Int. J. Anal. Appl. (2022), 20:29 Applications of Spherical Fuzzy Sets in Ternary Semigroups Wasitthirawat Krailoet1, Ronnason Chinram1,∗ Montakarn Petapirak1, 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 introduce the notions of spherical fuzzy ternary subsemigroups and spherical fuzzy ideals in ternary semigroups by using the concepts of ternary subsemigroups and ideals in ternary semigroups. We investigate their properties. Moreover, we study roughness of spherical fuzzy ideals in ternary semigroups. 1. Introduction The theory of ternary algebraic system was investigated by Lehmer [8] in 1932, but earlier such structures were studied by Kasner [5] who gave the idea of n-ary algebras. Furthermore, the ideal theory in ternary semigroups was established by Sioson [12]. In 1965, the notion of fuzzy sets was initiated by Zadeh [14]. The fuzzy set is an extension of classical sets and represented by using a generalization of the indicator of classical sets that is called a membership function. Later, the concept of fuzzy set was applied to study in many algebraic structures. In 1981, Kuroki [6] provided some properties of fuzzy ideals. In 2013, Iampan [4] gave the definition and characterized the properties of ideal extensions in ternary semigroups. After the introduction of ordinary fuzzy sets, the concept of rough sets was given by Pawlak [10] in 1982 which is defined depending on some equivalence relation on a universal finite set. The combination of theories of fuzzy sets and rough sets has been discussed in many research papers through all the years until 1990, when Dubois and Prade [3] proposed the notion of Received: May 10, 2022. 2010 Mathematics Subject Classification. 03E72. Key words and phrases. ideals; fuzzy ideals; rough fuzzy sets; spherical fuzzy sets; rough ideals; rough spherical fuzzy ideals; ternary semigroups. https://doi.org/10.28924/2291-8639-20-2022-29 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-29 2 Int. J. Anal. Appl. (2022), 20:29 rough fuzzy sets. In 2009, Petchkhaew and Chinram [11] studied fuzzy, rough and rough fuzzy ternary subsemigroups (left ideals, right ideals, lateral ideals, ideals) of ternary semigroups. Later, in 2012, Kar and Sarkar [7] focused on studying fuzzy ideals of ternary semigroups and their related properties. In 2016, Wang and Zhan [13] established the rough semigroups and the rough fuzzy semigroups based on fuzzy ideals. In 2019, Ashraf et al. [1] introduced the notion of spherical fuzzy set, which is a generalization of the picture fuzzy sets, intuitionistic fuzzy sets and Pythagorean fuzzy sets when the degree of abstinence is involved, as it provides enlargement of the space of degrees of truthfulness (membership), abstinence (hesitancy) and falseness (non-membership). Recently, in 2020, Chinram and Panityakul [2] introduced rough Pythagorean fuzzy ideals in ternary semigroups and gave some remarkable properties. Our aim of this paper is to study spherical fuzzy ternary subsemigroups and spherical fuzzy ideals in ternary semigroups by using the concepts of ternary subsemigroups and ideals in ternary semigroups. Moreover, we study roughness of spherical fuzzy sets and spherical fuzzy ideals in ternary semigroups. 2. Preliminaries In this section, we shall recall some basic definitions that will be used in this paper. 2.1. Ternary Semigroups. A non-empty set T together with a ternary operation, called ternary multiplication, denoted by juxtaposition, is said to be a ternary semigroup if (abc)de = a(bcd)e = ab(cde) for all a,b,c,d,e ∈ T. For any three non-empty subsets A,B and C of a ternary semigroup T , a product ABC is defined by ABC = {abc | a ∈ A,b ∈ B and c ∈ C}. Example 2.1. (1) The following example (Banach’s Example) shows that a ternary semigroup does not necessarily reduce an ordinary semigroup. Let T = {−i,0, i} be a ternary semigroup under ternary multiplication over C. We obtain that T is not a semigroup under multiplication over C. (2) Let Z− be the set of all negative integers. Then Z− is a ternary semigroup under ternary multiplication over Z. We obtain that Z− is not a semigroup under multiplication over Z. (3) The set of all odd permutation is a ternary semigroup under ternary composition. It is not a semigroup under composition. A non-empty subset S of a ternary semigroup T is called a ternary subsemigroup of T if S3 ⊆ S. Let I be a non-empty subset of a ternary semigroup T . Then I is called a left ideal of T if TTI ⊆ I, a lateral ideal of T if TIT ⊆ I and a right ideal of T if ITT ⊆ I. A non-empty subset I of a ternary semigroup T is called an ideal of T if I is a left ideal, a lateral ideal and a right ideal of T. An ideal I of a ternary semigroup T is called a proper ideal if I 6= T . Int. J. Anal. Appl. (2022), 20:29 3 2.2. Fuzzy sets. A fuzzy subset of a set S is a function S × S → [0,1]. Let f and g be any two fuzzy subsets of any set S. (1) f ⊆ g if f (a)≤ g(a) for all a ∈ S. (2) (f ∩g)(a)=min{f (a),g(a)} for all a ∈ S. (3) (f ∪g)(a)=max{f (a),g(a)} for all a ∈ S. A fuzzy subset f of a ternary semigroup T is called a fuzzy ternary subsemigroup of T if f (xyz)≥min{f (x), f (y), f (z)} for all x,y,z ∈ T. A fuzzy subset f of T is called a fuzzy left ideal of T if f (xyz) ≥ f (z) for all x,y,z ∈ T , a fuzzy lateral ideal of T if f (xyz) ≥ f (y) for all x,y,z ∈ T and a fuzzy right ideal of T if f (xyz) ≥ f (x) for all x,y,z ∈ T . A fuzzy subset f of a ternary semigroup T is called a fuzzy ideal of T if it is a fuzzy left ideal, a fuzzy lateral ideal and a fuzzy right ideal of T, i.e., f (xyz)≥max{f (x), f (y), f (z)} for all x,y,z ∈ T. For any three fuzzy sets f1, f2 and f3 of a ternary semigroup T. The product f1◦ f2◦ f3 of f1, f2 and f3 is defined by (f1◦ f2◦ f3)(y)=  supy=y1y2y3 min{f1(y1), f2(y2), f3(y3)} if y ∈ T 3, 0 otherwise. It is obvious that the product f1◦ f2◦ f3 of fuzzy subsets f1, f2 and f3 of a ternary semigroup T is also a fuzzy subset of T . Let F(T) be the set of all fuzzy subsets of a ternary semigroup T. Then F(T) is a ternary semigroup under this product. 2.3. Spherical Fuzzy Sets. Let S be a universal set. A spherical fuzzy set on S S := {< x,µS(x),ηS(x),νS(x) >| x ∈ S} where µS : S → [0,1], ηS : S → [0,1] and νS : S → [0,1] represent the degree of membership, the degree of hesitancy and the degree of non-membership of x ∈ S with the condition 0 ≤ (µS(x))2 + (ηS(x)) 2 +(νS(x)) 2 ≤ 1. We may also denote a spherical fuzzy set S by S =(µS,ηS,νS). Example 2.2. Let f be any fuzzy subset of a set S. Let µS : S → [0,1], ηS : S → [0,1] and νS : S → [0,1] be defined by µS(x)= f (x),ηS(x)=0 and νS(x)=1− f (x). Then S := {< x,µS(x),ηS(x),νS(x) >| x ∈ S} is a spherical fuzzy set on S. Let S1 = (µS1,ηS1,νS1) and S2 = (µS2,ηS2,νS2) be any two spherical fuzzy set of a universal set S. We say that S1 ⊆S2 if and only if µS1(x)≤ µS2(x),ηS1(x)≤ ηS1(x) and νS1(x)≥ νS1(x) for all x ∈ S. 4 Int. J. Anal. Appl. (2022), 20:29 3. Main Results 3.1. Spherical fuzzy ideals in ternary semigroups. We define spherical fuzzy ternary subsemigroups and spherical fuzzy ideals in ternary semigroups as follows: Definition 3.1. A spherical fuzzy set S =(µS,ηS,νS) on a ternary semigroup T is called a spherical fuzzy ternary subsemigroup of T if, for all a,b,c ∈ T (1) µS(abc)≥min{µS(a),µS(b),µS(c)}, (2) ηS(abc)≥min{ηS(a),ηS(b),ηS(c)}, (3) νS(abc)≤max{νS(a),νS(b),νS(c)}. Definition 3.2. A spherical fuzzy set S =(µS,ηS,νS) on a ternary semigroup T is called (1) a spherical fuzzy left ideal of T if for all a,b,c ∈ T , µS(abc)≥ µS(c), ηS(abc)≥ ηS(c) and νS(abc)≤ νS(c), (2) a spherical fuzzy lateral ideal of T if for all a,b,c ∈ T, µS(abc)≥ µS(b), ηS(abc)≥ ηS(b) and νS(abc)≤ νS(b), (3) a spherical fuzzy right ideal of T if for all a,b,c ∈ T, µS(abc)≥ µS(a), ηS(abc)≥ ηS(a) and νS(abc)≤ νS(a), (4) a spherical fuzzy ideal of T if for all a,b,c ∈ T , µS(abc)≥max{µS(a),µS(b),µS(c)}, ηS(abc)≥max{ηS(a),ηS(b),ηS(c)} and νS(abc)≤min{νS(a),νS(b),νS(c)}. Next, we define the product of three spherical fuzzy sets. Definition 3.3. Let S1, S2 and S3 be any three spherical fuzzy sets on a ternary semigroup T. The product S1◦S2◦S3 of S1, S2 and S3 is defined by S1◦S2◦S3 =((µS1◦µS2◦µS3),(ηS1◦ηS2◦ηS3),(νS1◦νS2◦νS3)) where (µS1◦µS2◦µS3)(x)=   sup x=abc min{µS1(a),µS2(b),µS3(c)}, if x ∈ T 3; 0, otherwise, (ηS1◦ηS2◦ηS3)(x)=   sup x=abc min{ηS1(a),ηS2(b),ηS3(c)}, if x ∈ T 3; 0, otherwise, Int. J. Anal. Appl. (2022), 20:29 5 and (νS1◦νS2◦νS3)(x)=   inf x=abc max{νS1(a),νS2(b),νS3(c)}, if x ∈ T 3; 1, otherwise. Theorem 3.1. Let S =(µS,ηS,νS) be a spherical fuzzy set on a ternary semigroup T . Then S is a spherical fuzzy ternary subsemigroup of T if and only if S◦S◦S ⊆S. Proof. Assume that S is a spherical fuzzy ternary subsemigroup of T . Let x ∈ T . If x /∈ T3, we obtain that (µS◦µS◦µS)(x)=0≤ µS(x), (ηS◦ηS◦ηS)(x)=0≤ ηS(x) and (νS◦νS◦νS)(x)=1≥ νS(x). Now, assume that x ∈ T3, we obtain that (µS◦µS◦µS)(x)= sup x=abc min{µS(a),µS(b),µS(c)}≤ sup x=abc µS(abc)= µS(x), (ηS◦ηS◦ηS)(x)= sup x=abc min{ηS(a),ηS(b),ηS(c)}≤ sup x=abc ηS(abc)= ηS(x) and (νS◦νS◦νS)(x)= inf x=abc max{νS(a),νS(b),νS(c)}≥ inf x=abc νS(abc)= νS(x). Hence S◦S◦S ⊆S. Conversely, let a,b,c ∈ T . µS(abc) ≥ (µS◦µS◦µS)(abc) = sup abc=x1x2x3 min{µS(x1),µS(x2),µS(x3)} ≥ min{µS(a),µS(b),µS(c)}, ηS(abc) ≥ (ηS◦ηS◦ηS)(abc) = sup abc=x1x2x3 min{ηS(x1),ηS(x2),ηS(x3)} ≥ min{ηS(a),ηS(b),ηS(c)} and νS(abc) ≤ (νS◦νS◦νS)(abc) = inf abc=x1x2x3 max{νS(x1),νS(x2),νS(x3)} ≤ max{νS(a),νS(b),νS(c)}. This implies that S is a spherical fuzzy ternary subsemigroup of T . � 6 Int. J. Anal. Appl. (2022), 20:29 Let T := (µT ,ηT ,νT ) be a spherical fuzzy set on a ternary semigroup T defined by µT (x)=1 and ηT (x)= νT (x)=0 for all x ∈ T . The following theorem holds. Theorem 3.2. Let S = (µS,ηS,νS) be a spherical fuzzy set on a ternary semigroup T. If S is a spherical fuzzy left ideal of T , then T ◦T ◦S ⊆S. Proof. Assume that S is a spherical fuzzy left ideal of T. If x /∈ T3, we obtain that (µT ◦µT ◦µS)(x)=0≤ µS(x), (ηT ◦ηT ◦ηS)(x)=0≤ ηS(x) and (νT ◦νT ◦νS)(x)=1≥ νS(x). Now, assume that x ∈ T3, we obtain that (µT ◦µT ◦µS)(x)= sup x=abc min{µT (a),µT (b),µS(c)}= sup x=abc µS(c)≤ µS(x), (ηT ◦ηT ◦ηS)(x)= sup x=abc min{ηT (a),ηT (b),ηS(c)}=0≤ ηS(x) and (νT ◦νT ◦νS)(x)= inf x=abc max{νT (a),νT (b),νS(c)}= inf x=abc νS(c)≥ νS(x). Hence T ◦T ◦S ⊆S. � Theorem 3.3. Let S = (µS,ηS,νS) be a spherical fuzzy set on a ternary semigroup T . If S is a spherical fuzzy lateral ideal of T , then T ◦S◦T ⊆S. Proof. The proof is similar to that of Theorem 3.2. � Theorem 3.4. Let S = (µS,ηS,νS) be a spherical fuzzy set on a ternary semigroup T . If S is a spherical fuzzy right ideal of T , then S◦T ◦T ⊆S. Proof. The proof is similar to that of Theorem 3.2. � 3.2. Rough Spherical Fuzzy Sets in Ternary Semigroups. The aims of this subsection is to connect rough set theory and spherical fuzzy sets of ternary semigroups. Definition 3.4. An equivalence relation ρ on a ternary semigroup T is called a congruence if for all x1,x2,x3,y1,y2,y3 ∈ T (x1,y1),(x2,y2),(x3,y3)∈ ρ ⇒ (x1x2x3,y1y2y3)∈ ρ. The congruence class of x ∈ T is denoted by [x]ρ. A congruence ρ on T is called complete if [y1]ρ[y2]ρ[y3]ρ = [y1y2y3]ρ for all y1,y2,y3 ∈ T. Int. J. Anal. Appl. (2022), 20:29 7 Definition 3.5. Let ρ be a congruence on a ternary semigroup T and S =(µS,ηS,νS) be the spherical fuzzy set on a ternary semigroup T . (1) The lower approximation is defined as App(S)= {< y,µS(y),ηS(y),νS(y) >| y ∈ T}, where µS(y) = inf y ′∈[y]ρ µS(y ′), ηS(y) = inf y ′∈[y]ρ ηS(y ′) and νS(y) = sup y ′∈[y]ρ νS(y ′) with the condition that 0≤ (µS(y))2 +(ηS(y))2 +(νS(y))2 ≤ 1. (2) The upper approximation is defined as App(S)= {< y,µS(y),ηS(y),νS(y) >| y ∈ T}, where µS(y) = sup y ′∈[y]ρ µS(y ′), ηS(y) = sup y ′∈[y]ρ ηS(y ′) and νS(y) = inf y ′∈[y]ρ νS(y ′) with the condition that 0≤ (µS(y))2 +(ηS(y))2 +(νS(y))2 ≤ 1. (3) The rough spherical fuzzy set of T is defined by App(S)= (App(S),App(S)). Theorem 3.5. Let ρ be a congruence on a ternary semigroup T and S1 = (µS1,ηS1,νS1) and S2 = (µS2,ηS2,νS2) be any two spherical fuzzy sets on T. The following statements hold. (1) If S1 ⊆S2, then App(S1)⊆ App(S2) and App(S1)⊆ App(S2). (2) App(S1 ∩S2)⊆ App(S1)∩App(S2). (3) App(S1 ∪S2)= App(S1)∪App(S2). (4) App(S1 ∩S2)= App(S1)∩App(S2). (5) App(S1)∪App(S2)⊆ App(S1 ∪S2). Proof. (1) Assume that S1 ⊆S2. Then µS1 ≤ µS2, ηS2 ≤ ηS1 and νS2 ≥ νS1. Thus for all y ∈ T, we have µS1(y)= sup y ′∈[y]ρ µS1(y ′)≤ sup y ′∈[y]ρ µS2(y ′)= µS2(y), ηS1(y)= sup y ′∈[y]ρ ηS1(y ′)≤ sup y ′∈[y]ρ ηS2(y ′)= ηS2(y) and νS1(y)= inf y ′∈[y]ρ νS1(y ′)≥ inf y ′∈[y]ρ νS2(y ′)= νS2(y). This implies that App(S1)⊆ App(S2). Similarly, we have App(S1)⊆ App(S2). (2) Since S1 ∩S2 ⊆S1 and S1 ∩S2 ⊆S2, App(S1 ∩S2)⊆ App(S1)∩App(S2) by (1). (3) Note that App(S1)∪App(S2)= (µS1 ∪µS2,ηS1 ∩ηS2,νS1 ∩νS2) 8 Int. J. Anal. Appl. (2022), 20:29 and App(S1 ∪S2)= (µS1∪S2,ηS1∪S2,νS1∪S2). Let y ∈ T . Then (µS1 ∪µS2)(y)=max{µS1(y),µS2(y)} =max{ sup y ′∈[y]ρ µS1(y ′), sup y ′∈[y]ρ µS2(y ′)} = sup y ′∈[y]ρ max{µS1(y ′),µS2(y ′)} = sup y ′∈[y]ρ µS1∪S2(y ′) = µS1∪S2(y), (ηS1 ∪ηS2)(y)=max{ηS1(y),ηS2(y)} =max{ sup y ′∈[y]ρ ηS1(y ′), sup y ′∈[y]ρ ηS2(y ′)} = sup y ′∈[y]ρ max{ηS1(y ′),ηS2(y ′)} = sup y ′∈[y]ρ ηS1∪S2(y ′) = ηS1∪S2(y) and (νS1 ∩νS2)(y)=min{νS1(y),νS2(y)} =min{ inf y ′∈[y]ρ νS1(y ′), inf y ′∈[y]ρ νS2(y ′)} = inf y ′∈[y]ρ min{νS1(y ′),νS2(y ′)} = inf y ′∈[y]ρ νS1∪S2(y ′) = νS1∪S2(y). (4) Note that App(S1)∩App(S2)= (µS1 ∩µS2,ηS1 ∩ηS2,νS1 ∪νS2) and App(S1 ∩S2)= (µS1∩S2,ηS1∩S2,νS1∩S2). Int. J. Anal. Appl. (2022), 20:29 9 Let y ∈ T . Then (µS1 ∩µS2)(y)=min{µS1(y),µS2(y)} =min{ inf y ′∈[y]ρ µS1(y ′), inf y ′∈[y]ρ µS2(y ′)} = inf y ′∈[y]ρ min{µS1(y ′),µS2(y ′)} = inf y ′∈[y]ρ µS1∩S2(y ′) = µS1∩S2(y), (ηS1 ∩ηS2)(y)=min{ηS1(y),ηS2(y)} =min{ inf y ′∈[y]ρ ηS1(y ′), inf y ′∈[y]ρ ηS2(y ′)} = inf y ′∈[y]ρ min{ηS1(y ′),ηS2(y ′)} = inf y ′∈[y]ρ ηS1∩S2(y ′) = ηS1∩S2(y) and (νS1 ∪νS2)(y)=max{νS1(y),νS2(y)} =max{ sup y ′∈[y]ρ νS1(y ′), sup y ′∈[y]ρ νS2(y ′)} = sup y ′∈[y]ρ max{νS1(y ′),νS2(y ′)} = sup y ′∈[y]ρ νS1∩S2(y ′) = νS1∩S2(y). (5) Since S1 ⊆S1 ∪S2 and S2 ⊆S1 ∪S2,App(S1)∪App(S2)⊆ App(S1 ∪S2) by (1). � Theorem 3.6. Let ρ be a congruence relation on a ternary semigroup T and S be a spherical fuzzy set on T . Then App(S) is also a spherical fuzzy set on T. 10 Int. J. Anal. Appl. (2022), 20:29 Proof. Let y ∈ T . Then (µS(y)) 2 +(ηS(y)) 2 +(νS(y)) 2 =( inf y ′∈[y]ρ µS(y ′))2 +( inf y ′∈[y]ρ ηS(y ′))2 +( sup y ′∈[y]ρ νS(y ′))2 = inf y ′∈[y]ρ (µS(y ′))2 + inf y ′∈[y]ρ (ηS(y ′))2 + sup y ′∈[y]ρ (νS(y ′))2 ≤ inf y ′∈[y]ρ (µS(y ′))2 + inf y ′∈[y]ρ (ηS(y ′))2 + sup y ′∈[y]ρ (1− (µS(y ′))2 − (ηS(y ′))2) ≤ inf y ′∈[y]ρ (µS(y ′))2 + inf y ′∈[y]ρ (ηS(y ′))2 +1− inf y ′∈[y]ρ (µS(y ′))2 − inf y ′∈[y]ρ (ηS(y ′))2 =1. This implies that 0 ≤ (µS(y))2 +(ηS(y))2 +(νS(y))2 ≤ 1. Therefore, App(S) is a spherical fuzzy set on T. � Let S be a spherical fuzzy set on a ternary semigroup T . Note that App(S) need not be a spherical fuzzy set on T , as can be seen in the following example. Example 3.1. Let T = {i,−i} be the ternary semigroup under the ternary multiplication, ρ = T ×T and S be a spherical fuzzy set on T defined by µS(i)=1,ηS(i)=0,νS(i)=0 and µS(−i)=0,ηS(−i)=1,νS(−i)=0. Then µS(i)= µS(−i)=1,ηS(i)= ηS(−i)=1,νS(i)= νS(−i)=0. In this example, we have that App(S) is not a spherical fuzzy set on T . 3.3. Rough Spherical Fuzzy Ideals in Ternary Semigroups. The aims of this subsection is to con- nect rough set theory and spherical fuzzy ideals of ternary semigroups. Theorem 3.7. Let ρ be a complete congruence relation on a ternary semigroup T . If S is a spherical fuzzy left ideal [spherical fuzzy lateral ideal, spherical fuzzy right ideal] of T , then App(S) is a spherical fuzzy left ideal [spherical fuzzy lateral ideal, spherical fuzzy right ideal] of T. Proof. Let y1,y2,y3 ∈ T. µS(y1y2y3)= inf y∈[y1y2y3]ρ µS(y) = inf y∈[y1]ρ[y2]ρ[y3]ρ µS(y)= inf abc∈[y1]ρ[y2]ρ[y3]ρ µS(abc) ≥ inf abc∈[y1]ρ[y2]ρ[y3]ρ µS(c)= inf c∈[y3] µS(c)= µS(y3), Int. J. Anal. Appl. (2022), 20:29 11 ηS(y1y2y3)= inf y∈[y1y2y3]ρ ηS(y) = inf y∈[y1]ρ[y2]ρ[y3]ρ ηS(y)= inf abc∈[y1]ρ[y2]ρ[y3]ρ ηS(abc) ≥ inf abc∈[y1]ρ[y2]ρ[y3]ρ ηS(c)= inf c∈[y3] ηS(c)= ηS(y3) and νS(y1y2y3)= sup y∈[y1y2y3]ρ νS(y) = sup y∈[y1]ρ[y2]ρ[y3]ρ νS(y)= sup abc∈[y1]ρ[y2]ρ[y3]ρ νS(abc) ≤ sup abc∈[y1]ρ[y2]ρ[y3]ρ νS(c)= sup c∈[y3] νS(c)= νS(y3). This implies that µS(y1y2y3)≥ µS(y3), ηS(y1y2y3)≥ ηS(y3) and νS(y1y2y3)≤ νS(y3). Then App(S) is a spherical fuzzy left ideal of T. The proofs of other cases are similar. � Corollary 3.1. Let ρ be a complete congruence relation on a ternary semigroup T . If S is a spherical fuzzy ideal of T, then App(S) is a spherical fuzzy ideal of T . Proof. This follows from Theorem 3.7. � Theorem 3.8. Let ρ be a congruence relation on a ternary semigroup T. If S is a spherical fuzzy left ideal [spherical fuzzy lateral ideal, spherical fuzzy right ideal] of T and App(S) is a spherical fuzzy set of T , then App(S) is a spherical fuzzy left ideal [spherical fuzzy lateral ideal, spherical fuzzy right ideal] of T . Proof. Let y1,y2,y3 ∈ T. µS(y1y2y3)= sup y∈[y1y2y3]ρ µS(y) ≥ sup y∈[y1]ρ[y2]ρ[y3]ρ µS(y)= sup abc∈[y1]ρ[y2]ρ[y3]ρ µS(abc) ≥ sup abc∈[y1]ρ[y2]ρ[y3]ρ µS(c)= sup c∈[y3]ρ µS(c)= µS(y3), ηS(y1y2y3)= sup y∈[y1y2y3]ρ ηS(y) ≥ sup y∈[y1]ρ[y2]ρ[y3]ρ ηS(y)= sup abc∈[y1]ρ[y2]ρ[y3]ρ ηS(abc) ≥ sup abc∈[y1]ρ[y2]ρ[y3]ρ ηS(c)= sup c∈[y3]ρ ηS(c)= ηS(y3) 12 Int. J. Anal. Appl. (2022), 20:29 and νS(y1y2y3)= inf y∈[y1y2y3]ρ νS(y) ≤ inf y∈[y1]ρ[y2]ρ[y3]ρ νS(y)= inf abc∈[y1]ρ[y2]ρ[y3]ρ νS(abc) ≤ inf c∈[y3]ρ[y2]ρ[y3]ρ νS(c)= inf c∈[y3]ρ νS(c)= νS(y3). This implies that µS(y1y2y3)≥ µS(y3), ηS(y1y2y3)≥ ηS(y3) and νS(y1y2y3)≤ νS(y3). Then App(S) is a spherical fuzzy left ideal of T. The proofs of other cases are similar. � Corollary 3.2. Let ρ be a congruence relation on a ternary semigroup T. If S is a spherical fuzzy ideal of T and App(S) is a spherical fuzzy set of T , then App(S) is a spherical fuzzy ideal of T. Proof. 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