EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6529 ISSN 1307-5543 – ejpam.com Published by New York Business Global Weak Filters of Sheffer Stroke Hilbert Algebras Based on the Intuitionistic Fuzzy Set Sun Shin Ahn1,∗, Young Joo Seo2, Young Bae Jun3 1 Department of Mathematics Education, Dongguk University, Seoul 04620, Korea 2 Research Institute for Natural Sciences, Department of Mathematics, Hanyang University, Seoul 04763, Korea 3 Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea Abstract. Using the concept of intuitionistic fuzzy points, the weak filter in Sheffer stroke Hilbert algebras is addressed. The notion of intuitionistic fuzzy weak filters in Sheffer stroke Hilbert alge- bras is introduced, and their properties are investigated. Conditions under which the intuitionistic fuzzy set becomes an intuitionistic fuzzy weak filter are examined. Characterizations of intu- itionistic fuzzy weak filters are considered, and conditions under which the intuitionistic fuzzy set becomes an intuitionistic fuzzy weak filter are discussed. The (0, 1)-set for the intuitionistic fuzzy set is established, and the phases in which it can be a weak filter are explored. Conditions for an intuitionistic level set and an intuitionistic q-set to be weak filters are provided. 2020 Mathematics Subject Classifications: 03B05, 03G25, 06F35, 08A72 Key Words and Phrases: Weak filter, intuitionistic fuzzy point, intuitionistic level set, intu- itionistic q-set, (0, 1)-set, intuitionistic fuzzy weak filter 1. Introduction The Sheffer operation (or, Sheffer stroke) is a logical operation in Boolean algebra that produces a false result only when both of its inputs are true. It is also known as the NAND operation, and is often symbolized as “|” or sometimes as “↑”. The Sheffer stroke has been applied to several algebraic structures, for example, Boolean algebra, BCK-algebra, MV-algebra, BL-algebra, and ortholattices, etc., and it is also being dealt with in the fuzzy environment (see [1–11]). In 2021, Oner et al. [5] applied the Sheffer stroke to Hilbert algebras. They introduced Sheffer stroke Hilbert algebra and investigated several properties. In [4], Oner et al. introduced the notion of deductive system and filter of Sheffer stroke Hilbert algebras, and dealt with their fuzzification. Oner et al. [5] ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6529 Email addresses: sunshine@dongguk.edu (S. S. Ahn), bejesus@hanyang.ac.kr (Y. J. Seo) skywine@gmail.com (Y. B. Jun) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 2 of 16 also introduced the concept of ideal and examined its properties in Sheffer stroke Hilbert algebras. The intuitionistic fuzzy set, which is a generalization of fuzzy sets, is introduced by K. Athanasov in 1986, and it is a useful tool for better modeling uncertainties and ambiguity. The fuzzy set only considers the degree of membership of the element, while the intuitionistic fuzzy set deals with membership and non-membership simultaneously, along with the degree of hesitation (or uncertainty) about the element. An intuitionistic fuzzy point (see [12]) is an extension of the classical concept of a point in set theory, which is adapted to the framework of the intuitionistic fuzzy set and plays an important role in intuitionistic fuzzy sets. Jun et al. [13] introduced the concept of weak filters that have weakened the filter conditions in the Sheffer stroke Hilbert algebra and investigated several properties. They presented how to make weak filters using ideals, and examined the shape of the weak filter in the Cartesian product of Sheffer stroke Hilbert algebras. The purpose of this paper is to study weak filters in Sheffer stroke Hilbert algebras using the concept of intuitionistic fuzzy points. We introduce the notion of intuitionistic fuzzy weak filters in Sheffer stroke Hilbert algebras, and investigates their properties. We examine the conditions under which the intuitionistic fuzzy set becomes an intuitionistic fuzzy weak filter. We discuss the characterization of intuitionistic fuzzy weak filters, and consider the conditions under which the intuitionistic fuzzy set becomes an intuitionistic fuzzy weak filter. We build a (0, 1)-set for the intuitionistic fuzzy set, and discuss the phases in which it can be a weak filter. It provides conditions for an intuitionistic level set and an intuitionistic q-set to be weak filters. 2. Preliminaries Definition 1 ([14]). Let B := (B, |) be a groupoid. Then the operation “|” is said to be Sheffer stroke or Sheffer operation if it satisfies: (s1) (∀a, b ∈ B) (a|b = b|a), (s2) (∀a, b ∈ B) ((a|a)|(a|b) = a), (s3) (∀a, b, c ∈ B) (a|((b|c)|(b|c)) = ((a|b)|(a|b))|c), (s4) (∀a, b, c ∈ B) ((a|((a|a)|(b|b)))|(a|((a|a)|(b|b))) = a). Let X := (X, |) be a groupoid. For every element a ∈ X, consider the following mapping: ða : X → X, b 7→ a|(b|b). Definition 2 ([5]). A Sheffer stroke Hilbert algebra is a groupoid X := (X, |) with a Sheffer stroke “|” that satisfies: (sH1) (a|(ðb(c)|ðb(c)))|((ða(b)|(ða(c)|ða(c)))|(ða(b)|(ða(c)|ða(c)))) = ða(a), (sH2) ða(b) = ðb(a) = ða(a) ⇒ a = b S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 3 of 16 for all a, b, c ∈ X. Recall that every Sheffer stroke Hilbert algebra X := (X, |) satisfies ða(a) = ðb(b) for all a, b ∈ X. It means that X := (X, |) has an algebraic constant which is denoted by “1” (see [5]). Let X := (X, |) be a Sheffer stroke Hilbert algebra. Then the order relation “⪯” on X is defined as follows: (∀a, b ∈ X)(a ⪯ b ⇔ ða(b) = 1). (1) We observe that the relation “ ⪯ ” is a partial order in a Sheffer stroke Hilbert algebra X := (X, |) (see [5]). Proposition 1 ([5]). Every Sheffer stroke Hilbert algebra X := (X, |) satisfies: ða(a) = 1, ða(1) = 1, ð1(a) = a, (2) a ⪯ ðb(a), (3) ða(b)|(b|b) = ðb(a)|(a|a), (4) (ða(b)|(b|b))|(b|b) = ða(b), (5) a|(ðb(c)|ðb(c)) = b|(ða(c)|ða(c)), (6) a ⪯ b ⇒ ðc(a) ⪯ ðc(b), ðb(c) ⪯ ða(c), (7) a|(ðb(c)|ðb(c)) = (ða(b)|(ða(c)|ða(c)), (8) for all a, b, c ∈ X. By (2), we know that the algebraic constant 1 is the greatest element in X := (X, |) with respect to the order ⪯. Proposition 2. Let X := (X, |) be a Sheffer stroke Hilbert algebra with the smallest element 0. Then 0|0 = 1, 1|1 = 0, (9) ð1(0) = 0, ð0(0) = 1. (10) Definition 3 ([4]). Let X := (X, |) be a Sheffer stroke Hilbert algebra. A subset F of X is called a filter of X := (X, |) if it satisfies: 1 ∈ F, (11) (∀a, b ∈ X)(b ∈ F ⇒ ða(b) ∈ F ), (12) (∀a, b, c ∈ X)(b, c ∈ F ⇒ (a|(b|c))|(b|c) ∈ F ). (13) Let X := (X, |) be a Sheffer stroke Hilbert algebra. If a subset F of X satisfies (11) and (12), we say that F is a weak filter of X := (X, |) (see [13]). S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 4 of 16 Let X be a set. An intuitionistic fuzzy set B∗ in X (see [15]) is an object having the form B∗ := {⟨a, fB(a), gB(a)⟩ | fB(a) + gB(a) ≤ 1, a ∈ X}, which is simply denoted by B∗ := (X; fB, gB) where fB and gB are fuzzy sets in X, The intuitionistic fuzzy set B∗ := (X; fB, gB) in X can be represented as follows: B∗ := (X; fB, gB) : X → [0, 1]× [0, 1], a 7→ (fB(a), gB(a)) such that fB(a) + gB(a) ≤ 1. An intuitionistic fuzzy set B∗ := (X; fB, gB) in a set X of the form B∗ := (X; fB, gB) : X → [0, 1]× [0, 1], b 7→ { (s, t) ∈ (0, 1]× [0, 1) if b = a, (0, 1) if b ̸= a, is said to be an intuitionistic fuzzy point with support a and value (s, t) such that s+t ≤ 1, and is denoted by a(s,t). Given an intuitionistic fuzzy set B∗ := (X; fB, gB) and intuitionistic fuzzy point a(s,t) in X, we say a(s,t) ∈ B∗ if fB(a) ≥ s and gB(a) ≤ t. (14) a(s,t) q B∗ if fB(a) + s > 1 and gB(a) + t < 1. (15) a(s,t) ∈∨q B∗ if a(s,t) ∈ B∗ or a(s,t) q B∗. (16) Given (s, t) ∈ (0, 1] × [0, 1) and an intuitionistic fuzzy set B∗ := (X; fB, gB) in X, consider the following sets: (fB, s)∈ := {a ∈ X | fB(a) ≥ s}, (gB, t)∈ := {a ∈ X | gB(a) ≤ t}, (fB, s)q := {a ∈ X | fB(a) + s > 1}, (gB, t)q := {a ∈ X | gB(a) + t < 1}. (fB, s)∈∨q := {a ∈ X | fB(a) ≥ s or fB(a) + s > 1}. (gB, t)∈∨q := {a ∈ X | gB(a) ≤ t or gB(a) + t < 1}. Also, we consider the sets below. (B∗, (s, t))∈ := (fB, s)∈ ∩ (gB, t)∈, (B∗, (s, t))q := (fB, s)q ∩ (gB, t)q, (B∗, (s, t))∈∨q := (fB, s)∈∨q ∩ (gB, t)∈∨q, which are called the intuitionistic level set, intuitionistic q-set and intuitionistic ∈∨q-set of B∗ := (X; fB, gB), respectively. S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 5 of 16 Definition 4 ([16]). An intuitionistic fuzzy set B∗ := (X; fB, gB) in a Sheffer stroke Hilbert algebra X := (X, |) is called an intuitionistic fuzzy filter of X := (X, |) if it satisfies: (∀x ∈ X)(fB(1) ≥ fB(x), gB(1) ≤ gB(x)), (17) (∀x, y ∈ X)(fB(ðx(y)) ≥ fB(y), gB(ðx(y)) ≤ gB(y)), (18) (∀x, y, z ∈ X) ( fB((x|(y|z))|(y|z)) ≥ min{fB(y), fB(z)} gB((x|(y|z))|(y|z)) ≤ max{gB(y), gB(z)} ) . (19) 3. Intuitionistic fuzzy weak filters In what follows, X := (X, |) stands for a Sheffer stroke Hilbert algebra, unless otherwise stated. Definition 5. An intuitionistic fuzzy set B∗ := (X; fB, gB) in X is called an intuitionistic fuzzy weak filter of X := (X, |) if it satisfies: (∀x ∈ X)(∀(s, t) ∈ (0, 1]× [0, 1)) ( x(s,t) ∈ B∗ ⇒ 1(s,t) ∈ B∗) , (20) (∀x, y ∈ X)(∀(s, t) ∈ (0, 1]× [0, 1)) ( y(s,t) ∈ B∗ ⇒ ðx(y)(s,t) ∈ B∗) . (21) Example 1. Let X = {c0, c1, c2, c3, c4, c5, c6, c7} be a set with the following Hasse diagram: rc3 rc2 r c0 rc4 rc5 rc6 rc7 rc1 �� ��� HH HHH ��� �� HHH HH ��� ��� ��� � HHH HH H HHH H Define a Sheffer stroke “|” on X by Table 1 Then X := (X, |) is a Sheffer stroke Hilbert algebra where the algebraic constant is c1 (see [5]). Let B∗ := (X; fB, gB) be an intuitionistic fuzzy set in X where fB : X → [0, 1], x 7→  0.66 if x = c1, 0.52 if x ∈ {c5, c6}, 0.44 if x = c7, 0.35 otherwise. and gB : X → [0, 1], x 7→  0.27 if x = c1, 0.32 if x ∈ {c6, c7}, 0.34 otherwise. It is routine to verify that B∗ := (X; fB, gB) is an intuitionistic fuzzy weak filter of X := (X, |). S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 6 of 16 Table 1: Cayley table for the Sheffer stroke “|” | c0 c2 c3 c4 c5 c6 c7 c1 c0 c1 c1 c1 c1 c1 c1 c1 c1 c2 c1 c7 c1 c1 c7 c7 c1 c7 c3 c1 c1 c6 c1 c6 c1 c6 c6 c4 c1 c1 c1 c5 c1 c5 c5 c5 c5 c1 c7 c6 c1 c4 c7 c6 c4 c6 c1 c7 c1 c5 c7 c3 c5 c3 c7 c1 c1 c6 c5 c6 c5 c2 c2 c1 c1 c7 c6 c5 c4 c3 c2 c0 It is obvious that every intuitionistic fuzzy filter is an intuitionistic fuzzy weak filter, but the converse may not be true. In fact, the intuitionistic fuzzy weak filter B∗ := (X; fB, gB) in Example 1 is not an intuitionistic fuzzy filter of X := (X, |) because of gB((c4|(c6|c7))|(c6|c7)) = gB(c4) = 0.34 ≰ 0.32 = max{gB(c6), gB(c7)}. Theorem 1. An intuitionistic fuzzy set B∗ := (X; fB, gB) in X is an intuitionistic fuzzy weak filter of X := (X, |) if and only if it satisfies: (∀x ∈ X)(fB(1) ≥ fB(x), gB(1) ≤ gB(x)), (22) (∀x, y ∈ X)(fB(ðx(y)) ≥ fB(y), gB(ðx(y)) ≤ gB(y)). (23) Proof. Assume that B∗ := (X; fB, gB) in X is an intuitionistic fuzzy weak filter of X := (X, |). Since x(s,t) ∈ B∗ for s = fB(x) and t = gB(x), we have 1(s,t) ∈ B∗ by (20). Hence fB(1) ≥ s = fB(x) and gB(1) ≤ t = gB(x), i.e., (22) is valid. Since y(fB(y),gB(y)) ∈ B∗ for all y ∈ X, it follows from (21) that ðx(y)(fB(y),gB(y)) ∈ B∗ for all x ∈ X. Hence fB(ðx(y)) ≥ fB(y) and gB(ðx(y)) ≤ gB(y), i.e., (23) is valid. Conversely, let B∗ := (X; fB, gB) be an intuitionistic fuzzy set in X that satisfies (22) and (23). Let x ∈ X and (s, t) ∈ (0, 1] × [0, 1) be such that x(s,t) ∈ B∗. Then fB(x) ≥ s and gB(x) ≤ t. If 1(s,t) ∈B∗, then fB(1) < s ≤ fB(x) or gB(1) > t ≥ gB(x) which is a contradiction. Thus 1(s,t) ∈ B∗. Let x, y ∈ X and (s, t) ∈ (0, 1] × [0, 1) be such that y(s,t) ∈ B∗. Then fB(y) ≥ s and gB(y) ≤ t. If ðx(y)(s,t) ∈B∗, then fB(ðx(y)) < s ≤ fB(y) or gB(ðx(y)) > t ≥ gB(y) which is a contradiction. Thus ðx(y)(s,t) ∈ B∗. Therefore B∗ := (X; fB, gB) is an intuitionistic fuzzy weak filter of X := (X, |). Theorem 2. An intuitionistic fuzzy set B∗ := (X; fB, gB) in X is an intuitionistic fuzzy weak filter of X := (X, |) if and only if the nonempty sets (fB, s)∈ and (gB, t)∈ are weak filters of X := (X, |) for all (s, t) ∈ (0, 1]× [0, 1). Proof. Assume that B∗ := (X; fB, gB) is an intuitionistic fuzzy weak filter of X := (X, |) and let (s, t) ∈ (0, 1] × [0, 1) be such that (fB, s)∈ ̸= ∅ ̸= (gB, t)∈, say x ∈ (fB, s)∈ and S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 7 of 16 a ∈ (gB, t)∈. Then fB(1) ≥ fB(x) ≥ s and gB(1) ≤ gB(a) ≤ t by (22). Hence 1 ∈ (fB, s)∈ and 1 ∈ (gB, t)∈. Let y ∈ (fB, s)∈ and b ∈ (gB, t)∈. Then fB(ðx(y)) ≥ fB(y) ≥ s and ða(b) ≤ gB(b) ≤ t for all x, a ∈ X by (23). It follows that ðx(y) ∈ (fB, s)∈ and ða(b) ∈ (gB, t)∈. Therefore (fB, s)∈ and (gB, t)∈ are weak filters of X := (X, |). Conversely, suppose that the nonempty sets (fB, s)∈ and (gB, t)∈ are weak filters of X := (X, |) for all (s, t) ∈ (0, 1] × [0, 1). If (22) is not valid, then fB(1) < fB(a) or gB(1) > gB(b) for some a, b ∈ X. It follows that 1 /∈ (fB, s)∈ or 1 /∈ (gB, t)∈ where s := fB(a) and t := gB(b). This is a contradiction, and so (22) is valid. Suppose that (23) is not valid. Then fB(ðx(y)) < fB(y) or gB(ða(b)) > gB(b). If we take s := fB(y) and t := gB(b), then ðx(y) /∈ (fB, s)∈ or ða(b) /∈ (gB, t)∈, a contradiction. Thus (23) is valid. Consequently, B∗ := (X; fB, gB) is an intuitionistic fuzzy weak filter of X := (X, |) by Theorem 1. Corollary 1. If B∗ := (X; fB, gB) is an intuitionistic fuzzy weak filter of X := (X, |), then its nonempty intuitionistic level set (B∗, (s, t))∈ is a weak filter of X := (X, |) for all (s, t) ∈ (0, 1]× [0, 1). We examine the conditions under which the intuitionistic fuzzy set becomes an intu- itionistic fuzzy weak filter. Theorem 3. If an intuitionistic fuzzy set B∗ := (X; fB, gB) in X satisfies (20) and x(s1,t1) ∈ B∗, ðx(y)(s2,t2) ∈ B∗ ⇒ y(min{s1,s2},max{t1,t2}) ∈ B∗ (24) for all x, y ∈ X and (s1, t1), (s2, t2) ∈ (0, 1]× [0, 1), then B∗ := (X; fB, gB) is an intuition- istic fuzzy weak filter of X := (X, |). Proof. Let B∗ := (X; fB, gB) be an intuitionistic fuzzy set in X that satisfies (20) and (24). We first show that the condition (24) is equivalent to the following facts. (∀x, y ∈ X) ( fB(y) ≥ min{fB(x), fB(ðx(y))} gB(y) ≤ max{gB(x), gB(ðx(y))} ) . (25) Suppose that B∗ := (X; fB, gB) satisfies (24) and let x, y ∈ X. If we take (s1, t1) = (fB(x), gB(x)) and (s2, t2) = (fB(ðx(y)), gB(ðx(y))), then x(s1,t1) ∈ B∗ and ðx(y)(s2,t2) ∈ B∗. It follows from (24) that y(min{s1,s2},max{t1,t2}) ∈ B∗. Hence fB(y) ≥ min{s1, s2} = min{fB(x), fB(ðx(y))} and gB(y) ≤ max{t1, t2} = max{gB(x), gB(ðx(y))}. Now, assume that (25) is valid and let x(s1,t1) ∈ B∗ and ðx(y)(s2,t2) ∈ B∗ for all (s1, t1), (s2, t2) ∈ (0, 1] × [0, 1). Then fB(x) ≥ s1, gB(x) ≤ t1, fB(ðx(y)) ≥ s2, and gB(ðx(y)) ≤ t2. Using (25), we have fB(y) ≥ min{fB(x), fB(ðx(y))} ≥ min{s1, s2} S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 8 of 16 and gB(y) ≤ max{gB(x), gB(ðx(y))} ≤ max{t1, t2}. Hence y(min{s1,s2},max{t1,t2}) ∈ B∗, and therefore (24) is valid. The combination of (1) and (3) induces y|(ðx(y)|ðx(y)) = 1 for all x, y ∈ X. It follows from (22) and (25) that fB(ðx(y)) ≥ min{fB(y), fB(y|(ðx(y)|ðx(y)))} = min{fB(y), fB(1)} = fB(y) and gB(ðx(y)) ≤ max{gB(y), gB(y|(ðx(y)|ðx(y)))} = max{gB(y), gB(1)} = gB(y) for all x, y ∈ X. Therefore B∗ := (X; fB, gB) is an intuitionistic fuzzy weak filter of X := (X, |) by Theorem 1. In the following theorem, we use weak filters to form intuitionistic fuzzy weak filters. Theorem 4. For every nonempty subset F of X, consider an intuitionistic fuzzy set B∗ F := (X; fF B , gFB) in X which is given by B∗ F := (X; fF B , gFB) : X → [0, 1]× [0, 1], x 7→ { (s1, t1) if x ∈ F , (s2, t2) otherwise where (s1, t1), (s2, t2) ∈ (0, 1] × [0, 1) with s1 > s2 and t1 < t2. Then B∗ F := (X; fF B , gFB) is an intuitionistic fuzzy weak filter of X := (X, |) if and only if F is a weak filter of X := (X, |). Proof. Let (s1, t1), (s2, t2) ∈ (0, 1]×[0, 1) be such that s1 > s2 and t1 < t2. Suppose that B∗ F := (X; fF B , gFB) is an intuitionistic fuzzy weak filter of X := (X, |). Since fB(1) = s1 and gB(1) = t1 by (22), we have 1 ∈ F . Let x ∈ X and y ∈ F . Using (23), we have fB(ðx(y)) ≥ fB(y) = s1 and gB(ðx(y)) ≤ gB(y) = t1. Thus fB(ðx(y)) = s1 and gB(ðx(y)) = t1 which shows that ðx(y) ∈ F . Hence F is a weak filter of X := (X, |). Conversely, assume that F is a weak filter of X := (X, |). Then 1 ∈ F, and so fB(1) = s1 ≥ fB(x) and gB(1) = t1 ≤ gB(x) for all x ∈ X. Let x, y ∈ X. If y /∈ F , then fB(y) = s2 ≤ fB(ðx(y)) and gB(y) = t2 ≥ gB(ðx(y)). If y ∈ F , then ðx(y) ∈ F , and thus fB(ðx(y)) = s1 = fB(y) and gB(ðx(y)) = t1 = gB(y). It follows from Theorem 1 that B∗ F := (X; fF B , gFB) is an intuitionistic fuzzy weak filter of X := (X, |). The example below illustrates Theorem 4. Example 2. Consider the Sheffer stroke Hilbert algebra X := (X, |) in Example 1. We can observe that F := {c1, c5, c6, c7} is a weak filter of X := (X, |). Hence the intuitionistic fuzzy set B∗ F := (X; fF B , gFB) given by B∗ F := (X; fF B , gFB) : X → [0, 1]× [0, 1], x 7→ { (0.7n , 0.52n ) if x ∈ F , (0.72n , 0.5 n ) otherwise, where n is a natural number, is an intuitionistic fuzzy weak filter of X := (X, |). S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 9 of 16 Corollary 2. For every a ∈ X and (s1, t1), (s2, t3) ∈ (0, 1]×[0, 1) with s1 > s2 and t1 < t2, consider an intuitionistic fuzzy set B∗ a⃗ := (X; f a⃗ B, g a⃗ B) in X which is defined by B∗ a⃗ := (X; f a⃗ B, g a⃗ B) : X → [0, 1]× [0, 1], x 7→ { (s1, t1) if x ∈ a⃗, (s2, t2) otherwise is an intuitionistic fuzzy weak filter of X := (X, |) where a⃗ := {x ∈ X | ða(x) = 1}. Proof. Since a⃗ is a weak filter of X := (X, |) for all a ∈ X (see [13]), it follows from Theorem 4 that B∗ a⃗ := (X; f a⃗ B, g a⃗ B) is an intuitionistic fuzzy weak filter of X := (X, |) for all a ∈ X. Corollary 3. For every b ∈ X and (s1, t1), (s2, t3) ∈ (0, 1]×[0, 1) with s1 > s2 and t1 < t2, consider an intuitionistic fuzzy set B∗ Xb := (X; fXb B , gX b B ) in X which is defined by B∗ Xb := (X; fXb B , gX b B ) : X → [0, 1]× [0, 1], x 7→ { (s1, t1) if x ∈ Xb, (s2, t2) otherwise is an intuitionistic fuzzy weak filter of X := (X, |) where Xb := {x ∈ X | ðb(x) = x}. Proof. Note that Xb is a weak filter of X := (X, |) for all b ∈ X (see [13]). Hence B∗ Xb := (X; fXb B , gX b B ) is an intuitionistic fuzzy weak filter of X := (X, |) for all b ∈ X by Theorem 4. Corollary 4. Let F be a subset of X. For every b ∈ X and (s1, t1), (s2, t2) ∈ (0, 1]× [0, 1) with s1 > s2 and t1 < t2, let B∗ Fb := (X; fFb B , gFb B ) be an intuitionistic fuzzy set in X which is defined by B∗ Fb := (X; fFb B , gFb B ) : X → [0, 1]× [0, 1], x 7→ { (s1, t1) if x ∈ Fb, (s2, t2) otherwise where Fb := {z ∈ X | ðb(x) = z, x ∈ F}. If F is a weak filter of X := (X, |), then B∗ Fb := (X; fFb B , gFb B ) is an intuitionistic fuzzy weak filter of X := (X, |). Proof. If F is a weak filter of X := (X, |), then Fb is a weak filter of X := (X, |) (see [13]). Thus B∗ Fb := (X; fFb B , gFb B ) is an intuitionistic fuzzy weak filter of X := (X, |) by Theorem 4. Corollary 5. Let G be a subset of X. For every (s1, t1), (s2, t2) ∈ (0, 1] × [0, 1) with s1 > s2 and t1 < t2, let B∗ G := (X; fG B , gGB) in X which is given by B∗ G := (X; fG B , gGB) : X → [0, 1]× [0, 1], x 7→ { (s1, t1) if x ∈ G∗, (s2, t2) otherwise where G∗ := {x ∈ X | (∀y ∈ G)(ðx(y) = 1 ⇒ y = 1)}. If G is a weak filter of X := (X, |), then B∗ G := (X; fG B , gGB) is an intuitionistic fuzzy weak filter of X := (X, |). S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 10 of 16 Proof. If G is a weak filter of X := (X, |), then G∗ is a weak filter of X := (X, |) (see [13]). Hence B∗ G := (X; fG B , gGB) is an intuitionistic fuzzy weak filter of X := (X, |) by Theorem 4. Theorem 5. Let {Fi | i ∈ Γ ⊆ (0, 1]} and {Gi | i ∈ Λ ⊆ [0, 1)} be collections of weak filters of X := (X, |) that satisfy ⋃ i∈Γ Fi = X = ⋃ i∈Λ Gi, and Fj ⊂ Fi ⇔ i < j ⇔ Gj ⊂ Gi for all i, j ∈ Γ ∪ Λ. If we define an intuitionistic fuzzy set B∗ := (X; fB, gB) in X by fB(x) = sup{i ∈ Γ | x ∈ Fi} and gB(x) = inf{i ∈ Λ | x ∈ Gi} for all x ∈ X, then it is an intuitionistic fuzzy weak filter of X := (X, |). Proof. It is sufficient to show that (fB, s)∈ and (gB, t)∈ are weak filters of X := (X, |) for all (s, t) ∈ (0, f(1)] × [gB(1), 1) according to Theorem 2. If s = sup{sa ∈ Γ | sa < s}, then x ∈ (fB, s)∈ ⇔ (∀sa < s)(x ∈ Fsa) ⇔ x ∈ ⋂ sa 0 such that (s − εa, s) ∩ Γ = ∅. Hence x /∈ Fsa for all sa > s − εa, and so if x ∈ Fsa then sa ≤ s − εa. Thus fB(x) ≤ s − εa < s, that is, x /∈ (fB, s)∈. Therefore (fB, s)∈ = ⋃ sa≥s Fsa , and it is a weak filter of X := (X, |). In order to show that (gB, t)∈ is a weak filter of X := (X, |), we need to consider the following two cases: t ̸= inf{tb ∈ Λ | tb > t} and t = inf{tb ∈ Λ | tb > t}. If the first case is valid, then (t, t+ εb) and Λ are disjoint for some εb > 0. We will verify that (gB, t)∈ = ⋃ tb≤t Gtb . If y ∈ ⋃ tb≤t Gtb , then y ∈ Gtb for some tb ≤ t. It follows that gB(y) = inf{t ∈ Λ | y ∈ Gt} ≤ tb ≤ t, i.e., y ∈ (gB, t)∈. If y /∈ ⋃ tb≤t Gtb , then y /∈ Gtb for all tb ≤ t < t+ εb. This shows that if y ∈ Gtb , then tb ≥ t+ εb > t, i.e., y /∈ (gB, t)∈. Hence (gB, t)∈ = ⋃ tb≤t Gtb which is a weak filter of X := (X, |). For the second case, we have y ∈ (gB, t)∈ ⇔ (∀tb > t)(y ∈ Gtb) ⇔ y ∈ ⋂ tb>t Gtb . S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 11 of 16 Hence (gB, t)∈ = ⋂ tb>t Gtb is a weak filter of X := (X, |). This completes the proof. Given an intuitionistic fuzzy set B∗ := (X; fB, gB) in X, consider the following set: X(0,1) := {x ∈ X | fB(x) ̸= 0, gB(x) ̸= 1} which is called the (0, 1)-set of B∗ := (X; fB, gB). We explore the conditions under which the nonempty (0, 1)-set is a weak filter. Theorem 6. If B∗ := (X; fB, gB) is an intuitionistic fuzzy weak filter of X := (X, |), then its nonempty (0, 1)-set is a weak filter of X := (X, |). Proof. Let B∗ := (X; fB, gB) be an intuitionistic fuzzy weak filter of X := (X, |). Suppose X(0,1) ̸= ∅, say x ∈ X(0,1). Then fB(1) ≥ fB(x) ̸= 0 and gB(1) ≤ gB(x) ̸= 1 by (22). Thus 1 ∈ X(0,1). Let x ∈ X and y ∈ X(0,1). Then fB(ðx(y)) ≥ fB(y) ̸= 0 and gB(ðx(y)) ≤ gB(y) ̸= 1 by (23). Hence ðx(y) ∈ X(0,1), and therefore X(0,1) is a weak filter of X := (X, |). In the following example, we can observe that the converse of Theorem 6 is not true in general. Example 3. Let X := (X, |) be the Sheffer stroke Hilbert algebra described in Example 1 and let B∗ := (X; fB, gB) be an intuitionistic fuzzy set in X given as follows: B∗ := (X; fB, gB) :X → [0, 1]× [0, 1], b 7→  ( 0.45 n , 0.872n ) if b = c1,( 0.69 n , 0.562n ) if b ∈ {c5, c6}, (0.00, 1.00) otherwise where n is a natural number. Then X(0,1) = {c1, c5, c6} which is a weak filter of X := (X, |). We can observe that fB(c1) = 0.45 n < 0.69 n = fB(c5) and/or gB(c1) = 0.87 2n > 0.56 2n = gB(c6), that is, (22) is not valid. Hence B∗ := (X; fB, gB) is not an intuitionistic fuzzy weak filter of X := (X, |). Theorem 7. If an intuitionistic fuzzy set B∗ := (X; fB, gB) in X satisfies: (∀x ∈ X)(∀(s, t) ∈ (0, 1]× [0, 1)) ( x(s,t) ∈ B∗ ⇒ 1(s,t) q B∗) , (26) (∀x, y ∈ X)(∀(s, t) ∈ (0, 1]× [0, 1)) ( y(s,t) ∈ B∗ ⇒ ðx(y)(s,t) q B∗) , (27) then its nonempty (0, 1)-set is a weak filter of X := (X, |). Proof. Let B∗ := (X; fB, gB) be an intuitionistic fuzzy set in X that satisfies (26) and (27). Suppose that X(0,1) ̸= ∅ and say x ∈ X(0,1). Then fB(x) ̸= 0 and gB(x) ̸= 1. Since x(fB(x),gB(x)) ∈ B∗, we have 1(fB(x),gB(x)) ∈ B∗ by (26). Hence fB(1) ≥ fB(x) ̸= 0 and gB(1) ≤ gB(x) ̸= 1, and so 1 ∈ X(0,1). Let x ∈ X and y ∈ X(0,1). Then fB(y) ̸= 0, gB(y) ̸= 1 and y(fB(y),gB(y)) ∈ B∗. It follows from (27) that ðx(y)(fB(y),gB(y)) ∈ B∗. Thus fB(ðx(y)) ≥ fB(y) ̸= 0 and gB(ðx(y)) ≤ gB(y) ̸= 1 which imply that ðx(y) ∈ X(0,1). Therefore X(0,1) is a weak filter of X := (X, |). S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 12 of 16 Theorem 8. If an intuitionistic fuzzy set B∗ := (X; fB, gB) in X satisfies: (∀x ∈ X)(∀(s, t) ∈ (0, 1]× [0, 1)) ( x(s,t) q B∗ ⇒ 1(s,t) ∈ B∗) , (28) (∀x, y ∈ X)(∀(s, t) ∈ (0, 1]× [0, 1)) ( y(s,t) q B∗ ⇒ ðx(y)(s,t) ∈ B∗) , (29) then its nonempty (0, 1)-set is a weak filter of X := (X, |). Proof. Let B∗ := (X; fB, gB) be an intuitionistic fuzzy set in X that satisfies (28) and (29). Suppose that X(0,1) ̸= ∅ and say x ∈ X(0,1). Then fB(x) ̸= 0 and gB(x) ̸= 1. Thus fB(x)+1 > 1 and gB(x)+0 < 1, i.e., x(1,0) q B∗. It follows from (28) that 1(1,0) ∈ B∗, that is, fB(1) ≥ 1 and gB(1) ≤ 0. Thus 1 ∈ X(0,1). If x ∈ X and y ∈ X(0,1), then fB(y) ̸= 0 and gB(y) ̸= 1, and so fB(y)+1 > 1 and gB(y)+0 < 1, i.e., y(1,0) q B∗. Using (29) induces ðx(y)(1,0) ∈ B∗. Hence fB(ðx(y)) ≥ 1 and gB(ðx(y)) ≤ 0 which shows that ðx(y) ∈ X(0,1). Therefore X(0,1) is a weak filter of X := (X, |). Theorem 9. If an intuitionistic fuzzy set B∗ := (X; fB, gB) in X satisfies: (∀x ∈ X)(∀(s, t) ∈ (0, 1]× [0, 1)) ( x(s,t) q B∗ ⇒ 1(s,t) q B∗) , (30) (∀x, y ∈ X)(∀(s, t) ∈ (0, 1]× [0, 1)) ( y(s,t) q B∗ ⇒ ðx(y)(s,t) q B∗) , (31) then its nonempty (0, 1)-set is a weak filter of X := (X, |). Proof. Let B∗ := (X; fB, gB) be an intuitionistic fuzzy set in X that satisfies (30) and (31). Suppose that X(0,1) ̸= ∅ and say x ∈ X(0,1). Then fB(x) ̸= 0 and gB(x) ̸= 1. Thus fB(x) + 1 > 1 and gB(x) + 0 < 1, i.e., x(1,0) q B∗. Using (30) induces 1(1,0) q B∗, that is, fB(1)+ 1 > 1 and gB(1)+ 0 < 1. Thus fB(1) ̸= 0 and gB(1) ̸= 1, i.e., 1 ∈ X(0,1). If x ∈ X and y ∈ X(0,1), then fB(y) ̸= 0 and gB(y) ̸= 1, and so fB(y) + 1 > 1 and gB(y) + 0 < 1, i.e., y(1,0) q B∗. It follows from (31) that ðx(y)(1,0) q B∗. Hence fB(ðx(y)) + 1 > 1 and gB(ðx(y)) + 0 < 1, and so ðx(y) ∈ X(0,1). Therefore X(0,1) is a weak filter of X := (X, |). We provide conditions for the intuitionistic level set and intuitionistic q-set to be weak filters. Theorem 10. If an intuitionistic fuzzy set B∗ := (X; fB, gB) in X satisfies: fB(x) ≤ max{fB(1), 0.5}, gB(x) ≥ min{gB(1), 0.5}, (32) fB(y) ≤ max{fB(ðx(y)), 0.5}, gB(y) ≥ min{gB(ðx(y)), 0.5} (33) for all x, y ∈ X, then its nonempty intuitionistic level set (B∗, (s, t))∈ is a weak filter of X := (X, |) for all (s, t) ∈ (0.5, 1]× [0, 0.5). Proof. Let (s, t) ∈ (0.5, 1]× [0, 0.5) be such that (B∗, (s, t))∈ ̸= ∅, say x ∈ (B∗, (s, t))∈. Then x ∈ (fB, s)∈ ∩ (gB, t)∈, which implies from (32) that max{fB(1), 0.5} ≥ fB(x) ≥ s > 0.5 S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 13 of 16 and min{gB(1), 0.5} ≤ gB(x) ≤ t < 0.5. Hence fB(1) ≥ s and gB(1) ≤ t, and so 1 ∈ (fB, s)∈ ∩ (gB, t)∈ = (B∗, (s, t))∈. If x ∈ X and y ∈ (B∗, (s, t))∈, then fB(y) ≥ s and gB(y) ≤ t. It follows from (33) that 0.5 < s ≤ fB(y) ≤ max{fB(ðx(y)), 0.5} and 0.5 > t ≥ gB(y) ≥ min{gB(ðx(y)), 0.5}. Hence fB(ðx(y)) ≥ s and gB(ðx(y)) ≤ t, which imply that ðx(y) ∈ (fB, s)∈ ∩ (gB, t)∈ = (B∗, (s, t))∈. Therefore (B∗, (s, t))∈ is a weak filter of X := (X, |). Theorem 11. If B∗ := (X; fB, gB) is an intuitionistic fuzzy weak filter of X := (X, |), then its nonempty intuitionistic q-set (B∗, (s, t))q is a weak filter of X := (X, |) for all (s, t) ∈ (0, 1]× [0, 1). Proof. Let (s, t) ∈ (0, 1] × [0, 1) be such that (B∗, (s, t))q ̸= ∅. Since fB(1) ≥ fB(x) and gB(1) ≤ gB(x) for x ∈ (B∗, (s, t))q, we have fB(1) ≥ fB(x) > 1 − s and gB(1) ≤ gB(x) < 1 − t. Hence 1 ∈ (fB, s)q ∩ (gB, t)q = (B∗, (s, t))q. If y ∈ (B∗, (s, t))q, then fB(ðx(y)) ≥ fB(y) > 1 − s and gB(ðx(y)) ≤ gB(y) < 1 − t for all x ∈ X by (23). Thus ðx(y) ∈ (fB, s)q ∩ (gB, t)q = (B∗, (s, t))q. Therefore (B∗, (s, t))q is a weak filter of X := (X, |). Proposition 3. Let B∗ := (X; fB, gB) be an intuitionistic fuzzy set in X. If its intuition- istic q-set (B∗, (s, t))q is a weak filter of X := (X, |) for all (s, t) ∈ (0, 0.5] × [0.5, 1), then 1 ∈ (B∗, (s, t))∈ and y ∈ (B∗, (s, t))q ⇒ ðx(y) ∈ (B∗, (s, t))∈ for all x, y ∈ X and (s, t) ∈ (0, 0.5]× [0.5, 1). Proof. Let x, y ∈ X and (s, t) ∈ (0, 0.5] × [0.5, 1). Assume that (B∗, (s, t))q is a weak filter of X := (X, |). Then 1 ∈ (B∗, (s, t))q and so fB(1) > 1− s ≥ s and gB(1) < 1− t ≤ t. Hence 1 ∈ (fB, s)∈ ∩ (gB, t)∈ = (B∗, (s, t))∈. If y ∈ (B∗, (s, t))q, then ðx(y) ∈ (B∗, (s, t))q. It follows that fB(ðx(y)) > 1− s ≥ s and gB(ðx(y)) < 1− t ≤ t. Thus ðx(y) ∈ (fB, s)∈ ∩ (gB, t)∈ = (B∗, (s, t))∈. Proposition 4. Given an intuitionistic fuzzy set B∗ := (X; fB, gB) in X, if its intuition- istic q-set (B∗, (s, t))q is a weak filter of X := (X, |) for all (s, t) ∈ (0.5, 1] × [0, 0.5), then the following is valid. y ∈ (B∗, (s, t))∈ ⇒ ðx(y) ∈ (B∗, (s, t))q for all x, y ∈ X and (s, t) ∈ (0.5, 1]× [0, 0.5). Proof. Let x, y ∈ X and (s, t) ∈ (0.5, 1] × [0, 0.5). If y ∈ (B∗, (s, t))∈, then fB(y) ≥ s > 1 − s and gB(y) ≤ t < 1 − t. Thus y ∈ (fB, s)q ∩ (gB, t)q = (B∗, (s, t))q, and so ðx(y) ∈ (B∗, (s, t))q. S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 14 of 16 Theorem 12. If an intuitionistic fuzzy set B∗ := (X; fB, gB) in X satisfies: x(s,t) q B∗ ⇒ 1(s,t) ∈∨q B∗, (34) y(s,t) q B∗ ⇒ ðx(y)(s,t) ∈∨q B∗, (35) for all x, y ∈ X and (s, t) ∈ (0.5, 1] × [0, 0.5), then its nonempty intuitionistic q-set (B∗, (s, t))q is a weak filter of X := (X, |) for all (s, t) ∈ (0.5, 1]× [0, 0.5) Proof. Let (s, t) ∈ (0.5, 1]× [0, 0.5) be such that (B∗, (s, t))q ̸= ∅, say x ∈ (B∗, (s, t))q. Then x(s,t) q B∗, and so 1(s,t) ∈ ∨q B∗ by (34), i.e., 1(s,t) ∈ B∗ or 1(s,t) q B∗. If 1(s,t) q B∗, then 1 ∈ (B∗, (s, t))q. If 1(s,t) ∈ B∗, then fB(1) ≥ s > 1 − s and gB(1) ≤ t < 1 − t. Hence 1 ∈ (fB, s)q ∩ (gB, t)q = (B∗, (s, t))q. Let y ∈ (B∗, (s, t))q. Then y(s,t) q B∗, and so ðx(y)(s,t) ∈∨q B∗ by (35), that is, ðx(y)(s,t) ∈ B∗ or ðx(y)(s,t) q B∗. If ðx(y)(s,t) q B∗, then ðx(y) ∈ (B∗, (s, t))q. If ðx(y)(s,t) ∈ B∗, then fB(ðx(y)) ≥ s > 1 − s and gB(ðx(y)) ≤ t < 1− t. Hence ðx(y) ∈ (fB, s)q ∩ (gB, t)q = (B∗, (s, t))q. Consequently, (B∗, (s, t))q is a weak filter of X := (X, |). Theorem 13. Given a weak filter F of X := (X, |), if an intuitionistic fuzzy set B∗ := (X; fB, gB) in X satisfies B∗(x) = (0, 1), i.e., fB(x) = 0 and gB(x) = 1, for x ∈ X \ F and x ∈ (B∗, (0.5, 0.5))∈ for x ∈ F , then its nonempty intuitionistic q-set (B∗, (s, t))q is a weak filter of X := (X, |) for all (s, t) ∈ (0.5, 1]× [0, 0.5). Proof. Let (s, t) ∈ (0.5, 1]× [0, 0.5) be such that (B∗, (s, t))q ̸= ∅, say y ∈ (B∗, (s, t))q. Then y(s,t) q B∗, and so fB(y) + s > 1 and gB(y) + t < 1. If y ∈ X \ F , then 1 < fB(y) + s = 0 + s = s and 1 > gB(y) + t = 1 + s which is a contradiction. Hence y ∈ F , and thus y ∈ (B∗, (0.5, 0.5))∈, that is, gB(y) ≤ 0.5 ≤ fB(y). Since 1 ∈ F , we have 1 ∈ (B∗, (0.5, 0.5))∈, that is, fB(1) ≥ 0.5 and gB(1) ≤ 0.5. If 1(s,t) ∈B∗, then fB(1) < s or gB(1) > t. At this time, the following three cases should be considered. (i) fB(1) < s and gB(1) > t. (ii) fB(1) < s and gB(1) ≤ t. (iii) fB(1) ≥ s and gB(1) > t. The first case induces fB(1) + s > 2fB(1) ≥ 1 and gB(1) + t < 2gB(1) ≤ 1. For the case (ii), we get fB(1) + s > 2fB(1) ≥ 1 and gB(1) + t < 2t ≤ 1. The third case implies that fB(1) + s > 2s ≥ 1 and gB(1) + t < 2gB(1) ≤ 1. This shows that 1(s,t) q B∗ and consequently 1(s,t) ∈∨q B∗. Since y ∈ F , we have ðx(y) ∈ F for all x ∈ X because F is a weak filter of X := (X, |). Thus ðx(y) ∈ (B∗, (0.5, 0.5))∈, that is, fB(ðx(y)) ≥ 0.5 and gB(ðx(y)) ≤ 0.5. If ðx(y)(s,t) ∈B∗, then fB(ðx(y)) < s or gB(ðx(y)) > t. At this time, the following three cases should be considered. (iv) fB(ðx(y)) < s and gB(ðx(y)) > t. (v) fB(ðx(y)) < s and gB(ðx(y)) ≤ t. S. S. Ahn, Y. J. Seo, Y. B. Jun / Eur. J. Pure Appl. Math, 18 (3) (2025), 6529 15 of 16 (vi) fB(ðx(y)) ≥ s and gB(ðx(y)) > t. The case (iv) induces fB(ðx(y))+s > 2fB(ðx(y)) ≥ 1 and gB(ðx(y))+t < 2gB(ðx(y)) ≤ 1. For the case (v), we hace fB(ðx(y))+s > 2fB(ðx(y)) ≥ 1 and gB(ðx(y))+ t < 2t ≤ 1. The case (vi) implies that fB(ðx(y)) + s > 2s ≥ 1 and gB(ðx(y)) + t < 2gB(ðx(y)) ≤ 1. This shows that ðx(y)(s,t) q B∗ and consequently ðx(y)(s,t) ∈∨q B∗. It follows from Theorem 12 that (B∗, (s, t))q is a weak filter of X := (X, |). 4. Conclusion This work aims to advance the theoretical framework of Schaeffer stroke Hilbert al- gebras using intuitionistic fuzzy points to develop weak filters. We have introduced the concept of intuitionistic fuzzy weak filters in Sheffer stroke Hilbert algebras, and have investigated several properties. We have explored the conditions under which the intu- itionistic fuzzy set becomes an intuitionistic fuzzy weak filter. We have discussed the characterization of intuitionistic fuzzy weak filters. 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