Int. J. Anal. Appl. (2022), 20:68 Fuzzy Subalgebras and Ideals With Thresholds of Hilbert Algebras Aiyared Iampan1,∗, P. Jayaraman2, S. D. Sudha2, N. Rajesh3 1Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 2Department of Mathematics, Bharathiyar University, Coimbatore 641046, Tamilnadu, India 3Department of Mathematics, Rajah Serfoji Government College, Thanjavur 613005, Tamilnadu, India ∗Corresponding author: aiyared.ia@up.ac.th Abstract. The concepts of fuzzy subalgebras and ideals with thresholds of Hilbert algebras are pre- sented, some of their features are explained, and their extensions are demonstrated using the theory of fuzzy sets as a foundation. We also talk about the connections between fuzzy subalgebras (ideals) with thresholds and their level subsets. The homomorphic images and inverse images of fuzzy subal- gebras and ideals with thresholds in Hilbert algebras are also studied and some related properties are investigated. 1. Introduction The concept of fuzzy sets was proposed by Zadeh [25]. The theory of fuzzy sets has several applica- tions in real-life situations, and many scholars have researched fuzzy set theory. After the introduction of the concept of fuzzy sets, several research studies were conducted on the generalizations of fuzzy sets. The integration between fuzzy sets and some uncertainty approaches such as soft sets and rough sets has been discussed in [1,3,6]. The idea of intuitionistic fuzzy sets suggested by Atanassov [2] is one of the extensions of fuzzy sets with better applicability. Applications of intuitionistic fuzzy sets appear in various fields, including medical diagnosis, optimization problems, and multicriteria decision- making [12–14]. The concept of Hilbert algebras was introduced in early 50-ties by Henkin [15] for Received: Nov. 7, 2022. 2010 Mathematics Subject Classification. 03G25, 03E72. Key words and phrases. Hilbert algebra; fuzzy subalgebra with thresholds; fuzzy ideal with thresholds. https://doi.org/10.28924/2291-8639-20-2022-68 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-68 2 Int. J. Anal. Appl. (2022), 20:68 some investigations of implication in intuitionistic and other non-classical logics. In 60-ties, these al- gebras were studied especially by Diego [8] from algebraic point of view. Diego [8] proved that Hilbert algebras form a variety which is locally finite. Hilbert algebras were treated by Busneag [4, 5] and Jun [16] and some of their filters forming deductive systems were recognized. Dudek [10] considered the fuzzification of subalgebras/ideals and deductive systems in Hilbert algebras. The concept of fuzzy sets with thresholds is presented in several articles as seen in [9,17–21,23,24]. This study builds on the theory of fuzzy sets to introduce fuzzy subalgebras and ideals with thresholds of Hilbert algebras, explain some of their characteristics, and show how they may be extended. We also talk about the connections between fuzzy subalgebras (ideals) with thresholds and their level subsets. Additionally, the homomorphic images and inverse images of fuzzy subalgebras and ideals with thresholds in Hilbert algebras are investigated and some associated features are examined. 2. Preliminaries Let’s go through the idea of Hilbert algebras as it was introduced by Diego [8] in 1966 before we start. Definition 2.1. [8] A Hilbert algebra is a triplet with the formula X =(X, ·,1), where X is a nonempty set, · is a binary operation, and 1 is a fixed member of X that is true according to the axioms stated below: (1) (∀x,y ∈ X)(x · (y ·x)=1), (2) (∀x,y,z ∈ X)((x · (y ·z)) · ((x ·y) · (x ·z))=1), (3) (∀x,y ∈ X)(x ·y =1,y ·x =1⇒ x = y). In [10], the following conclusion was established. Lemma 2.1. Let X =(X, ·,1) be a Hilbert algebra. Then (1) (∀x ∈ X)(x ·x =1), (2) (∀x ∈ X)(1 ·x = x), (3) (∀x ∈ X)(x ·1=1), (4) (∀x,y,z ∈ X)(x · (y ·z)= y · (x ·z)). In a Hilbert algebra X =(X, ·,1), the binary relation ≤ is defined by (∀x,y ∈ X)(x ≤ y ⇔ x ·y =1), which is a partial order on X with 1 as the largest element. Definition 2.2. [26] A nonempty subset D of a Hilbert algebra X = (X, ·,1) is called a subalgebra of X if x ·y ∈ D for all x,y ∈ D. Int. J. Anal. Appl. (2022), 20:68 3 Definition 2.3. [7] A nonempty subset D of a Hilbert algebra X =(X, ·,1) is called an ideal of X if the following conditions hold: (1) 1∈ D, (2) (∀x,y ∈ X)(y ∈ D ⇒ x ·y ∈ D), (3) (∀x,y1,y2 ∈ X)(y1,y2 ∈ D ⇒ (y1 · (y2 ·x)) ·x ∈ D). A fuzzy set [25] in a nonempty set X is defined to be a function µ : X → [0,1], where [0,1] is the unit closed interval of real numbers. Definition 2.4. [22] A fuzzy set µ in a Hilbert algebra X =(X, ·,1) is said to be a fuzzy subalgebra of X if the following condition holds: (∀x,y ∈ X)(µ(x ·y)≥min{µ(x),µ(y)}). Definition 2.5. [11] A fuzzy set µ in a Hilbert algebra X =(X, ·,1) is said to be a fuzzy ideal of X if the following conditions hold: (1) (∀x ∈ X)(µ(1)≥ µ(x)), (2) (∀x,y ∈ X)(µ(x ·y)≥ µ(y)), (3) (∀x,y1,y2 ∈ X)(µ((y1 · (y2 ·x)) ·x)≥min{µ(y1),µ(y2)}). 3. Fuzzy subalgebras and ideals with thresholds For ε,δ ∈ [0,1] with ε < δ, we introduce the concepts of fuzzy subalgebras and ideals with thresholds ε and δ of Hilbert algebras and investigate some related properties. Definition 3.1. A fuzzy set µ in a Hilbert algebra X = (X, ·,1) is called a fuzzy subalgebra with thresholds ε and δ (FSTδε) of X, where ε,δ ∈ [0,1] with ε < δ if the following condition holds: (∀x,y ∈ X)(max{µ(x ·y),ε}≥min{µ(x),µ(y),δ}). Example 3.1. Let X = {1,x,y,z,0} with the following Cayley table: · 1 x y z 0 1 1 x y z 0 x 1 1 y z 0 y 1 x 1 z z z 1 1 y 1 y 0 1 1 1 1 1 Then X is a Hilbert algebra. We define a fuzzy set µ : A → [0,1] as follows: µ(1)=0.9,µ(x)=0.6,µ(y)=0.8,µ(z)=0.4,µ(0)=0.2. Then µ is a FST0.90.8 of X. 4 Int. J. Anal. Appl. (2022), 20:68 Proposition 3.1. If µ is a FSTδε of a Hilbert algebra X =(X, ·,1), then (∀x ∈ X)(max{µ(1),ε}≥min{µ(x),δ}). Proof. For all x ∈ X, we have max{µ(1),ε}=max{µ(x ·x),ε}≥min{µ(x),µ(x),δ}=min{µ(x),δ}. � Definition 3.2. A fuzzy set µ in a Hilbert algebra X =(X, ·,1) is called a fuzzy ideal with thresholds ε and δ (FITδε) of X if the following conditions hold: (1) (∀x ∈ X)(max{µ(1),ε}≥min{µ(x),δ}), (2) (∀x,y ∈ X)(max{µ(x ·y),ε}≥min{µ(y),δ}), (3) (∀x,y1,y2 ∈ X)(max{µ((y1 · (y2 ·x)) ·x),ε}≥min{µ(y1),µ(y2),δ}). Example 3.2. Let X = {1,x,y,z,0} with the following Cayley table: · 1 x y z 0 1 1 x y z 0 x 1 1 y z 0 y 1 x 1 z z z 1 1 y 1 y 0 1 1 1 1 1 Then X is a Hilbert algebra. We define a fuzzy set µ : A → [0,1] as follows: µ(1)=0.9,µ(x)=0.5,µ(y)=0.4,µ(z)=0.3,µ(0)=0.2. Then µ is a FIT0.90.8 of X. Proposition 3.2. If µ is FITδε of a Hilbert algebra X =(X, ·,1), then (∀x,y ∈ X)(max{µA((y ·x) ·x),ε)≥min{µ(1),µ(y),δ}). Proof. Let x,y ∈ X. Then max{µ((y ·x) ·x),ε}=max{µ((1 · (y ·x)) ·x),ε}≥min{µ(1),µ(y),δ}. � Lemma 3.1. If µ is a FITδε of a Hilbert algebra X =(X, ·,1), then (∀x,y ∈ X)(x ≤ y ⇒min{µ(1),µ(x),δ}≤max{µ(y),ε}). (3.1) Int. J. Anal. Appl. (2022), 20:68 5 Proof. Let x,y ∈ X be such that x ≤ y. Then x ·y =1 and so max{µ(y),ε} = max{µ(1 ·y),ε} = max{µ(((x ·y) · (x ·y)) ·y),ε} ≥ min{µ(x ·y),µ(x),δ} = min{µ(1),µ(x),δ}. � Theorem 3.1. Every FITδε of a Hilbert algebra X =(X, ·,1) is a FST δ ε. Proof. Assume that µ is a FITδε of X. Let x,y ∈ X. Then max{µ(x ·y),ε}≥min{µ(y),δ}≥min{µ(x),µ(y),δ}. Hence, µ is a FSTδε of X. � Theorem 3.2. Let ε,δ ∈ [0,1] with ε < δ. If a fuzzy set µ in a Hilbert algebra X = (X, ·,1) is such that µ(x)≤ ε for all x ∈ X, then it is a FITδε (resp., FST δ ε) of X. Proof. Let x ∈ X. Then max{µ(1),ε} = ε ≥ µ(x) ≥ min{µ(x),δ}. Let x,y ∈ X. Then max{µ(x · y),ε} = ε ≥ µ(y) ≥ min{µ(y),δ}. Let x,y1,y2 ∈ X. Then max{µ((y1 · (y2 · x)) · x),ε} = ε ≥ µ(y1)≥min{µ(y1),µ(y2),δ}. Hence, µ is a FITδε of X. Let x,y ∈ X. Then max{µ(x · y),ε} = ε ≥ µ(x) ≥ min{µ(x),µ(y),δ}. Hence, µ is a FSTδε of X. � By Theorem 3.2, we get the following theorem. Theorem 3.3. Let ε,δ ∈ [0,1] with ε < δ. If a fuzzy set µ in a Hilbert algebra X = (X, ·,1) is such that µ(x)≥ δ for all x ∈ X, then it is a FITδε (resp., FST δ ε) of X. Theorem 3.4. Let µ be a fuzzy set in a Hilbert algebra X = (X, ·,1) and ε,δ ∈ [0,1] with ε < δ. Then µ is a FSTδε of X if and only if for all t ∈ (ε,δ],U(µ,t) := {x ∈ X | µ(x)≥ t} is a subalgebra of X if it is nonempty. Proof. Assume that µ is a FSTδε of X. Let t ∈ (ε,δ] be such that U(µ,t) 6= ∅ and let x,y ∈ U(µ,t). Then µ(x)≥ t, µ(y)≥ t, and δ ≥ t. Thus t is a lower bound of {µ(x),µ(y),δ}. Since µ is a FSTδε of X, we have max{µ(x ·y),ε}≥min{µ(x),µ(y),δ}≥ t > ε. So, max{µ(x ·y),ε}= µ(x ·y). Since max{µ(x ·y),ε}≥ t, we get µ(x ·y)≥ t. Hence, x ·y ∈ U(µ,t). Therefore, U(µ,t) is a subalgebra of X. Conversely, assume that for all t ∈ (ε,δ], U(µ,t) is a subalgebra of X if it is nonempty. Let x,y ∈ X. Then µ(x),µ(y) ∈ [0,1]. Choose t = min{µ(x),µ(y)}. Then µ(x) ≥ t and µ(y) ≥ t. Thus x,y ∈ U(µ,t) 6= ∅. By the assumption, we have U(µ,t) is a subalgebra of X. So, x ·y ∈ U(µ,t), that is, µ(x · y) ≥ t = min{µ(x),µ(y)}. Thus max{µ(x · y),ε} ≥ µ(x · y) ≥ min{µ(x),µ(y)} ≥ min{µ(x),µ(y),δ}. Hence, µ is a FSTδε of X. � 6 Int. J. Anal. Appl. (2022), 20:68 The following theorem can be proved similarly to Theorem 3.4. Theorem 3.5. Let µ be a fuzzy set in a Hilbert algebra X = (X, ·,1) and ε,δ ∈ [0,1] with ε < δ. Then µ is a FITδε of X if and only if for all t ∈ (ε,δ],U(µ,t) is an ideal of X if it is nonempty. Definition 3.3. Let f be a function from a nonempty set X to a nonempty set Y . If µ is a fuzzy set in X, then the fuzzy set β in Y defined by β(y)=   sup t∈f−1(y) {µ(t)} if f−1(y) 6= ∅ 0 otherwise is said to be the image of µ under f . Similarly, if α is a fuzzy set in Y , then the fuzzy set f−1(α)= α◦f in X (i.e., the fuzzy set is defined by f−1(α)(x) = α(f (x)) for all x ∈ X) is called the pre-image of α under f . Definition 3.4. A fuzzy set µ in a Hilbert algebra X =(X, ·,1) is said to have the sup property if for any nonempty subset T of X, there exists t0 ∈ T such that µ(t0)= supt∈T µ(t). Definition 3.5. Let X and Y be any two nonempty sets and let f : X → Y be any function. A fuzzy set µ in X is said to be f -invariant if (∀x,y ∈ X)(f (x)= f (y)⇒ µ(x)= µ(y)). Lemma 3.2. Let (X, ·,1X) and (Y,?,1Y ) be Hilbert algebras and let f : X → Y be a surjective homomorphism. Let µ be an f -invariant fuzzy set in X with sup property. For any x,y ∈ Y , there exist x0 ∈ f−1(x) and y0 ∈ f−1(y) such that β(x)= µ(x0),β(y)= µ(y0), and β(x ? y)= µ(x0 ·y0). Proof. Let x,y ∈ Y . Since f is surjective, we have f−1(x), f−1(y), and f−1(x·y)are nonempty subsets of X. Since µ has sup property, there exist elements x0 ∈ f−1(x),y0 ∈ f−1(y), and z0 ∈ f−1(x ? y) such that β(x)= sups∈f−1(x){µ(s)}= µ(x0), β(y)= sups∈f−1(y){µ(s)}= µ(y0), β(x ? y)= sups∈f−1(x?y){µ(s)}= µ(z0). Since f (z0)= x ? y = f (x0)? f (y0)= f (x0 ·y0), and µ is f -invariant, it follows that β(x ? y)= µ(z0)= µ(x0 ·y0). � Int. J. Anal. Appl. (2022), 20:68 7 Theorem 3.6. Let (X, ·,1X) and (Y,?,1Y ) be Hilbert algebras and let f : X → Y be a surjective homomorphism. Then the following statements hold: (1) if µ is an f -invariant FSTδε of X with sup property, then β is a FST δ ε of Y , (2) if µ is an f -invariant FITδε of X with sup property, then β is a FIT δ ε of Y . Proof. (1) Assume that µ is an f -invariant FSTδε of X with sup property. Let a,b ∈ Y . Then by Lemma 3.2, there exist a0 ∈ f−1(a) and b0 ∈ f−1(b) such that β(a) = µ(a0), β(b) = µ(b0), and β(a ? b) = µ(a0 · b0). Thus max{β(a ? b),ε} = max{µ(a0 · b0),ε} ≥ min{µ(a0),µ(b0),δ} = min{β(a),β(b),δ}. Hence, β is a FSTδε of Y . (2) Assume that µ is an f -invariant FITδε of X with sup property. Since f (1X) = 1Y , we have f−1(1Y ) 6= ∅. By Lemma 3.2, there exists x1 ∈ f−1(1Y ) such that µ(x1) = β(1Y ). Thus f (x1) = 1Y = f (1X). Since µ is f -invariant, we have µ(x1) = µ(1X). So, β(1Y ) = µ(1X). Let y ∈ Y . Since f is surjective, we have f−1(y) 6= ∅. By Lemma 3.2, there exists x ∈ f−1(y) such that µ(x)= β(y). Thus max{β(1Y ),ε} = max{µ(1X),ε} ≥ min{µ(x),δ} = min{β(y),δ}. Now, let a,b ∈ Y . Then by Lemma 3.2, there exist a0 ∈ f−1(a) and b0 ∈ f−1(b) such that µ(a0) = β(a), µ(b0) = β(b), and µ(a0 ·b0)= β(a ? b). Thus max{β(a ? b),ε}=max{µ(a0 ·b0),ε}≥min{µ(b0),δ}=min{β(b),δ}. Next, let a,b,y ∈ Y . By Lemma 3.2, there exist a0 ∈ f−1(a), b0 ∈ f−1(b) and x ∈ f−1(y) such that µ(a0) = β(a), µ(b0) = β(b), µ(y) = β(x) and µ((a0 · (b0 · x)) · x) = β((a ? (b ? x)) ? x). Thus max{β((a ? (b ? x)) ? x),ε} = max{µ((a0 · (b0 · x)) · x) · x),ε} ≥ min{µ(a0),µ(b0),δ} = min{β(a),β(b),δ}. Hence, β is a FITδε of Y . � Theorem 3.7. Let (X, ·,1X) and (Y,?,1Y ) be Hilbert algebras and let f : X → Y be a homomorphism. Then the following statements hold: (1) if α is a FSTδε of Y , then f −1(α) is a FSTδε of X, (2) if α is a FITδε of Y , then f −1(α) is a FITδε of X. Proof. (1) Assume that α is a FSTδε of Y . Let x,y ∈ X. Then max{f−1(α)(x ·y),ε} = max{(α◦ f )(x ·y),ε} = max{(α(f (x ·y)),ε} = max{(α(f (x)? f (y)),ε} ≥ min{α(f (x)),α(f (y)),δ} = min{(α◦ f )(x),(α◦ f )(y),δ} = min{f−1(α)(x), f−1(α)(y),δ}. Hence, f−1(α) is a FSTδε of X. 8 Int. J. Anal. Appl. (2022), 20:68 (2) Assume that α is a FITδε of Y . Let x ∈ X. Then max{f−1(α)(1X),ε} = max{(α◦ f )(1X),ε} = max{(α(f (1X)),ε} = max{α(1Y ),ε} ≥ min{α(f (x)),δ} = min{(α◦ f )(x),δ} = min{f−1(α)(x),δ}. Let x,y ∈ X. Then max{f−1(α)(x ·y),ε} = max{(α◦ f )(x ·y),ε} = max{α(f (x ·y)),ε} ≥ min{α(f (y)),ε} = min{(α◦ f )(y),δ} = min{f−1(α)(y),δ}. Let x,y1,y2 ∈ X. Then max{f−1(α)((y1 · (y2 ·x)) ·x),ε} = max{(α◦ f )((y1 · (y2 ·x)) ·x),ε} = max{α(f ((y1 · (y2 ·x)) ·x)),ε} = max{α(f (y1)? (f (y2)? f (x))? f (x)),ε} ≥ min{α(f (y1)),α(f (y2)),δ} = min{(α◦ f )(y1),(α◦ f )(y2),δ} = min{f−1(α)(y1), f−1(α)(y2),δ}. Hence, f−1(α) is a FITδε of X. � Definition 3.6. Let f be a function from a nonempty set X to a nonempty set Y . If µ is a fuzzy set in X, then the fuzzy set η in Y defined by η(y)=   inft∈f−1(y){µ(t)} if f −1(y) 6= ∅ 1 otherwise is said to be the image of µ under f . Definition 3.7. A fuzzy set µ in a Hilbert algebra X =(X, ·,1) is said to have the inf property if for any nonempty subset T of X, there exists t0 ∈ T such that µ(t0)= inft∈T µ(t). The following lemma can be proved similarly to Lemma 3.2. Lemma 3.3. Let (X, ·,1X) and (Y,?,1Y ) be Hilbert algebras and let f : X → Y be a surjective homomorphism. Let µ be an f -invariant fuzzy set in X with inf property. For any x,y ∈ Y , there exist x0 ∈ f−1(x) and y0 ∈ f−1(y) such that η(x)= µ(x0),η(y)= µ(y0), and η(x ? y)= µ(x0 ·y0). Int. J. Anal. Appl. (2022), 20:68 9 The following theorem can be proved similarly to Theorem 3.6. Theorem 3.8. Let (X, ·,1X) and (Y,?,1Y ) be Hilbert algebras and let f : X → Y be a surjective homomorphism. Then the following statements hold: (1) if µ is an f -invariant FSTδε of X with inf property, then η is a FST δ ε of Y , (2) if µ is an f -invariant FITδε of X with inf property, then η is a FIT δ ε of Y . 4. Conclusion In the present paper, we have introduced the concepts of fuzzy subalgebras and ideals with thresholds of Hilbert algebras. The relationship between fuzzy subalgebras (ideals) and their level subsets is described. Moreover, the homomorphic images and inverse images of fuzzy subalgebras and ideals with thresholds in Hilbert algebras are also studied and some related properties are investigated. Acknowledgment This research project was supported by the Thailand Science Research and Innovation Fund and the University of Phayao (Grant No. FF66-RIM032). 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Math. 29 (2005), 813-826. https://doi.org/10.1007/s10489-018-1152-z https://doi.org/10.22111/ijfs.2018.4159 https://eudml.org/doc/213228 https://eudml.org/doc/213228 https://cir.nii.ac.jp/crid/1571417124616097792 https://cir.nii.ac.jp/crid/1571417124616097792 https://doi.org/10.4134/CKMS.2007.22.2.173 https://doi.org/10.4134/BKMS.2005.42.4.703 https://doi.org/10.1088/1742-6596/1879/2/022108 https://doi.org/10.1155/2010/838392 https://doi.org/10.1134/s1995080209010053 https://doi.org/10.7508/ijmsi.2009.02.002 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x 1. Introduction 2. Preliminaries 3. Fuzzy subalgebras and ideals with thresholds 4. Conclusion Acknowledgment References