EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6199 ISSN 1307-5543 – ejpam.com Published by New York Business Global Sheffer Stroke Hilbert Algebras in Connection with Crossing Cubic Structures Anas Al-Masarwah1,∗, Noor Bani Abd Al-Rahman1, Mohammed Alqahtani2,∗, Majdoleen Abuqamar3 1 Department of Mathematics, Faculty of Science, Ajloun National University, P. O. Box 43, Ajloun 26810, Jordan 2 Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, P.O. Box 93499, Riyadh 11673, Saudi Arabia 3 Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid 21110, Jordan Abstract. The idea of Sheffer stroke Hilbert algebra is studied from the perspective of the struc- ture of the cubic structure. The filter and deductive system of the Sheffer stroke Hilbert algebra are defined and examined through the crossing cubic structure, which is an extension of the fuzzi- ness of these substructures, verifying their many characteristics. Moreover, conditions suitable for the crossing cubic structure are established to be a crossing cubic filter and several characteriza- tion theorems are reached. Accordingly, the relationship between crossing cubic filters and filters of Sheffer stroke Hilbert algebras is explained such that the crossing cubic deductive system can handle all of the results for the crossing cubic filter covered above in the same way. 2020 Mathematics Subject Classifications: 03E72, 03G25, 28E10, 03B52 Key Words and Phrases: Sheffer stroke Hilbert algebras, filters, deductive systems, crossing cubic filters, crossing cubic deductive systems 1. Introduction In mathematical logic, there are certain logical operations such that conjunction (and), disjunction (or) and negation (not). These logical operations are represented by the sym- bols “∧”, “∨” and “∼”, respectively. In a conjunction operator, if anyone of the statement is false, then the output is false. In a disjunction operator, if anyone of the statements is true, then the output is true. A negation operator gives the opposite result, i.e., if the input is true, then the output is false. Table 1 describes the truth table for the conjunction “∧” and disjunction “∨” operations. ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6199 Email addresses: anas.almasarwah@anu.edu.jo (A. Al-Masarwah), noorbaniabdalrahman@gmail.com (N. Bani Abd Al-Rahman), m.alqahtani@seu.edu.sa (M. Alqahtani), mjabuqamar@gmail.com (M. Abuqamar) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 2 of 17 Table 1: The truth table for the conjunction “∧” and disjunction “∨” operations. Input(A) Input(B) Output(A ∧B) Output(A ∨B) T T T T T F F T F T F T F F F F A Sheffer stroke, represented by the symbol “|”, is a two-input logical operation that produces an incorrect output only when both inputs are true i.e., the Sheffer stroke oper- ation A | B is equivalent to ∼ (A ∧ B). Table 2 describes the truth table for the Sheffer stroke operation “|”. Table 2: The truth table for the Sheffer stroke “|” . Input(A) Input(B) Output(A | B) T T F T F T F T T F F T Here, we will make an example on the truth values of Sheffer stroke operation for the given two statements A and B. Consider the following two statements: A : p is divisible by 2, B : p is divisible by 3. Then, the truth table of the Sheffer stroke “|” between the two statements A and B is presented by Table 3. Table 3: The truth table for the Sheffer stroke “|” . Values of p A B A | B p = 12 T T F p = 4 T F T p = 9 F T T p = 7 F F T Sheffer [1] used the Sheffer stroke “|” (the so-called Sheffer operation) in 1913 as a fun- damental axiomatization of Boolean algebra and demonstrated that all Boolean functions may be generated from a single binary operation, as term operations. After that, McCune et al. [2] demonstrated that this operation can represent any Boolean operation. Many researchers were drawn to this operation since it is the only way to define any Boolean operation or function [2]. As a result, many algebraic structures, axioms or formulas, are reduced. Because simple and fewer axiom systems are obtained, it is simple to verify some attributes and notations for a new algebraic structure. Thus, many scientists seek A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 3 of 17 to apply such a reduction to explore algebraic structures using the Sheffer operation such as BG-algebras [3], BM-algebra [4], ortholattices [5], and INK-algebra [6], etc. Henkin [7] proposed Hilbert algebras in the 1950’s as an algebraic equivalent of Hilbert’s positive implicative propositional calculus [8], for use in intuitionistic and other non- classical logic studies. These algebraic structures, which include the implication and the distinct element 1, can be understood as parts of propositional logic. In particular, Diego [9] analyzed Hilbert algebras, including their deductive systems and different features, and showed that a variety of Hilbert algebras exist. The idea of a deductive system, also known as implicative filters, is crucial to the overall development of Hilbert algebras because they can be employed to symbolize several logical theories. Furthermore, Busneag [10, 11] and Jun [12] analyzed Hilbert algebras, associated concepts, and deductive systems. The Shef- fer stroke was used in Hilbert algebras by Oner et al. in 2021, and they presented its several properties, deductive systems, and ideals [13]. Fuzzy mathematics is a new field in which ordinary notions are being transferred to uncertainty cases. Zadeh [14] adopted the concept of fuzzy structures through his popular paper as a development of standard structure, which only allows elements to be fully in or fully out of a set. This concept is a significant mathematical framework for dealing with uncertainty and ambiguity in data. Zadeh [15] made an extension of the theory of fuzziness concept by an interval-valued fuzziness concept. As another extension of uncertainty sets, Zhang [16] presented the notion bipolar fuzzy sets as a very useful tool for considering positive and negative data at the same time. By ignore the positive part of a bipolar uncertainty set, Jun et al. [17] proposed a new concept, namely N-structures. After that, there were many extensions of fuzziness structures, such as the intuitionistic fuzzy, neutrosophic set and Lukasiewicz fuzzy set, see [18–20]. By incorporating the concept of interval-valued fuzziness sets with N -structures, which creates an expansion of a bipolar-valued fuzziness set to produce the thought of crossing cubic structures [21], that floated for the first time in 2021 by Jun and Song [22], where they applied it to BCK/BCI-algebra. Additionally, Ozturk et al. [23] developed the con- cept of crossing cubic on semigroup structures, and commutative ideal in BCK-algebras. Al-Masarwah et al. [24] established the idea of crossing cubic Lie subalgebra in the con- text of Lie algebras. They utilized the notions of Lie ideals, homomorphisms and some fundamental properties of Lie algebras. In the fuzzification of the Sheffer stroke Hilbert algebras, the concept of fuzzy (uncer- tainty) sets was connected with a filter and a deductive system of Sheffer stroke Hilbert algebra by Oner et al. [25]. Vasuki et al. [26] presented the concept of anti-Q-fuzzy deductive systems of Hilbert algebras and proved some results. In the connection between intuitionistic fuzzy sets and Sheffer stroke Hilbert algebras, Saeid et al. [27] implemented the concept of intuitionistic fuzzy deductive system and intuitionistic fuzzy filter in Shef- fer stroke Hilbert algebras, and investigated some of its features. Oner et al. [28] studied the idea of Sheffer stroke Hilbert algebras via neutrosophic N-structures. They discussed neutrosophic N-subalgebras and neutrosophic N-ideals in Sheffer stroke Hilbert algebras. Concentrating on exploring the filter and deductive system of the Sheffer stroke Hilbert algebra using the crossing cubic structure, we propose the idea of crossing cubic filter and A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 4 of 17 crossing cubic deductive system, give examples, and then discuss certain features. We create a crossing cubic filter by applying appropriate conditions to a specific crossing cubic structure. We study characterizations of crossing cubic filters. We construct crossing cubic filters that are associated with filters. Eventually, we prove that crossing cubic deductive system and crossing cubic filter are a complementary notion. 2. Preliminaries This section covers some fundamental definitions and results of Sheffer stroke Hilbert algebras, filters, deductive systems, crossing cubic structures, and some results that will be utilized throughout the study. Definition 1 ([1]). A groupoid is denoted by φ◦ := (φ, |). Then, the operation “|” is called a Sheffer stroke (Sheffer operation) if the following identities hold: ∀n, v,w,∈ φ, (1) n | v = v | n; (2) (n | n) | (n | v) = n; (3) n | ((v | w) | (v | w)) = ((n | v) | (n | v)) | w; (4) (n | ((n | n) | (v | v))) | (n | ((n | n) | (v | v))) = n. Definition 2 ([13]). A Sheffer stroke Hilbert algebra with a Sheffer stroke “|” is a groupoid ϑ◦ := (ϑ, |) that satisfies the axioms, ∀n, v,w ∈ ϑ : (1) (n | ((X) | (X))) | (((Y ) | ((Z) | (Z))) | ((Y ) | ((Z) | (Z)))) = n | (n | n), where X := v | (w | w), Y := n | (v | v) and Z := n | (w | w); (2) n | (v | v) = v | (n | n) = n | (n | n) ⇒ n = v. For a Sheffer stroke Hilbert algebra ϑ◦ := (ϑ, |), see[13] a relation “≤”, which is a partial order on ϑ, is defined as (A)(n ≤ v ⇔ n | (v | v) = 1, ∀n, v ∈ ϑ). Proposition 1 ([13]). If ϑ◦ := (ϑ, |) is a Sheffer stroke Hilbert algebra, then ϑ satisfies: ∀n, v,w ∈ ϑ, (1) n | (n | n) = 1; (2) n | (1 | 1) = 1; (3) 1 | (n | n) = n; (4) n ≤ v | (n | n); (5) (n | (v | v)) | (v | v) = (v | (n | n)) | (n | n); A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 5 of 17 (6) ((n | (v | v)) | (v | v)) | (v | v) = n | (v | v); (7) n | ((v | (w | w)) | (v | (w | w))) = v | ((n | (w | w)) | (n | (w | w))). Definition 3 ([25]). A subset Ψ of a Sheffer stroke Hilbert algebra ϑ◦ := (ϑ, |) is called a filter of ϑ if the conditions are satisfied: ∀n, v,w ∈ ϑ (1) 1 ∈ Ψ; (2) v ∈ Ψ ⇒ n | (v | v) ∈ Ψ; (3) v,w ∈ Ψ ⇒ (n | (v | w)) | (v | w) ∈ Ψ. Definition 4 ([13]). A subset Ψ of a Sheffer stroke Hilbert algebra ϑ◦ := (ϑ, |) is called a deductive system of ϑ if the conditions are satisfied: ∀n, v ∈ ϑ (1) 1 ∈ Ψ; (2) n ∈ Ψ, n | (v | v) ∈ Ψ ⇒ v ∈ Ψ. Definition 5 ([15]). Let ϑ ̸= ϕ be a set. An interval valued fuzzy set on ϑ is an object of the following form: Υ̃Π = {(n, ∓ ΥΠ (n)) : n ∈ ϑ} = {(n, [Υ− Π(n),Υ+ Π(n)]) : n ∈ ϑ}, where ∓ ΥΠ: ϑ → I[0, 1] is a function and I[0, 1] denotes the set containing every closed subinterval of [0, 1]. The members of I[0, 1] are called interval numbers. If we take an interval number υ̃ = [υ−, υ+] ∈ I[0, 1], then 0 ≤ υ− ≤ υ+ ≤ 1. For each two interval numbers υ̃ = [υ−, υ+] and σ̃ = [σ−, σ+] in I[0, 1], we define (1) υ̃ ⪯ σ̃( or σ̃ ⪰ υ̃) ⇔ υ− ≤ σ−, υ+ ≤ σ+; (2) υ̃ = σ̃ ⇔ υ̃ ⪯ σ̃ and σ̃ ⪯ υ̃; (3) m̃in{υ̃, σ̃} = [min{υ−, σ−},min{υ+, σ+}]. Definition 6 ([17]). An N-structure ϖΠ of ϑ is an object of the following form: ϖΠ = {(n, ϖ−(n)) : n ∈ ϑ}, where ϖ− : ϑ → [−1, 0]. Definition 7 ([21]). Let ϑ ̸= ϕ be a set. A crossing cubic structure of ϑ is an object having the following form: Π = {(n, Υ̃Π(n), ϖΠ(n)) | n ∈ ϑ}, A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 6 of 17 where Υ̃Π = [Υ− Π,Υ + Π] : ϑ → I[0, 1] is an interval valued fuzzy set on ϑ and ϖΠ : ϑ → [−1, 0] is an N-structure on ϑ. For the purpose of simplicity, we will employ the symbol Π = (Υ̃Π, ϖΠ) for the crossing cubic structure Π = {(n, Υ̃Π(n), ϖΠ(n)) | n ∈ ϑ}. Definition 8 ([21]). Let Π = (Υ̃Π, ϖΠ) be a crossing cubic structure over ϑ. Then, (τ̃ , κ)-level of Π = (Υ̃Π, ϖΠ) is the crisp set in ϑ denoted by Γ ( Υ̃Π; (τ̃ , κ) ) and is defined as Γ ( Υ̃Π; (τ̃ , κ) ) = { n ∈ ϑ | Υ̃Π(n) ⪰ τ̃ , ϖΠ(n) ≤ κ } , where τ̃ ∈ I[0, 1] and κ ∈ [−1, 0]. For τ̃ ∈ I[0, 1] and κ ∈ [−1, 0]. The τ̃ -level Γ ( Υ̃Π; τ̃ ) and κ-level Γ ( ϖΩ;κ ) subsets of ∏ can be defined as: Γ ( Υ̃Π; τ̃ ) = { n ∈ ϑ | Υ̃Π(n) ⪰ τ̃ } and Γ ( ϖΠ;κ ) = { n ∈ ϑ | ϖΠ(n) ≤ κ } . 3. Crossing Cubic Filters Here, we present the idea of crossing cubic filters. Then, we study certain charac- terizations of it. Also, we discuss a relation between a crossing cubic filter and a crisp filter. Definition 9. A crossing cubic structure Π = (Υ̃Π, ϖΠ) over ϑ is called a crossing cubic filter of ϑ if it satisfies the conditions: ∀n, v, w ∈ ϑ, (i) (∀n ∈ ϑ) ( Υ̃Π(1) ⪰ Υ̃Π(n), ϖΠ(1) ≤ ϖΠ(n) ) ; (ii) (∀n, v ∈ ϑ) ( Υ̃Π(n | (v | v)) ⪰ Υ̃Π(v), ϖΠ(n | (v | v)) ≤ ϖΠ(v) ) ; (iii) (∀n, v, w ∈ ϑ) ( Υ̃Π((n | (v | w)) | (v | w)) ⪰ m̃in{Υ̃Π(v), Υ̃Π(w)}, ϖΠ((n | (v | w)) | (v | w)) ≤ max{ϖΠ(v), ϖΠ(w)} ) . Example 1. Let ϑ = {0, 1, 2ϑ, 3ϑ, 4ϑ, 5ϑ, 6ϑ, 7ϑ} be a set. The Hasse diagram and Sheffer stroke “|” on ϑ, which are shown in Figure 1 and Table 4, respectively. A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 7 of 17 Figure 1: Hasse Diagram Table 4: Cayley table for the Sheffer stroke “|” | 0 2ϑ 3ϑ 4ϑ 5ϑ 6ϑ 7ϑ 1 0 1 1 1 1 1 1 1 1 2ϑ 1 7ϑ 1 1 7ϑ 7ϑ 1 7ϑ 3ϑ 1 1 6ϑ 1 6ϑ 1 6ϑ 6ϑ 4ϑ 1 1 1 5ϑ 1 5ϑ 5ϑ 5ϑ 5ϑ 1 7ϑ 6ϑ 1 4ϑ 7ϑ 6ϑ 4ϑ 6ϑ 1 7ϑ 1 5ϑ 7ϑ 3ϑ 5ϑ 3ϑ 7ϑ 1 1 6ϑ 5ϑ 6ϑ 5ϑ 2ϑ 2ϑ 1 1 7ϑ 6ϑ 5ϑ 4ϑ 3ϑ 2ϑ 0 Then, ϑ◦ := (ϑ, |) is a Sheffer stroke Hilbert algebra (see [13]). Let Π = (Υ̃Π, ϖΠ) be a crossing cubic structure in ϑ, which is shown in Table 5. Table 5: Π = (Υ̃Π, ϖΠ) is represented tabularly ϑ Υ̃Π(n) ϖΠ(n) 0 [0.28, 0.65] -0.41 2ϑ [0.28, 0.65] -0.55 3ϑ [0.28, 0.65] -0.41 4ϑ [0.32, 0.73] -0.41 5ϑ [0.28, 0.65] -0.63 6ϑ [0.38, 0.76] -0.55 7ϑ [0.32, 0.73] -0.41 1 [0.42, 0.91] -0.71 It is customary to check that Π = (Υ̃Π, ϖΠ) is a crossing cubic filter of ϑ. Proposition 2. Every crossing cubic filter Π = (Υ̃Π, ϖΠ) of ϑ satisfies: (i) (∀n, v ∈ ϑ) ( Υ̃Π((n | (v | v)) | (v | v)) ⪰ Υ̃Π(n), ϖΠ((n | (v | v)) | (v | v)) ≤ ϖΠ(n) ) ; A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 8 of 17 (ii) (∀n, v ∈ ϑ) ( n ≤ v ⇒ { Υ̃Π(v) ⪰ Υ̃Π(n), ϖΠ(v) ≤ ϖΠ(n) ) . Proof. Let Π = (Υ̃Π, ϖΠ) be a crossing cubic filter of ϑ. Then, by using (5) of Proposition 1 and (iii) of Definition 9, ∀n, v ∈ ϑ, we have Υ̃Π((n | (v | v)) | (v | v)) = Υ̃Π((v | (n | n)) | (n | n)) ⪰ m̃in{Υ̃Π(n), Υ̃Π(n)} = Υ̃Π(n) and ϖΠ((n | (v | v)) | (v | v)) = ϖΠ((v | (n | n)) | (n | n)) ≤ max{ϖΠ(n), ϖΠ(n)} = ϖΠ(n). Therefore, the proof of (i) is completed. Now, let n, v ∈ ϑ be such that n ≤ v. Then n | (v | v) = 1 by condition (A). Using (3) of Proposition 1 and (i) of Proposition 2, we have Υ̃Π(v) = Υ̃Π(1 | (v | v)) = Υ̃Π((n | (v | v)) | (v | v)) ⪰ Υ̃Π(n) and ϖΠ(v) = ϖΠ(1 | (v | v)) = ϖΠ((n | (v | v)) | (v | v)) ≤ ϖΠ(n). This completes the proof of (ii). Next example illustrates that if a crossing cubic structure Π = (Υ̃Π, ϖΠ) satisfies condition (ii) of Proposition 2, then it is not enough for a crossing cubic filter of ϑ. Example 2. Let ϑ = {0, 1, 2ϑ, 3ϑ} be a set. The Hasse diagram and Sheffer stroke “|” on ϑ, which are shown in Figure 2 and Table 6, respectively. Figure 2: Hasse Diagram A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 9 of 17 Table 6: Cayley table for the Sheffer stroke “|” | 1 2ϑ 3ϑ 0 1 0 3ϑ 2ϑ 1 2ϑ 3ϑ 3ϑ 1 1 3ϑ 2ϑ 1 2ϑ 1 0 1 1 1 1 Then, ϑ◦ := (ϑ, |) is a Sheffer stroke Hilbert algebra (see [13]). Let Π = (Υ̃Π, ϖΠ) be a crossing cubic structure in ϑ which is shown in Table 7. Table 7: Π = (Υ̃Π, ϖΠ) is represented tabularly ϑ Υ̃Π(n) ϖΠ(n) 0 [0.29,0.63] -0.13 2ϑ [0.32,0.67] -0.38 3ϑ [0.36,0.75] -0.55 1 [0.47,0.89] -0.82 It is customary to check that Π = (Υ̃Π, ϖΠ) in ϑ satisfies condition (ii) of Proposition 2, but it is not a crossing cubic filter of ϑ since Υ̃Π((0 | (2ϑ | 3ϑ)) | (2ϑ | 3ϑ)) = Υ̃Π(0) = [0.29, 0.63] ⪰̸ [0.32, 0.67] = m̃in{Υ̃Π(2ϑ), Υ̃Π(3ϑ)} or ϖΠ((0 | (2ϑ | 3ϑ)) | (2ϑ | 3ϑ)) = ϖΠ(0) = −0.13 ⩽̸ −0.38 = max{ϖΠ(2ϑ), ϖΠ(3ϑ)}. Next, we propose conditions under which a crossing cubic structure Π = (Υ̃Π, ϖΠ) is a crossing cubic filter of ϑ. Theorem 1. Let Π = (Υ̃Π, ϖΠ) be a crossing cubic structure in ϑ. Then, Π = (Υ̃Π, ϖΠ) is a crossing cubic filter of ϑ if and only if it satisfies (ii) of Proposition 2 and condition (B), where (B) (∀n, v ∈ ϑ) ( Υ̃Π((n | v) | (n | v)) ⪰ m̃in{Υ̃Π(n), Υ̃Π(v)}, ϖΠ((n | v) | (n | v)) ≤ max{ϖΠ(n), ϖΠ(v)} ) . Proof. Let Π = (Υ̃Π, ϖΠ) be a crossing cubic filter of ϑ. Proposition 2 establishes the validity of condition (ii). Now, using (1) and (2) of Definition 1, as well as (2) and (3) of Proposition 1, we get ((1 | 1) | (n | v)) | (n | v) = ((n | v) | (1 | 1)) | (n | v) = 1 | (n | v) = 1 | (((n | v) | (n | v)) | ((n | v) | (n | v))) = (n | v) | (n | v) A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 10 of 17 ∀n, v ∈ ϑ, it follows from (iii) of Definition 9 that Υ̃Π((n | v) | (n | v)) = Υ̃Π(((1 | 1) | (n | v)) | (n | v)) ⪰ m̃in{Υ̃Π(n), Υ̃Π(v)} and ϖΠ((n | v) | (n | v)) = ϖΠ(((1 | 1) | (n | v)) | (n | v)) ≤ max{ϖΠ(n), ϖΠ(v)}. In contrast, let us assume that a crossing cubic structure Π = (Υ̃Π, ϖΠ) satisfies (ii) of Proposition 2 and the condition (B). Since n ≤ 1 and v ≤ n | (v | v)∀n, v ∈ ϑ, using (ii) of Proposition 2, we have (∀n, v ∈ ϑ) ( Υ̃Π(1) ⪰ Υ̃Π(n), ϖΠ(1) ≤ ϖΠ(n) ) and ( Υ̃Π(n | (v | v)) ⪰ Υ̃Π(v), ϖΠ(n | (v | v)) ≤ ϖΠ(v) ) , respectively. In (4) of Proposition 1, if n = (v | w) | (v | w) and v = n | (v | w), and use (2) of Definition 1, then ∀n, v, w ∈ ϑ, (v | w) | (v | w) ≤ (n | (v | w)) | (((v | w) | (v | w)) | ((v | w) | (v | w))) = (n | (v | w)) | (v | w). Using (ii) of Proposition 2 and the condition (B), we have Υ̃Π((n | (v | w)) | (v | w)) ⪰ Υ̃Π((v | w) | (v | w)) ⪰ m̃in{Υ̃Π(v), Υ̃Π(w)} and ϖΠ((n | (v | w)) | (v | w)) ≤ ϖΠ((v | w) | (v | w)) ≤ max{ϖΠ(v), ϖΠ(w)}. Therefore, Π = (Υ̃Π, ϖΠ) is a crossing cubic filter of ϑ. Theorem 2. If Π = (Υ̃Π, ϖΠ) is a crossing cubic filter of ϑ, then the nonempty sets ξ ( Υ̃Π; τ̃ ) and ξ ( ϖΠ;κ ) are filters of ϑ, for all τ̃ = [τ−, τ+] ∈ I[0, 1] and κ ∈ [−1, 0]. Proof. Suppose that Π = (Υ̃Π, ϖΠ) is a crossing cubic filter of ϑ. Let τ̃ = [τ−, τ+] ∈ I[0, 1] and κ ∈ [-1, 0] be such that ξ ( Υ̃Π; τ̃ ) and ξ ( ϖΠ;κ ) are nonempty. Clearly, 1 ∈ ξ ( Υ̃Π; τ̃ ) ∩ ξ ( ϖΠ;κ ) by (i) of Definition 9. Let n ∈ ϑ and v ∈ ξ ( Υ̃Π; τ̃ ) ∩ ξ ( ϖΠ;κ ) . Then, Υ̃Π(v) ⪰ τ̃ and ϖΦ(v) ≤ κ. Using (ii) of Definition 9, we have Υ̃Π(n | (v | v)) ⪰ Υ̃Π(v) ⪰ τ̃ and ϖΠ(n | (v | v)) ≤ ϖΠ(v) ≤ κ. Hence, n | (v | v) ∈ ξ ( Υ̃Π; τ̃ ) ∩ ξ ( ϖΠ;κ ) . Let n ∈ ϑ and v, w ∈ ξ ( Υ̃Π; τ̃ ) ∩ξ ( ϖΠ;κ ) . Then Υ̃Π(v) ⪰ τ̃ , ϖΠ(v) ≤ κ, Υ̃Π(w) ⪰ τ̃ and ϖΠ(w) ≤ κ. It follows from (iii) of Definition 9 that Υ̃Π((n | (v | w)) | (v | w)) ⪰ m̃in{Υ̃Π(v), Υ̃(w)} ⪰ τ̃ A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 11 of 17 and ϖΠ((n | (v | w)) | (v | w)) ≤ max{ϖΠ(v), ϖΠ(w)} ≤ κ. Thus, (n | (v | w)) | (v | w) ∈ ξ ( Υ̃Π; τ̃ ) ∩ ξ ( ϖΠ;κ ) . Therefore, ξ ( Υ̃Π; τ̃ ) and ξ ( ϖΠ;κ ) are filters of ϑ. Theorem 3. Given a crossing cubic structure Π = (Υ̃Π, ϖΠ) in ϑ. If the nonempty sets ξ ( Υ̃Π; τ̃ ) and ξ ( ϖΠ;κ ) are filters of ϑ for all τ̃ = [τ−, τ+] ∈ I[0, 1] and κ ∈ [−1, 0], then Π = (Υ̃Π, ϖΠ) is a crossing cubic filter of ϑ. Proof. Assume that ξ ( Υ̃Π; τ̃ ) and ξ ( ϖΠ;κ ) are nonempty filters of ϑ for all τ̃ = [τ−, τ+] ∈ I[0, 1] and κ ∈ [−1, 0]. Let n ∈ ϑ be such that Υ̃Π(n) = τ̃ . Then, n ∈ ξ ( Υ̃Π; τ̃ ) i.e., ξ ( Υ̃Π; τ̃ ) ̸= ϕ, and so 1 ∈ ξ ( Υ̃Π; τ̃ ) . Hence, Υ̃Π(1) ⪰ τ̃ = Υ̃Π(n). If there is n ∈ ϑ such that ϖΠ(1) ≰ ϖΠ(n), then ϖΠ(1) > ϖΠ(n). Hence, n ∈ ξ ( ϖΠ;ϖΠ(n) ) and 1 /∈ ξ ( ϖΠ;ϖΠ(n) ) , a contradiction. Let v ∈ ϑ be such that Υ̃Π(v) = τ̃ . Then, v ∈ ξ ( Υ̃Π; τ̃ ) , which implies that n | (v | v) ∈ ξ ( Υ̃Π; τ̃ ) ∀n ∈ ϑ. Hence, Υ̃Π(n | (v | v)) ⪰ τ̃ = Υ̃Π(v). If there are n, v ∈ ϑ such that ϖΠ(n | (v | v)) ≰ ϖΠ(v), then ϖΠ(n | (v | v)) > ϖΠ(v). It follows that v ∈ ξ ( ϖΠ;ϖΠ(v) ) and n | (v | v) /∈ ξ ( ϖΠ;ϖΠ(v) ) , a contradiction. Let n, v,w ∈ ϑ be such that m̃in{Υ̃Π(v), Υ̃Π(w)} = τ̃ . Then v,w ∈ ξ ( Υ̃Π; τ̃ ) . It follows that (n | (v | w)) | (v | w) ∈ ξ ( Υ̃Π; τ̃ ) . Therefore Υ̃Π((n | (v | w)) | (v | w)) ⪰ τ̃ = m̃in{Υ̃Π(v), Υ̃Π(w)}. If there are n, v,w ∈ ϑ such that ϖΠ((n | (v | w)) | (v | w)) ≰ max{ϖΠ(v), ϖΠ(w)}, then ϖΠ((n | (v | w)) | (v | w)) > max{ϖΠ(v), ϖΠ(w)}. If we take κ := max{ϖΠ(v), ϖΠ(w)}, then v,w ∈ ξ ( ϖΠ;κ ) and (n | (v | w)) | (v | w) /∈ ξ ( ϖΠ;κ ) . This is a contradiction. Consequently, Π = (Υ̃Π, ϖΠ) is a crossing cubic filter of ϑ. Theorem 4. Given a nonempty subset Ψ of ϑ, let ΠΨ = (Υ̃ΠΨ , ϖΠΨ ) be a crossing cubic structure in ϑ defined by ΠΨ = (Υ̃ΠΨ , ϖΠΨ ) : ϑ → I[0, 1] × [−1, 0], n 7→ { (τ̃ , κ) if n ∈ Ψ, (α̃, β) otherwise, where (τ̃ , κ), (α̃, β) ∈ I[0, 1]× [−1, 0] such that τ̃ ≻ α̃ and κ < β. Then, ΠΨ = (Υ̃ΠΨ , ϖΠΨ ) is a crossing cubic filter of ϑ if and only if Ψ is a filter of ϑ. Also, Ψ = ϑΠΨ , where ϑΠΨ = {n ∈ ϑ | Υ̃ΠΨ (n) = Υ̃ΠΨ (1), and ϖΠΨ (n) = ϖΠΨ (1)}. A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 12 of 17 Proof. Assume that ΠΨ = (Υ̃ΠΨ , ϖΠΨ ) is a crossing cubic filter of ϑ. Then, Υ̃ΠΨ (1) = τ̃ and ϖΠΨ (1) = κ by (i) of Definition 9, and so 1 ∈ Ψ. Let n, v ∈ ϑ be such that v ∈ Ψ. Then, Υ̃ΠΨ (v) = τ̃ and ϖΠΨ (v) = κ. It follows from (ii) of Definition 9 that Υ̃ΠΨ (n | (v | v)) ⪰ Υ̃ΠΨ (v) = τ̃ and ϖΠΨ (n | (v | v)) ≤ ϖΠΨ (v) = κ. Hence, Υ̃ΠΨ (n | (v | v)) = τ̃ and ϖΠΨ (n | (v | v)) = κ. This illustrates that n | (v | v) ∈ Ψ. Let v, w ∈ Ψ. Then, Υ̃ΠΨ (v) = τ̃ = Υ̃ΠΨ (w) and ϖΠΨ (v) = κ = ϖΠΨ (w). Using (iii) of Definition 9 and for n ∈ ϑ, we have Υ̃ΠΨ ((n | (v | w)) | (v | w)) ⪰ m̃in{Υ̃ΠΨ (v), Υ̃ΠΨ (w)} = τ̃ and ϖΠΨ ((n | (v | w)) | (v | w)) ≤ max{ϖΠΨ (v), ϖΠΨ (w)} = κ. It follows that Υ̃ΠΨ ((n | (v | w)) | (v | w)) = τ̃ and ϖΠΨ ((n | (v | w)) | (v | w)) = κ. Hence, (n | (v | w)) | (v | w) ∈ Ψ. Therefore, Ψ is a filter of ϑ. Conversely, let Ψ be a filter of ϑ. Since 1 ∈ Ψ, we have Υ̃ΠΨ (1) = τ̃ ⪰ Υ̃ΠΨ (n) and ϖΠΨ (1) = κ ≤ ϖΠΨ (n) for all n ∈ ϑ. Let n, v ∈ ϑ. Then, we have two cases: Case(1). If v ∈ Ψ, then n | (v | v) ∈ Ψ and thus Υ̃ΠΨ (n | (v | v)) = τ̃ = Υ̃ΠΨ (v) and ϖΠΨ (n | (v | v)) = κ = ϖΠΨ (v). Case(2). If v /∈ Ψ, then Υ̃ΠΨ (v) = α̃ and ϖΠΨ (v) = β. Hence, Υ̃ΠΨ (n | (v | v)) ⪰ Υ̃ΠΨ (v) and ϖΠΨ (n | (v | v)) ≤ ϖΠΨ (v). Moreover, let n, v, w ∈ ϑ. Then, we have the following cases: Case(1). If v, w ∈ Ψ. Then, (n | (v | w)) | (v | w) ∈ Ψ. Thus, Υ̃ΠΨ ((n | (v | w)) | (v | w)) = τ̃ = m̃in{Υ̃ΠΨ (v), Υ̃ΠΨ (w)} and ϖΠΨ ((n | (v | w)) | (v | w)) = κ = max{ϖΠΨ (v), ϖΠΨ (w)}. Case(2). If v, w /∈ Ψ. Then, Υ̃ΠΨ ((n | (v | w)) | (v | w)) ⪰ m̃in{Υ̃ΠΨ (v), Υ̃ΠΨ (w)} = α̃ and ϖΠ((n | (v | w)) | (v | w)) ≤ max{ϖΠ(v), ϖΠ(w)} = β. Case(3). If v ∈ Ψ and w /∈ Ψ, then Υ̃ΠΨ (v) = τ̃ , ϖΠΨ (v) = κ, Υ̃ΠΨ (w) = α̃ and ϖΠΨ (w) = β. So, Υ̃ΠΨ ((n | (v | w)) | (v | w)) ⪰ m̃in{Υ̃ΠΨ (v), Υ̃ΠΨ (w)} = α̃ and ϖΠΨ ((n | (v | w)) | (v | w)) ≤ max{ϖΠΨ (v), ϖΠΨ (w)} = β. A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 13 of 17 Case(4). If v /∈ Ψ and w ∈ Ψ, then Υ̃ΠΨ (v) = α̃,ϖΠΨ (v) = β, Υ̃ΠΨ (w) = τ̃ and ϖΠΨ (w) = κ. So, Υ̃ΠΨ ((n | (v | w)) | (v | w)) ⪰ m̃in{Υ̃ΠΨ (v), Υ̃ΠΨ (w)} = α̃ and ϖΠΨ ((n | (v | w)) | (v | w)) ≤ max{ϖΠΨ (v), ϖΠΨ (w)} = β. Hence, ΠΨ = (Υ̃ΠΨ , ϖΠΨ ) is a crossing cubic filter of ϑ. Since Ψ is a filter of ϑ, we get ϑΠΨ = {n ∈ ϑ | Υ̃ΠΨ (n) = Υ̃ΠΨ (1), and ϖΠΨ (n) = ϖΠΨ (1)} = {n ∈ ϑ | Υ̃ΠΨ (n) = τ̃ , and ϖΠΨ (n) = κ} = {n ∈ ϑ|n ∈ Ψ} = Ψ. 4. Crossing Cubic Deductive Systems Here, we present the idea of crossing cubic deductive system and give an example on it. Then, we prove that a crossing cubic deductive system and a crossing cubic filter are a complementary notion. Definition 10. A crossing cubic structure Π = (Υ̃Π, ϖΠ) is called a crossing cubic de- ductive system of ϑ if it satisfies (i) of Definition 9 and the condition (C), where (C) (∀n, v ∈ ϑ) ( Υ̃Π(v) ⪰ m̃in{Υ̃Π(n), Υ̃Π(n | (v | v))}, ϖΠ(v) ≤ max{ϖΠ(n), ϖΠ(n | (v | v))} ) . Example 3. Consider the Sheffer stroke Hilbert algebra ϑ◦ := (ϑ, |) in Example 1 and let Π = (Υ̃Π, ϖΠ) be a crossing cubic structure in ϑ, which is displayed in Table 8. Table 8: Π = (Υ̃Π, ϖΠ) is represented tabularly ϑ Υ̃Π(n) ϖΠ(n) 0 [0.19, 0.54] -0.37 2ϑ [0.24, 0.62] -0.57 3ϑ [0.19, 0.54] -0.37 4ϑ [0.19, 0.54] -0.37 5ϑ [0.26, 0.63] -0.61 6ϑ [0.24, 0.62] -0.57 7ϑ [0.19, 0.54] -0.37 1 [0.41, 0.87] -0.69 It is routine to check that Π = (Υ̃Π, ϖΠ) is a crossing cubic deductive system of ϑ. A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 14 of 17 Next theorem proves that a crossing cubic deductive system and a crossing cubic filter are a complementary notion. Theorem 5. A crossing cubic structure Π = (Υ̃Π, ϖΠ) in ϑ is a crossing cubic deductive system of ϑ if and only if it is a crossing cubic filter of ϑ. Proof. Assume that Π = (Υ̃Π, ϖΠ) is a crossing cubic deductive system of ϑ and let n, v, w ∈ ϑ. Using the condition (A) and (4) of Proposition 1 induces v | ((n | (v | v)) | (n | (v | v))) = 1. It follows from (i) of Definition 9 and the condition (C) that Υ̃Π(n | (v | v)) ⪰ m̃in{Υ̃Π(v), Υ̃Π(v | ((n | (v | v)) | (n | (v | v))))} = m̃in{Υ̃Π(v), Υ̃Π(1)} = Υ̃Π(v) and ϖΠ(n | (v | v)) ≤ max{ϖΠ(v), ϖΠ(v | ((n | (v | v)) | (n | (v | v))))} = max{ϖΠ(v), ϖΠ(1)} = ϖΠ(v). Note that by (2) of Definition 1, (1) and (7) of Proposition 1, we have v | (((v | w) | w) | ((v | w) | w)) = v | (((v | w) | ((w | w) | (w | w))) | ((v | w) | ((w | w) | (w | w)))) = (v | w) | ((v | ((w | w) | (w | w))) | (v | ((w | w) | (w | w)))) = (v | w) | ((v | w) | (v | w)) = 1. Using (i) of Definition 9 and the condition (C), we get Υ̃Π((v | w) | w) ⪰ m̃in{Υ̃Π(v), Υ̃Π(v | (((v | w) | w) | ((v | w) | w)))} = m̃in{Υ̃Π(v), Υ̃Π(1)} = Υ̃Π(v) and ϖΠ((v | w) | w) ≤ max{ϖΠ(v), ϖΠ(v | (((v | w) | w) | ((v | w) | w)))} = max{ϖΠ(v), ϖΠ(1)} = ϖΠ(v). Since by (1) and (2) of Definition 1, we get w | (((v | w) | (v | w)) | ((v | w) | (v | w))) = w | (v | w) = (v | w) | w. Then, we obtain Υ̃Π((v | w) | (v | w)) ⪰ m̃in{Υ̃Π(w), Υ̃Π(w | (((v | w) | (v | w)) | ((v | w) | (v | w))))} = m̃in{Υ̃Π(w), Υ̃Π((v | w) | w)} ⪰ m̃in{Υ̃Π(w), Υ̃Π(v)} A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 15 of 17 and ϖΠ((v | w) | (v | w)) ≤ max{ϖΠ(w), ϖΠ(w | (((v | w) | (v | w)) | ((v | w) | (v | w))))} = max{ϖΠ(w), ϖΠ((v | w) | w)} ≤ max{ϖΠ(w), ϖΠ(v)}. Therefore Υ̃Π((n | (v | w)) | (v | w)) = Υ̃Π((n | (((v | w) | (v | w)) | ((v | w) | (v | w)))) | (((v | w) | (v | w)) | ((v | w) | (v | w)))) ⪰ Υ̃Π((v | w) | (v | w)) ⪰ m̃in{Υ̃Π(w), Υ̃Π(v)} and ϖΠ((n | (v | w)) | (v | w)) = ϖΠ((n | (((v | w) | (v | w)) | ((v | w) | (v | w)))) | (((v | w) | (v | w)) | ((v | w) | (v | w)))) ≤ ϖΠ((v | w) | (v | w)) ≤ max{ϖΠ(w), ϖΠ(v)} Consequently, Π = (Υ̃Π, ϖΠ) is a crossing cubic filter of ϑ. Conversely, suppose that Π = (Υ̃Π, ϖΠ) is a crossing cubic filter of ϑ. By (1), (2) and (3) of Definition 1; and (1), (2), (3), (5) and (6) of Proposition 1, and for every n, v ∈ ϑ, we have v = ((n | n) | (1 | 1)) | (v | v) = ((n | n) | ((v | (v | v)) | (v | (v | v)))) | (v | v) = ((((n | n) | v) | ((n | n) | v)) | (v | v)) | (v | v) = (v | ((n | n) | v)) | ((n | n) | v) = ((((n | n) | v) | v) | v) | (((n | n) | v) | v) = (v | (n | (n | (v | v)))) | (n | (n | (v | v))). It follows from (iii) of Definition 9 that Υ̃Π(v) = Υ̃Π((v | (n | (n | (v | v)))) | (n | (n | (v | v)))) ⪰ m̃in{Υ̃Π(n), Υ̃Π(n | (v | v))} and ϖΠ(v) = ϖΠ((v | (n | (n | (v | v)))) | (n | (n | (v | v)))) ≤ max{ϖΠ(n), ϖΠ(n | (v | v))}. Therefore Π = (Υ̃Π, ϖΠ) is a crossing cubic deductive system of ϑ. Remark 1. By Theorem 5, the crossing cubic deductive system can handle all of the results for the crossing cubic filter covered above in the same way. A. Al-Masarwah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6199 16 of 17 5. Conclusion The crossing cubic structure creates new opportunities to introduce new tools, allowing experiments and studies that were previously not possible through broad applications in various fields. Combining the interval-value fuzzy set and N-structure leads to acquiring the crossing cubic structure. We utilized the crossing cubic structure on filters and deduc- tive systems in Sheffer stroke Hilbert algebras. Also, we defined the notions of the crossing cubic filter and crossing cubic deductive system. Then, we verified many characteristics of these notions. We established conditions suitable for the crossing cubic structure to be a crossing cubic filter. The crossing cubic filters also received further characterization theorems. To create a relationship between crossing cubic filters and filters, we generated crossing cubic filters that are related with filters. We demonstrated that the crossing cubic deductive system can handle all of the results for the crossing cubic filter in the same way. This work is designed to stimulate more research on crossing cubic structures, resulting in fresh and unfamiliar findings. This opens up possibilities for additional research and applications. The concept of crossing cubic structures can be applied in the future. 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