Int. J. Anal. Appl. (2023), 21:26 The Prominentness of Fuzzy GE-Filters in GE-Algebras Sun Shin Ahn1,∗, Rajab Ali Borzooei2, Young Bae Jun3 1Department of Mathematics Education, Dongguk University, Seoul 04620, Korea 2Department of Mathematics, Shahid Beheshti University, Tehran 1983963113, Iran 3Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea ∗Corresponding author: sunshine@dongguk.edu Abstract. Based on the concept of fuzzy points, the notion of a prominent fuzzy GE-filter is defined, and the various properties involved are investigated. The relationship between a fuzzy GE-filter and a prominent fuzzy GE-filter is discussed, and the characterization of a prominent fuzzy GE-filter is considered. The conditions under which a fuzzy GE-filter can be a prominent fuzzy GE-filter are explored, and conditions for the trivial fuzzy GE-filter to be a prominent fuzzy GE-filter are provided. The conditions under which the ∈t-set and Qt-set can be prominent GE-filters are explored. Finally, the extension property for the prominent fuzzy GE-filter is discussed. 1. Introduction Henkin and Scolem introduced the concept of Hilbert algebra in the implication investigation in intuitionistic logics and other nonclassical logics. Diego [6] established that Hilbert algebras form a locally finite variety. Later several researchers extended the theory on Hilbert algebras (see [4,5,7,8]). The notion of BE-algebra was introduced by Kim et al. [9] as a generalization of a dual BCK-algebra. Rezaei et al. [13] discussed relations between Hilbert algebras and BE-algebras. As a generalization of Hilbert algebras, Bandaru et al. [2] introduced the notion of GE-algebras, and investigated several properties. Bandaru et al. [3] introduced the concept of bordered GE-algebra and investigated its properties. Later, Ozturk et al. [10] introduced the concept of strong GE-filters, GE-ideals of bordered GE-algebras and investigated its properties. Song et al. [14] introduced the concept of Imploring GE-filters of GE-algebras and discussed its properties. Rezaei et al. [12] introduced the concept of Received: Jan. 24, 2023. 2020 Mathematics Subject Classification. 03G25, 06F35, 08A72. Key words and phrases. (prominent) GE-filter; (prominent) fuzzy GE-filter; trivial fuzzy GE-filter; ∈t-set; Qt-set. https://doi.org/10.28924/2291-8639-21-2023-26 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-26 2 Int. J. Anal. Appl. (2023), 21:26 prominent GE-filters in GE-algebras and discussed its properties. Bandaru et al. [1] discussed the fuzzy notion of GE-filters in GE-algebras. The purpose of this paper is to define a prominent fuzzy GE-filter using the concept of fuzzy points and investigate the various properties involved. We consider the relationship between a fuzzy GE-filter and a prominent fuzzy GE-filter. We explore the conditions under which a fuzzy GE-filter can be a prominent fuzzy GE-filter. We discuss the characterization of a prominent fuzzy GE-filter. We provide conditions for the trivial fuzzy GE-filter to be a prominent fuzzy GE-filter. We explore the conditions under which the ∈t-set and Qt-set can be prominent GE-filters. We finally discuss the extension property for the prominent fuzzy GE-filter. 2. Preliminaries 2.1. Basics related to GE-algebras. Definition 2.1 ( [2]). By a GE-algebra we mean a set X with a constant “1” and a binary operation “∗” satisfying the following axioms: (GE1) a∗a =1, (GE2) 1∗a = a, (GE3) a∗ (b∗c)= a∗ (b∗ (a∗c)) for all a,b,c ∈ X. We denote the GE-algebra by X := (X,∗,1). A binary relation “≤ ” in a GE-algebra X := (X,∗,1) is defined by: (∀x,y ∈ X)(x ≤ y ⇔ x ∗y =1). (2.1) Definition 2.2 ( [2]). A GE-algebra X := (X,∗,1) is said to be • transitive if it satisfies: (∀a,b,c ∈ X)(a∗b ≤ (c ∗a)∗ (c ∗b)) . (2.2) • commutative if it satisfies: (∀a,b ∈ X)((a∗b)∗b =(b∗a)∗a) . (2.3) Note that every commutative GE-algebra is transitive and antisymmetric. Proposition 2.1 ( [2]). Every GE-algebra X := (X,∗,1) satisfies the following items. (∀a ∈ X)(a∗1=1) . (2.4) (∀a,b ∈ X)(a∗ (a∗b)= a∗b) . (2.5) (∀a,b ∈ X)(a ≤ b∗a) . (2.6) Int. J. Anal. Appl. (2023), 21:26 3 (∀a,b,c ∈ X)(a∗ (b∗c)≤ b∗ (a∗c)) . (2.7) (∀a ∈ X)(1≤ a ⇒ a =1) . (2.8) (∀a,b ∈ X)(a ≤ (a∗b)∗b) . (2.9) If X := (X,∗,1) is transitive, then (∀a,b,c ∈ X)(a ≤ b ⇒ c ∗a ≤ c ∗b, b∗c ≤ a∗c) . (2.10) (∀a,b,c ∈ X)(a∗b ≤ (b∗c)∗ (a∗c)) . (2.11) (∀a,b,c ∈ X)(a∗b ≤ (c ∗a)∗ (c ∗b)) . (2.12) Definition 2.3. A subset F of a GE-algebra X := (X,∗,1) is called • a GE-filter of X := (X,∗,1) (see [2]) if it satisfies: 1∈ F, (2.13) (∀a,b ∈ X)(a ∈ F, a∗b ∈ F ⇒ b ∈ F). (2.14) • a prominent GE-filter of X := (X,∗,1) (see [12]) if it satisfies (2.13) and (∀a,b,c ∈ X)(a∗ (b∗c)∈ F, a ∈ F ⇒ ((c ∗b)∗b)∗c ∈ F). (2.15) Lemma 2.1 ( [2]). Every GE-filter F of X := (X,∗,1) satisfies: (∀x,y ∈ X)(x ≤ y, x ∈ F ⇒ y ∈ F). (2.16) Lemma 2.2 ( [12]). Every prominent GE-filter is a GE-filter. 2.2. Basics related to fuzzy sets. A fuzzy set f in a set X of the form f (b) := { t ∈ (0,1] if b = a, 0 if b 6= a, is said to be a fuzzy point with support a and value t and is denoted by a t . For a fuzzy set f in a set X and t ∈ (0,1], we say that a fuzzy point a t is (i) contained in f , denoted by a t ∈ f , (see [11]) if f (a)≥ t. (ii) quasi-coincident with f , denoted by a t q f, (see [11]) if f (a)+ t > 1. If a t αf is not established for α ∈{∈,q}, it is denoted by a t αf . Given t ∈ (0,1] and a fuzzy set f in a set X, consider the following sets (f ,t)∈ := {x ∈ X | xt ∈ f} and (f ,t)q := {x ∈ X | x t q f} which are called an ∈t-set and Qt-set of f , respectively, in X. 4 Int. J. Anal. Appl. (2023), 21:26 Definition 2.4 ( [1]). A fuzzy set f in a GE-algebra X := (X,∗,1) is called a fuzzy GE-filter of X := (X,∗,1) if it satisfies: (∀t ∈ (0,1])((f ,t)∈ 6= ∅ ⇒ 1∈ (f ,t)∈) , (2.17) x ∗y ∈ (f ,tb)∈, x ∈ (f ,ta)∈ ⇒ y ∈ (f ,min{ta,tb})∈ (2.18) for all x,y ∈ X and ta,tb ∈ (0,1]. 3. The Prominentness of Fuzzy GE-Filters In what follows, let X := (X,∗,1) denote a GE-algebra unless otherwise specified. Definition 3.1. A fuzzy set f in X is called a prominent fuzzy GE-filter of X := (X,∗,1) if it satisfies (2.17) and (∀x,y,z ∈ X)(∀ta,tb ∈ (0,1]) ( x ∗ (y ∗z)∈ (f ,tb)∈, x ∈ (f ,ta)∈ ⇒ ((z ∗y)∗y)∗z ∈ (f ,min{ta,tb})∈ ) . (3.1) Example 3.1. Let X = {1,2,3,4,5,6,7} be a set with a binary operation “∗” given by Table 1. Table 1. Cayley table for the binary operation “∗” ∗ 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 2 1 1 1 4 6 6 1 3 1 2 1 5 5 5 7 4 1 1 3 1 1 1 1 5 1 2 1 1 1 1 7 6 1 2 3 1 1 1 1 7 1 2 3 6 5 6 1 Then X := (X,∗,1) is a GE-algebra (see [12]). Define a fuzzy set f in X as follows: f : X → [0,1], x 7→ { 0.85 if x ∈{1,2,3,7}, 0.37 otherwise. It is routine to verify that f is a prominent fuzzy GE-filter of X := (X,∗,1). We discuss the relationship between a fuzzy GE-filter and a prominent fuzzy GE-filter. Theorem 3.1. Every prominent fuzzy GE-filter is a fuzzy GE-filter. Proof. Let f be a prominent fuzzy GE-filter of X := (X,∗,1). Let x,y ∈ X and ta,tb ∈ (0,1] be such that x ∈ (f ,ta)∈ and x ∗ y ∈ (f ,tb)∈. Then x ∗ (1 ∗ y) = x ∗ y ∈ (f ,tb)∈ by (GE2), and so Int. J. Anal. Appl. (2023), 21:26 5 y = ((y ∗1) ∗1) ∗ y ∈ (f ,tb)∈ by (GE1), (GE2), (2.4) and (3.1). Hence f is a fuzzy GE-filter of X := (X,∗,1). � The following example shows that the converse of Theorem 3.1 may not be true. Example 3.2. Consider the GE-algebra X := (X,∗,1) in Example 3.1 and let f be a fuzzy set in X defined by f : X → [0,1], x 7→ { 0.79 if x ∈{1,3,7}, 0.46 otherwise. It is routine to verify that f is a fuzzy GE-filter of X := (X,∗,1). But it is not a prominent fuzzy GE-filter of X := (X,∗,1) since 3∈ (f ,0.67)∈ and 3∗(4∗2)=1∈ (f ,0.62)∈, but ((2∗4)∗4)∗2= 2 /∈ (f ,0.62)∈ =(f ,min{0.67,0.62})∈. We explore the conditions under which a fuzzy GE-filter can be a prominent fuzzy GE-filter. Theorem 3.2. Given a fuzzy GE-filter f of X := (X,∗,1), it is a prominent fuzzy GE-filter of X := (X,∗,1) if and only if it satisfies: (∀x,y ∈ X)(∀t ∈ (0,1])(x ∗y ∈ (f ,t)∈ ⇒ ((y ∗x)∗x)∗y ∈ (f ,t)∈). (3.2) Proof. Assume that f is a prominent fuzzy GE-filter of X := (X,∗,1) and let x,y ∈ X and t ∈ (0,1] be such that x ∗y ∈ (f ,t)∈. Then 1∗(x ∗y)= x ∗y ∈ (f ,t)∈ by (GE2). Since 1∈ (f ,t)∈, it follows from (3.1) that ((y ∗x)∗x)∗y ∈ (f ,t)∈. Conversely, let f be a fuzzy GE-filter of X := (X,∗,1) that satisfies the condition (3.2). Let x,y,z ∈ X and ta,tb ∈ (0,1] be such that x ∗ (y ∗ z) ∈ (f ,tb)∈ and x ∈ (f ,ta)∈. Then y ∗ z ∈ (f ,min{ta,tb})∈ by (2.18), and so ((z ∗ y) ∗ y) ∗ z ∈ (f ,min{ta,tb})∈ by (3.2). Therefore f is a prominent fuzzy GE-filter of X := (X,∗,1). � Lemma 3.1 ( [1]). Every fuzzy GE-filter f of X satisfies: (∀x,y ∈ X)(∀ta ∈ (0,1])(x ≤ y, x ∈ (f ,ta)∈ ⇒ y ∈ (f ,ta)∈) , (3.3) (∀x,y,z ∈ X)(∀ta,tb ∈ (0,1]) ( z ≤ y ∗x, y ∈ (f ,tb)∈, z ∈ (f ,ta)∈ ⇒ x ∈ (f ,min{ta,tb})∈ ) . (3.4) Theorem 3.3. In a commutative GE-algebra, every fuzzy GE-filter is a prominent fuzzy GE-filter. Proof. Let f be a prominent fuzzy GE-filter of X := (X,∗,1). It is sufficient to show that f satisfies the condition (3.1). Let x,y,z ∈ X and ta,tb ∈ (0,1] be such that x ∗ (y ∗ z) ∈ (f ,tb)∈ and 6 Int. J. Anal. Appl. (2023), 21:26 x ∈ (f ,ta)∈. Using (2.3), (2.7), and (2.12), we have 1= ((z ∗y)∗y)∗ ((y ∗z)∗z) ≤ (y ∗z)∗ (((z ∗y)∗y)∗z) ≤ (x ∗ (y ∗z))∗ (x ∗ (((z ∗y)∗y)∗z)) ≤ x ∗ ((x ∗ (y ∗z))∗ (((z ∗y)∗y)∗z)), and so x ∗((x ∗(y ∗z))∗(((z ∗y)∗y)∗z))=1, i.e., x ≤ (x ∗(y ∗z))∗(((z ∗y)∗y)∗z). It follows from Lemma 3.1 that ((z ∗y)∗y)∗z ∈ (f ,min{ta,tb})∈. Therefore f is a prominent fuzzy GE-filter of X := (X,∗,1). � Theorem 3.4. A fuzzy set f in X is a prominent fuzzy GE-filter of X := (X,∗,1) if and only if it satisfies: (∀x ∈ X)(f (1)≥ f (x)). (3.5) (∀x,y,z ∈ X)(f (((z ∗y)∗y)∗z)≥min{f (x), f (x ∗ (y ∗z))}). (3.6) Proof. Assume that f is a prominent fuzzy GE-filter of X := (X,∗,1). Suppose there exists a ∈ X such that f (1) < f (a). Let t0 = 1 2 (f (1)+ f (a)). Then f (1) < t0 and 0 < t0 < f (a) ≤ 1. Hence a ∈ (f ,t0)∈ and so (f ,t0)∈ 6= ∅. Thus 1∈ (f ,t0)∈, that is, f (1)≥ t0, which is contradiction. Hence f (1) ≥ f (x) for all x ∈ X. Let x,y,z ∈ X be such that f (x) = t1 and f (x ∗ (y ∗ z)) = t2. Then x ∈ (f ,t1)∈ and x ∗ (y ∗z)∈ (f ,t2)∈. It follows from (3.1) that ((z ∗y)∗y)∗z ∈ (f ,min{t1,t2})∈. Hence f (((z ∗y)∗y)∗z)≥min{t1,t2}=min(f (x), f (x ∗ (y ∗z))). Conversely, assume that f satisfies (3.5) and (3.6). Let t ∈ (0,1] and x ∈ (f ,t)∈. Then f (x)≥ t and hence f (1) ≥ f (x) ≥ t. Thus 1 ∈ (f ,t)∈. Let x,y,z ∈ X be such that x ∈ (f ,t1)∈ and x ∗ (y ∗ z) ∈ (f ,t2)∈. Then f (x) ≥ t1 and f (x ∗ (y ∗ z)) ≥ t2. Therefore f (((z ∗ y) ∗ y) ∗ z) ≥ min{f (x), f (x ∗ (y ∗z))}≥min{t1,t2} by (3.6). Hence ((z ∗y)∗y)∗z ∈ (f ,min{t1,t2})∈. Thus f is a prominent fuzzy GE-filter of X := (X,∗,1). � Theorem 3.5. Given an element b ∈ X, define a fuzzy set fb in X as follows: fb : X → [0,1], x 7→ { t1 if x ∈~b, t2 otherwise., where ~b := {x ∈ X | b ≤ x} and t1 > t2 in (0,1]. Then fb is a prominent fuzzy GE-filter of X := (X,∗,1) if and only if X := (X,∗,1) satisfies: (∀x,y,z ∈ X)(x ∈~b, x ∗ (y ∗z)∈~b ⇒ ((z ∗y)∗y)∗z ∈~b). (3.7) Proof. Assume that fb is a prominent fuzzy GE-filter of X := (X,∗,1) and let x,y,z ∈ X be such that x ∈~b and x ∗ (y ∗z)∈~b. Then fb(x)= t1 = fb(x ∗ (y ∗z)), which implies from (3.6) that fb(((z ∗y)∗y)∗z)≥min{fb(x), fb(x ∗ (y ∗z))}= t1. Int. J. Anal. Appl. (2023), 21:26 7 Hence fb(((z ∗y)∗y)∗z)= t1, and thus ((z ∗y)∗y)∗z ∈~b. Conversely, suppose that X := (X,∗,1) satisfies the condition (3.7). Since 1 ∈~b, we get fb(1) = t1 ≥ fb(x) for all x ∈ X. For every x,y,z ∈ X, if x /∈ ~b or x ∗ (y ∗ z) /∈ ~b, then fb(x) = t2 or fb(x ∗ (y ∗z))= t2. Hence fb(((z ∗y)∗y)∗z)≥ t2 =min{fb(x), fb(x ∗ (y ∗z))}. If x ∈~b and x ∗ (y ∗z)∈~b, then fb(x)= t1 and fb(x ∗ (y ∗z))= t1. Thus fb(((z ∗y)∗y)∗z)= t1 =min{fb(x), fb(x ∗ (y ∗z))}. Therefore fb is a prominent fuzzy GE-filter of X := (X,∗,1) by Theorem 3.4. � Consider a fuzzy set f in X which is given by f : X → [0,1], x 7→ { t1 if x =1, t2 otherwise, where t1 > t2 in (0,1]. It is clear that f is a fuzzy GE-filter of X := (X,∗,1), which is called the trivial fuzzy GE-filter of X := (X,∗,1). But it is not a prominent fuzzy GE-filter of X := (X,∗,1) as seen in the following example. Example 3.3. Consider the GE-algebra X := (X,∗,1) in Example 3.1 and let f be a fuzzy set in X defined by f : X → [0,1], x 7→ { 0.83 if x =1, 0.57 otherwise. Then f is a fuzzy GE-filter of X := (X,∗,1), but it is not a prominent fuzzy GE-filter of X := (X,∗,1) since 1∈ (f ,0.69)∈ and 1∗(4∗2)=1∈ (f ,0.64)∈, but ((2∗4)∗4)∗2=2 /∈ (f ,min{0.69,0.64})∈. We provide conditions for the trivial fuzzy GE-filter to be a prominent fuzzy GE-filter. Theorem 3.6. In a commutative GE-algebra, the trivial fuzzy GE-filter is a prominent fuzzy GE-filter. Proof. Let f be the trivial fuzzy GE-filter of a commutative GE-algebra X := (X,∗,1). Then (f ,t)∈ =   ∅ if t ∈ (t1,1], {1} if t ∈ (t2,t1], X if t ∈ (0,t2]. It is sufficient to show that (f ,t)∈ = {1} is a prominent GE-filter of X := (X,∗,1). Let x,y,z ∈ X be such that x ∈ {1} and x ∗ (y ∗ z) ∈ {1}. Using (GE2), (2.3) and (GE1), we get y ∗ z = 1, and thus ((z ∗ y)∗ y)∗ z = ((y ∗ z)∗ z)∗ z = (1∗ z)∗ z = z ∗ z = 1 ∈ {1}. Hence (f ,t)∈ = {1} is a prominent GE-filter of X := (X,∗,1), and therefore f is a prominent fuzzy GE-filter of X := (X,∗,1) by Theorem ??. � 8 Int. J. Anal. Appl. (2023), 21:26 We explore the conditions under which the ∈t-set and Qt-set can be prominent GE-filters. Theorem 3.7. Given a fuzzy set f in X, its ∈t-set (f ,t)∈ is a prominent GE-filter of X for all t ∈ (0.5,1] if and only if f satisfies: (∀x ∈ X)(f (x)≤max{f (1),0.5}), (3.8) (∀x,y ∈ X)(min{f (x), f (x ∗ (y ∗z))}≤max{f (((z ∗y)∗y)∗z),0.5}). (3.9) Proof. Assume that the ∈t-set (f ,t)∈ of f is a prominent GE-filter of X for all t ∈ (0.5,1]. If there exists a ∈ X such that f (a) � max{f (1),0.5}, then t := f (a) ∈ (0.5,1], a t ∈ f and 1 t ∈ f , that is, a ∈ (f ,t)∈ and 1 /∈ (f ,t)∈. This is a contradiction, and thus f (x)≤max{f (1),0.5} for all x ∈ X. If (3.9) is not valid, then min{f (a), f (a∗ (b∗c))} > max{f (((c ∗b)∗b)∗c),0.5} for some a,b,c ∈ X. If we take t := min{f (a), f (a ∗ (b ∗ c))}, then t ∈ (0.5,1], a t ∈ f and a∗(b∗c) t ∈ f . Hence a ∈ (f ,t)∈ and a ∗ (b ∗ c) ∈ (f ,t)∈, which imply that ((c ∗b)∗b)∗ c ∈ (f ,t)∈. Thus ((c∗b)∗b)∗c t ∈ f , and so f (((c ∗b)∗b)∗c)≥ t > 0.5 which is a contradiction. Therefore min{f (x), f (x ∗ (y ∗z))}≤max{f (((z ∗y)∗y)∗z),0.5} for all x,y ∈ X. Conversely, suppose that f satisfies (3.8) and (3.9). Let (f ,t)∈ 6= ∅ for all t ∈ (0.5,1]. Then there exists a ∈ (f ,t)∈ and thus at ∈ f , i.e., f (a)≥ t. It follows from (3.8) that max{f (1),0.5}≥ f (a)≥ t > 0.5. Thus 1 t ∈ f , i.e., 1 ∈ (f ,t)∈. Let t ∈ (0.5,1] and x,y,z ∈ X be such that x ∈ (f ,t)∈ and x ∗ (y ∗ z) ∈ (f ,t)∈. Then xt ∈ f and x∗(y∗z) t ∈ f , that is, f (x) ≥ t and f (x ∗ (y ∗ z)) ≥ t. Using (3.9), we get max{f (((z ∗y)∗y)∗z),0.5}≥min{f (x), f (x ∗ (y ∗z))}≥ t > 0.5 and so ((z∗y)∗y)∗z t ∈ f , i.e., ((z ∗y)∗y)∗z ∈ (f ,t)∈. Therefore (f ,t)∈ is a prominent GE-filter of X for all t ∈ (0.5,1]. � Lemma 3.2 ( [1]). A fuzzy set f in X is a fuzzy GE-filter of X if and only if the nonempty ∈t-set (f ,t)∈ of f in X is a GE-filter of X for all t ∈ (0,1]. Lemma 3.3 ( [12]). Let F be a GE-filter of X := (X,∗,1). Then it is a prominent GE-filter of X := (X,∗,1) if and only if it satisfies: (∀x,y ∈ X)(x ∗y ∈ F ⇒ ((y ∗x)∗x)∗y ∈ F). (3.10) Theorem 3.8. A fuzzy set f in X is a prominent fuzzy GE-filter of X := (X,∗,1) if and only if the nonempty ∈t-set (f ,t)∈ of f in X is a prominent GE-filter of X := (X,∗,1) for all t ∈ (0,1]. Int. J. Anal. Appl. (2023), 21:26 9 Proof. Assume that f is a prominent fuzzy GE-filter of X := (X,∗,1). Then f is a fuzzy GE-filter of X := (X,∗,1) (see Theorem 3.1), and so the nonempty ∈t-set (f ,t)∈ of f in X is a GE-filter of X := (X,∗,1) for all t ∈ (0,1] by Lemma 3.2. Let x,y ∈ X and t ∈ (0,1] be such that x ∗ y ∈ (f ,t)∈. Since f is a prominent fuzzy GE-filter of X := (X,∗,1), it follows from (3.2) that ((y ∗ x) ∗ x) ∗ y ∈ (f ,t)∈, and therefore (f ,t)∈ is a prominent GE-filter of X := (X,∗,1) for all t ∈ (0,1] by Lemma 3.3. Conversely, suppose that the nonempty ∈t-set (f ,t)∈ of f in X is a prominent GE-filter of X := (X,∗,1) for all t ∈ (0,1]. Then (f ,t)∈ is a GE-filter of X := (X,∗,1) by Lemma 2.2, and thus f is a fuzzy GE-filter of X := (X,∗,1) by Lemma 3.2. Let x,y ∈ X and t ∈ (0,1] be such that x ∗y ∈ (f ,t)∈. Then ((y ∗x)∗x)∗y ∈ (f ,t)∈ by Lemma 3.3. It follows from Theorem 3.2 that f is a prominent fuzzy GE-filter of X := (X,∗,1). � Theorem 3.9. If f is a prominent fuzzy GE-filter of X := (X,∗,1), then the nonempty Qt-set (f ,t)q of f is a prominent GE-filter of X := (X,∗,1) for all t ∈ (0,1]. Proof. Let f be a prominent fuzzy GE-filter of X := (X,∗,1) and assume that (f ,t)q 6= ∅ for all t ∈ (0,1]. Then there exists a ∈ (f ,t)q, and so at q f , i.e., f (a)+t > 1. Hence f (1)+t ≥ f (a)+t > 1, i.e., 1 ∈ (f ,t)q. Let x,y,z ∈ X be such that x ∈ (f ,t)q and x ∗ (y ∗ z) ∈ (f ,t)q. Then xt q f and x∗(y∗z) t q f , that is, f (x)+ t > 1 and f (x ∗ (y ∗z))+ t > 1. It follows from (3.6) that f (((z ∗y)∗y)∗z)+ t ≥min{f (x), f (x ∗ (y ∗z))}+ t =min{f (x)+ t, f (x ∗ (y ∗z))+ t} > 1. Hence ((z∗y)∗y)∗z t q f , and therefore ((z ∗ y) ∗ y) ∗ z ∈ (f ,t)q. Consequently, (f ,t)q is a prominent GE-filter of X := (X,∗,1) for all t ∈ (0,1]. � We finally discuss the extension property for the prominent fuzzy GE-filter. Question. Let f and g be fuzzy GE-filters of X := (X,∗,1) such that f ⊆ g, that is, f (x) ≤ g(x) for all x ∈ X. If f is a prominent fuzzy GE-filter of X := (X,∗,1), then is g also a prominent fuzzy GE-filter of X := (X,∗,1)? The example below provides a negative answer to the Question. Example 3.4. Let X = {1,2,3,4,5,6} be a set with a binary operation “∗” given by Table 2. Then X := (X,∗,1) is a GE-algebra (see [12]). Define a fuzzy set f in X as follows: f : X → [0,1], x 7→ { 0.65 if x =1, 0.37 otherwise. 10 Int. J. Anal. Appl. (2023), 21:26 Table 2. Cayley table for the binary operation “∗” ∗ 1 2 3 4 5 6 1 1 2 3 4 5 6 2 1 1 3 4 3 1 3 1 6 1 1 6 6 4 1 2 1 1 2 2 5 1 1 1 4 1 1 6 1 1 3 4 3 1 It is routine to verify that f is a prominent GE-filter of X := (X,∗,1). Now, we define a fuzzy set g in X as follows: g : X → [0,1], x 7→   0.73 if x =1, 0.67 if x ∈{2,6}, 0.48 otherwise. Then f (x) ≤ g(x) for all x ∈ X, that is, f ⊆ g, and g is a fuzzy GE-filter of X := (X,∗,1). Since 4∗5=2∈ (g,0.61)∈ and ((5∗4)∗4)∗5=5 /∈ (g,0.61)∈, we know that g is not a prominent fuzzy GE-filter of X := (X,∗,1) by Theorem 3.2. We provide conditions for the answer of Question above to be positive. Theorem 3.10. (Extension property for the prominent fuzzy GE-filter) Let f and g be fuzzy GE- filters of a transitive GE-algebra X := (X,∗,1) such that f ⊆ g, that is, f (x)≤ g(x) for all x ∈ X. If f is a prominent fuzzy GE-filter of X := (X,∗,1), then so is g. Proof. If f is a prominent fuzzy GE-filter of X := (X,∗,1), then it is a fuzzy GE-filter of X := (X,∗,1) by Theorem 3.1 and (f ,t)∈ is a prominent GE-filter of X := (X,∗,1) for all t ∈ (0,1] by Theorem 3.8. Let a := x ∗ y ∈ (g,t)∈ for all x,y ∈ X and t ∈ (0,1]. Then 1 ∈ (f ,t)∈ by (2.17) and 1= a∗(x∗y)≤ x∗(a∗y) by (GE1) and (2.7). Hence x∗(a∗y)∈ (f ,t)∈ by (3.3). Using assumption and Theorem 3.2 induces (((a∗y)∗x)∗x)∗ (a∗y)∈ (f ,t)∈ ⊆ (g,t)∈. Since (((a ∗ y) ∗ x) ∗ x) ∗ (a ∗ y) ≤ a ∗ ((((a ∗ y) ∗ x) ∗ x) ∗ y) by (2.7) and (g,t)∈ is a GE- filter of X := (X,∗,1), we have a ∗ ((((a ∗ y) ∗ x) ∗ x) ∗ y) ∈ (g,t)∈ by Lemma 2.1. Hence (((a∗y)∗x)∗x)∗y ∈ (g,t)∈ by (2.14). Since y ≤ a∗y by (2.6), we have (((a∗y)∗x)∗x)∗y ≤ ((y ∗x)∗x)∗y Int. J. Anal. Appl. (2023), 21:26 11 by running (2.10) three times. It follows from Lemma 2.1 that ((y ∗ x) ∗ x) ∗ y ∈ (g,t)∈. Hence (g,t)∈ is a prominent GE-filter of X := (X,∗,1) by Lemma 3.3, and therefore g is a prominent fuzzy GE-filter of X := (X,∗,1) by Theorem 3.8. � Corollary 3.1. Let X := (X,∗,1) be a transitive GE-algebra. Then the trivial fuzzy GE-filter f is a prominent fuzzy GE-filter of X := (X,∗,1) if and only if every fuzzy GE-filter is a prominent fuzzy GE-filter of X := (X,∗,1). Corollary 3.2. In a commutative GE-algebra, every fuzzy GE-filter is a prominent fuzzy GE-filter. The following example describes the extension property for the prominent fuzzy GE-filter. Example 3.5. Let X = {1,2,3,4,5,6} be a set with a binary operation “∗” given by Table 3. Table 3. Cayley table for the binary operation “∗” ∗ 1 2 3 4 5 6 1 1 2 3 4 5 6 2 1 1 3 4 4 6 3 1 2 1 5 5 6 4 1 1 1 1 1 6 5 1 1 1 1 1 6 6 1 2 3 4 5 1 Then X := (X,∗,1) is a GE-algebra (see [12]). Define a fuzzy set f in X as follows: f : X → [0,1], x 7→ { 0.59 if x ∈{1,2,3}, 0.36 otherwise. Then f is a prominent fuzzy GE-filter of X := (X,∗,1). If we take a fuzzy set g in X defined as follows: g : X → [0,1], x 7→ { 0.69 if x ∈{1,2,3,6}, 0.56 otherwise, then f ⊆ g and g is a prominent fuzzy GE-filter of X := (X,∗,1). 4. Conclusion Using the concept of fuzzy points, we have introduced the notion of a prominent fuzzy GE-filter in GE-algebras, and have investigated the various properties involved. We have considered the relationship between a fuzzy GE-filter and a prominent fuzzy GE-filter, and have discussed the characterization of a prominent fuzzy GE-filter. We have explored the conditions under which a fuzzy GE-filter can be a prominent fuzzy GE-filter. We have provided conditions for the trivial fuzzy GE-filter to be a 12 Int. J. Anal. Appl. (2023), 21:26 prominent fuzzy GE-filter, and have explored the conditions under which the ∈t-set and Qt-set can be prominent GE-filters. We finally have discussed the extension property for the prominent fuzzy GE-filter. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] R.K. Bandaru, T.G. Alemayehu, Y.B. Jun, Fuzzy GE-Filters of GE-Algebras, J. Algebra Related Topics. (submitted). [2] R. Bandaru, A.B. Saeid, Y.B. Jun, On GE-Algebras, Bull. Sect. Logic. 50 (2020), 81–96. https://doi.org/10. 18778/0138-0680.2020.20. [3] R.K. Bandaru, M.A. Öztürk, Y.B. Jun, Bordered GE-algebras, J. Algebraic Syst. In Press. [4] S. Celani, A Note on Homomorphisms of Hilbert Algebras, Int. J. Math. Math. 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Appl. 76 (1980), 571–599. https://doi.org/10.1016/0022-247x(80)90048-7. [12] A. Rezaei, R. Bandaru, A.B. Saeid, Y.B. Jun, Prominent GE-Filters and GE-Morphisms in GE-Algebras, Afr. Mat. 32 (2021), 1121–1136. https://doi.org/10.1007/s13370-021-00886-6. [13] A. Rezaei, A. Borumand Saeid, R.A. Borzooei, Relation Between Hilbert Algebras and BE-Algebras, Appl. Appl. Math.: Int. J. 8 (2013), 573–584. [14] S.Z. Song, R. Bandaru, Y.B. Jun, Imploring GE-Filters of GE-Algebras, J. Math. 2021 (2021), 1–7. https: //doi.org/10.1155/2021/6651531. https://doi.org/10.18778/0138-0680.2020.20 https://doi.org/10.18778/0138-0680.2020.20 https://doi.org/10.1155/s0161171202011134 https://doi.org/10.1155/s0161171202011134 http://dml.cz/dmlcz/119331 https://doi.org/10.1155/2021/5520023 https://doi.org/10.1016/0022-247x(80)90048-7 https://doi.org/10.1007/s13370-021-00886-6 https://doi.org/10.1155/2021/6651531 https://doi.org/10.1155/2021/6651531 1. Introduction 2. Preliminaries 2.1. Basics related to GE-algebras 2.2. Basics related to fuzzy sets 3. The Prominentness of Fuzzy GE-Filters 4. Conclusion References