α+β +γ ≤ 0. If TαN , I β N , and Fγ N are left ideals of S, then PN is a picture N-left ideal of S whenever it is nonempty. Proof. Let a,b ∈ S such that TN(ab) > TN(b). Then TN(ab) > tα ≥ TN(b) for some tα ∈ [−1,0). Thus b ∈ Ttα N (b) but ab /∈ Ttα N (b), this is a contradiction. So TN(ab)≤ TN(b). Similarly way we can get IN(ab)≥ IN(b) and FN(ab)≥ FN(b). Therefore PN is a picture N-left ideal of S. � Theorem 3.5. Let PN be a picture N-structure over a semigroup S and let α,β,γ ∈ [−1,0] be such that −1≤ α+β +γ ≤ 0. If TαN , I β N , and Fγ N are right ideals of S, then PN is a picture N-right ideal of S whenever it is nonempty. Proof. It is similar to Theorem 3.4. � Theorem 3.6. Let PN be a picture N-structure over a semigroup S and let α,β,γ ∈ [−1,0] be such that −1≤ α+β+γ ≤ 0. If TαN , I β N , and Fγ N are ideals of S, then PN is a picture N-ideal of S whenever it is nonempty. Proof. It follows from Theorem 3.4 and 3.5. � Theorem 3.7. Let S be a semigroup. Then intersection of two picture N-left ideals of S is also a picture N-left ideal of S. Proof. Let PN := S (TN,IN,FN) and PM := S (TM,IM,FM) be picture N-left ideals of S. Then TN∩M(xy)=max{TN(xy),TM(xy)}≤max{TN(y),TM(y)}= TN∩M(y), IN∩M(xy)=min{IN(xy), IM(xy)}≥min{IN(y), IM(y)}= IN∩M(y), FN∩M(xy)=min{FN(xy),FM(xy)}≥min{FN(y),FM(y)}= FN∩M(y) for all x,y ∈ S. Then PN∩M is a picture N-left ideal of S. � 6 Int. J. Anal. Appl. (2022), 20:41 Theorem 3.8. Let S be a semigroup. Then intersection of two picture N-right ideals of S is also a picture N-right ideal of S. Proof. It is similar to Theorem 3.7. � Theorem 3.9. Let S be a semigroup. Then intersection of two picture N-ideals of S is also a picture N-ideal of S. Proof. It follows from Theorem 3.7 and 3.8. � Theorem 3.10. For any nonempty subset A of a semigroup S, the following conditions are equivalent: (1) A is a left ideal of S. (2) The characteristic picture N-structure χA(PN) over S is a picture N-left ideal of S. Proof. Assume that A is a left ideal of S. Let x,y ∈ S. If y /∈ A, then χA(T)N(xy)≤ 0= χA(T)N(y), χA(I)N(xy)≥−1= χA(I)N(y), χA(F)N(xy)≥−1= χA(F)N(y). Otherwise y ∈ A. Then xy ∈ A, we have χA(T)N(xy)=−1= χA(T)N(y), χA(I)N(xy)=0= χA(I)N(y), χA(F)N(xy)=0= χA(F)N(y). Therefore χA(PN) is a picture N-left ideal of S. Conversely, suppose that χA(PN) is a picture N-left ideal of S. Let y ∈ A and x ∈ S. Then χA(T)N(xy)≤χA(T)N(y)=−1, χA(I)N(xy)≥χA(I)N(y)=0, χA(F)N(xy)≥χA(F)N(y)=0. Hence xy ∈ A. Therefore A is a left ideal of S. � Theorem 3.11. For any nonempty subset A of a semigroup S, the following conditions are equivalent: (1) A is a right ideal of S. (2) The characteristic picture N-structure χA(PN) over S is a picture N-right ideal of S. Proof. It is similar to Theorem 3.10. � Theorem 3.12. For any nonempty subset A of a semigroup S, the following conditions are equivalent: Int. J. Anal. Appl. (2022), 20:41 7 (1) A is an ideal of S. (2) The characteristic picture N-structure χA(PN) over S is a picture N-ideal of S. Proof. It follows from Theorem 3.11 and 3.12. � Theorem 3.13. Let χA(PN) and χB(PN) be characteristic picture N-structures over a semigroup S for subsets A and B of S. Then (1) χA(PN)∩χB(PN)= χA∩B(PN). (2) χA(PN)�χB(PN)= χAB(PN). Proof. (1) Let s ∈ S. If s ∈ A∩B, then s ∈ A and s ∈ B. Thus (χA(T)N ∩χB(T)N)(s)=max{χA(T)N(s),χB(T)N(s)}=−1= χA∩B(T)N(s), (χA(I)N ∩χB(I)N)(s)=min{χA(I)N(s),χB(I)N(s)}=0= χA∩B(I)N(s), (χA(F)N ∩χB(F)N)(s)=min{χA(F)N(s),χB(F)N(s)}=0= χA∩B(F)N(s). Hence χA(PN)∩χB(PN)= χA∩B(PN). If s /∈ A∩B, then s /∈ A or s /∈ B. Thus (χA(T)N ∩χB(T)N)(s)=max{χA(T)N(s),χB(T)N(s)}=0= χA∩B(T)N(s), (χA(I)N ∩χB(I)N)(s)=min{χA(I)N(s),χB(I)N(s)}=−1= χA∩B(I)N(s), (χA(F)N ∩χB(F)N)(s)=min{χA(F)N(s),χB(F)N(s)}=−1= χA∩B(F)N(s). Hence χA(PN)∩χB(PN)= χA∩B(PN). (2) Let x ∈ S. If x /∈ AB, then (χA(T)N ◦χB(T)N)(x)=0= χAB(T)N(x), (χA(I)N ◦χB(I)N)(x)=0= χAB(I)N(x), (χA(F)N ◦χB(F)N)(x)=0= χAB(F)N(x). If x ∈ AB, then x = ab for some a ∈ A and b ∈ B. We have (χA(T)N ◦χB(T)N)(x)= inf x=ab max{χA(T)N(a),χB(T)N(b)} ≤max{χA(T)N(a),χB(T)N(b)} =−1 = χAB(T)N(x), 8 Int. J. Anal. Appl. (2022), 20:41 (χA(I)N ◦χB(I)N)(x)= sup x=ab min{χA(I)N(a),χB(I)N(b)} ≥min{χA(I)N(a),χB(I)N(b)} =0 = χAB(I)N(x), (χA(F)N ◦χB(F)N)(x)= sup x=ab min{χA(F)N(a),χB(F)N(b)} ≥min{χA(F)N(a),χB(F)N(b)} =0 = χAB(F)N(x). Therefore χA(PN)�χB(PN)= χAB(PN). � Theorem 3.14. Let PM be a picture N-structure over a semigroup S. Then PM is a picture N-left ideal of S if and only if PN �PM ⊆ PM for any picture N-structure PN over S. Proof. Assume that PM is a picture N-left ideal of S and let s,t,u ∈ S. If s = tu, then we have (i) TM(s) = TM(tu) ≤ TM(u) ≤ max{TM(t),TM(u)} which implies TM(s) ≤ TN◦M(s). Other- wise s 6= tu. Then TM(s)≤ 0= TN◦M(s). (ii) IM(s) = IM(tu) ≥ IM(u) ≥ min{IM(t), IM(u)} which implies IM(s) ≥ IN◦M(s). Otherwise s 6= tu. Then IM(s)≥−1= IN◦M(s). (iii) FM(s)= FM(tu)≥ FM(u)≥min{FM(t),FM(u)} which implies FM(s)≥ FN◦M(s). Otherwise s 6= tu. Then FM(s)≥−1= FN◦M(s). Conversely, assume that PM is a picture N-structure over S such that PN � PM ⊆ PM for any picture N-structure PN over S. Let x,y ∈ S. If a = xy, then TM(xy)= TM(a) ≤ (χX(T)N ◦TM)(a) = inf a=st max{χX(T)N(s),TM(t)} ≤max{χX(T)N(x),TM(y)} = TM(y), IM(xy)= IM(a) ≥ (χX(I)N ◦ IM)(a) = sup a=st min{χX(I)N(s), IM(t)} Int. J. Anal. Appl. (2022), 20:41 9 ≥min{χX(I)N(x), IM(y)} = IM(y), FM(xy)= FM(a) ≥ (χX(F)N ◦FM)(a) = sup a=st min{χX(F)N(s),FM(t)} ≥max{χX(F)N(x),FM(y)} = FM(y). Therefore PM is a picture N-left ideal of S. � Theorem 3.15. Let PM be a picture N-structure over a semigroup S. Then PM is a picture N-right ideal of S if and only if PM �PN ⊆ PM for any picture N-structure PN over S. Proof. It is similar to Theorem 3.14. � Theorem 3.16. Let PM be a picture N-structure over S. Then PM is a picture N-ideal of S if and only if PM �PN ⊆ PM for any picture N-structure PN over S. Proof. It follows from Theorem 3.14 and 3.15. � Theorem 3.17. Let PM and PN be picture N-structure over S. If PM is a picture N-left ideal of S, then so is PM �PN. Proof. Assume that PM is a picture N-left ideal of S, and let x,y ∈ S. If there exist a,b ∈ S such that y = ab, then xy = x(ab)= (xa)b. We have (TN ◦TM)(y)= inf y=ab max{TN(a),TM(b)} ≤ inf xy=(xa)b max{TN(xa),TM(b)} = inf xy=cb max{TN(c),TM(b)} =(TN ◦TM)(xy), (IN ◦ IM)(y)= sup y=ab min{IN(a), IM(b)} ≥ sup xy=(xa)b min{IN(xa), IM(b)} = sup xy=cb min{IN(c), IM(b)} =(IN ◦ IM)(xy), 10 Int. J. Anal. Appl. (2022), 20:41 (FN ◦FM)(y)= sup y=ab min{FN(a),FM(b)} ≥ sup xy=(xa)b min{FN(xa),FM(b)} = sup xy=cb min{FN(c),FM(b)} =(FN ◦FM)(xy). Therefore PM �PN is a picture N-left ideal of S. � Theorem 3.18. Let PM and PN be picture N-structure over a semigroup S. If PM is a picture N-right ideal of S, then so is PM �PN. Proof. It is similar to Theorem 3.17. � Theorem 3.19. Let PM and PN be picture N-structure over a semigroup S. If PM is a picture N-ideal of S, then so is PM �PN. Proof. It follows from Theorem 3.17 and 3.18. � Acknowledgment: This work was supported by the PSU-TUYF Charitable Trust Fund, Prince of Songkla University, Contract no. 2-2564-01. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] K.T. Atanassov, Intuitionistic Fuzzy Sets, VII ITKR Session, Sofia, 20-23 June 1983 (Deposed in Centr. Sci.-Techn. Library of the Bulg. Acad. of Sci., 1697/84) (in Bulgarian). Reprinted: Int. J. Bioautom. 20(S1) (2016), S1–S6. https://www.biomed.bas.bg/bioautomation/2016/vol_20.s1/files/20.s1_02.pdf. [2] B.C. Cuong, Picture Fuzzy Sets, J. Computer Sci. Cybernetics, 30 (2014), 409–420. https://doi.org/10.15625/ 1813-9663/30/4/5032. [3] B. Elavarasan, F. Smarandache, Y.B. 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