EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6273 ISSN 1307-5543 – ejpam.com Published by New York Business Global Hybrid Ideals of a Near Algebra P. Narasimha Swamy1, Bhurgula Harika2, Ravikumar Bandaru3, Aiyared Iampan4,∗ 1 Department of Mathematics, GITAM Deemed to be University, Hyderabad, Telangana-502329, India 2 Malla Reddy College of Engineering and Technology, Humanities and Sciences, Hyderabad, Telangana-500100, India 3 Department of Mathematics, School of Advanced Sciences, VIT-AP University, Amaravati-522237, Andhra Pradesh, India 4 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand Abstract. This study aims to explore hybrid ideals in a near algebra. It concludes precise defini- tions and theorems regarding the hybrid ideal, near algebra homomorphism, the Cartesian product of hybrid ideals, and the coset of the hybrid ideal within the context of near algebras. It is demon- strated that an onto homomorphic image, and the Cartesian product of a hybrid ideal of a near algebra, is the hybrid ideal of a near algebra. 2020 Mathematics Subject Classifications: 16Y30, 03E72, 08A72 Key Words and Phrases: Hybrid structure, Near algebra, Ideal, Hybrid field 1. Introduction A near ring is an algebraic system equipped with two binary operations that satisfy all ring axioms, except possibly one of the distributive laws. The concept of a near ring was first presented in a monograph by Pilz [1]. A near algebra is defined as a near ring in which the right scalar domain is a field, and Brown [2] studied its foundational properties. According to Jordan, within the formalism of quantum mechanics, the set of operators forms only a near algebra, making the study of such structures relevant not only for purely axiomatic reasons but also due to their physical applications. In recent years, various generalizations of classical algebraic structures have emerged, especially in the context of non-associative algebras. These include alternative rings, Jor- dan algebras and Γ-rings serve as fertile ground for exploring additive and multiplicative mappings. The behavior of such mappings under weakened structural assumptions has ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6273 Email addresses: swamy.pasham@gmail.com (P. Narasimha Swamy), harika.burgula84@gmail.com (B. Harika), ravimaths83@gmail.com (R. Bandaru), aiyared.ia@up.ac.th (A. Iampan) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 2 of 16 been the subject of extensive research. Notable contributions include the investigation of n-multiplicative mappings on Γ-rings [3], the additivity of multiplicative maps on alter- native rings [4], and additive maps preserving generalized inverses on alternative division algebras [5]. Additionally, Breš ar’s monograph on zero product determined algebras [6] offers a comprehensive treatment of how structural constraints influence the nature of linear preservers and related mappings. In particular, Srinivas and Narasimha Swamy [7] introduced fuzzy near algebras over fuzzy fields, defining fuzzy ideals and examining their algebraic behavior under homomorphisms and direct sums. Parallel to these algebraic developments, fuzzy set theory—pioneered by Zadeh [8]—has evolved into a robust framework for dealing with uncertainty in mathematical modeling. This theory has found widespread applications in engineering, robotics, computer science, and decision theory. Later generalizations, such as hesitant fuzzy sets [9], vague sets, interval-valued sets, and rough sets, have extended this capacity. To address the limita- tions found in these frameworks, Molodtsov [10] introduced soft set theory, a flexible tool for modeling uncertainty, which has since been applied in diverse fields, including game theory, integration theory, and operations research. To unify the strengths of fuzzy and soft set theories, Jun et al. [11] introduced the notion of hybrid structures. These frameworks combine multiple uncertainty paradigms through parameterization over a universe set, allowing for more nuanced representations. Based on these ideas, hybrid ideals, hybrid fields, and hybrid subalgebras have been defined and studied in [12–14]. Building on this foundation, Bhurgula et al. [15] introduced the concept of hybrid near algebras, establishing key structural properties, including closure under homomorphisms and Cartesian products. Motivated by these developments, this work introduces the concept of hybrid ideals in near algebras over hybrid fields. Hybrid structures are then employed to examine structural aspects of near algebras. Throughout this paper, Y denotes a (right) near algebra over a field L. 2. Preliminaries This section provides the necessary background and notational framework for devel- oping hybrid ideals in near algebras. We begin by recalling essential definitions related to near algebras and hybrid structures, including hybrid sets, hybrid subalgebras, and related homomorphisms. These foundational concepts serve as the basis for formalizing the hybrid ideal structure introduced in subsequent sections. Definition 1. [1] Let U be a universal set, P (U) be the power set, L be the set of param- eters, and I be the unit interval. A mapping ξ̃λ := (ξ̃, λ) : L → P (U)× I; q 7→ (ξ̃(q), λ(q)), i.e., the image of q is signified by (ξ̃(q), λ(q)) is named a hybrid structure (HS) in L upon U , where ξ̃ : L → P (U) and λ : L → I are the mappings. Definition 2. [1] Let ξ̃λ be an HS in L upon U . Then the sets ξ̃λ[α, t] = {q ∈ L | ξ̃(q) ⊇ α, λ(q) ≤ t}, P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 3 of 16 ξ̃λ(α, t] = {q ∈ L | ξ̃(q) ⊃ α, λ(q) ≤ t}, ξ̃λ[α, t) = {q ∈ L | ξ̃(q) ⊇ α, λ(q) < t}, ξ̃λ(α, t) = {q ∈ L | ξ̃(q) ⊃ α, λ(q) < t}, are called the [α, t]-hybrid cut (HC), (α, t]-HC, [α, T )-HC, and (α, t)-HC of ξ̃λ corre- spondingly, where α ∈ P (U) and t ∈ I. Apparently, ξ̃λ(α, t) ⊆ ξ̃λ(α, t] ⊆ ξ̃λ[α, t] and ξ̃λ(α, t) ⊆ ξ̃λ[α, t) ⊆ ξ̃λ[α, t]. Definition 3. [13] Let L be a field. An HS ξ̃λ in L upon U is called a hybrid field (HF) of L upon U if the following conditions hold: (i) ξ̃(s+ a) ⊇ ξ̃(s) ∩ ξ̃(a), λ(s+ a) ≤ ∨ {λ(s), λ(a)}, ∀s, a ∈ L (ii) ξ̃(−s) ⊇ ξ̃(s), λ(−s) ≤ λ(s), ∀s ∈ L (iii) ξ̃(sa) ⊇ ξ̃(s) ∩ ξ̃(a), λ(sa) ≤ ∨ {λ(s), λ(a)},∀s, a ∈ L (iv) s ̸= 0 ⇒ ξ̃(s−1) ⊇ ξ̃(s), λ(s−1) ≤ λ(s), ∀s ∈ L. Definition 4. [3] Let ξ̃λ be an HF of a field L upon U and Y be an NA over L. An HS ϱ̃γ in Y upon U is called a hybrid near algebra (HNA) upon HF (ξ̃λ, L) if the resulting conditions hold: (i) ϱ̃(q + ς) ⊇ ϱ̃(q) ∩ ϱ̃(ς), γ(q + ς) ≤ ∨ {γ(q), γ(ς)}, ∀q, ς ∈ Y (ii) ϱ̃(sq) ⊇ ξ̃(s) ∩ ϱ̃(q), λ(sq) ≤ ∨ {λ(s), γ(q)},∀s ∈ L, q ∈ Y (iii) ϱ̃(qς) ⊇ ϱ̃(q) ∩ ϱ̃(ς), γ(qς) ≤ ∨ {γ(q), γ(ς)},∀q, ς ∈ Y (iv) ξ̃(1) ⊇ ϱ̃(q), λ(1) ≤ γ(q),∀q ∈ Y . Definition 5. [3] Let ϱ̃γ and h̃µ be two HSs in L upon U . Then the hybrid intersection of ϱ̃γ and h̃µ is an HS ϱ̃γ ⋒ h̃µ : L → P (U)× I; q 7→ ((ϱ̃∩̃h̃)(q), (γ ∨ µ)(q)),∀q ∈ L, where ϱ̃∩̃h̃ : L → P (U); q 7→ ϱ̃(q) ∩ h̃(q) and γ ∨ µ : L → I; q 7→ ∨{γ(q), µ(q)}. 3. Hybrid ideal of a near algebra The hybrid ideal of a near algebra is introduced in this section, along with some of its features over the hybrid field. Definition 6. Let ξ̃λ be an HF of a field L upon U and Y be a NA over L. An HS ϱ̃γ in Y upon U is called a hybrid ideal of a near algebra (HINA) upon the HF (ξ̃λ, L) if (i) ϱ̃(q + ς) ⊇ ϱ̃(q) ∩ ϱ̃(ς), γ(q + ς) ≤ ∨ {γ(q), γ(ς)}, ∀q, ς ∈ Y (ii) ϱ̃(sq) ⊇ ξ̃(s) ∩ ϱ̃(q), λ(sq) ≤ ∨ {λ(s), γ(q)},∀s ∈ L, q ∈ Y (iii) ξ̃(1) ⊇ ϱ̃(q), λ(1) ≤ γ(q),∀q ∈ Y P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 4 of 16 (iv) ϱ̃(qς) ⊇ ϱ̃(q), γ(qς) ≤ γ(q), ∀q, ς ∈ Y (v) ϱ̃(ς(q + i)− ςq) ⊇ ϱ̃(i), γ(ς(q + i)− ςq) ≤ γ(i),∀q, ς, i ∈ Y, 1 is the unity in L. If ϱ̃γ satisfies (i), (ii), (iii) and (iv) then ϱ̃γ is called a right HINA of Y . If ϱ̃γ satisfies (i), (ii), (iii) and (v) then ϱ̃γ is called a left HINA of Y . Example 1. Let L = Z2 = {0, 1}⊕2,⊗2 be a field. The HS ξ̃λ in L upon U = {u1, u2, u3, u4, u5} is given by L ξ̃ λ 0 {u1, u2, u3, u4} 0.4 1 {u3, u4, u5} 0.5 Then ξ̃λ is an HF in L upon U . Let Y = {0, a3, b3, c3} be a set with two binary operations + by + 0 a3 b3 c3 0 0 a3 b3 c3 a3 a3 0 c3 b3 b3 b3 c3 0 a3 c3 c3 b3 a3 0 . 0 a3 b3 c3 0 0 0 0 0 a3 a3 a3 a3 a3 b3 b3 b3 b3 b3 c3 c3 c3 c3 c3 Clearly, Y forms an NA over L. The HS ϱ̃γ in Y upon U = {u1, u2, u3, u4, u5} is given as follows: Y ϱ̃ γ 0 {u1, u4, u5} 0.5 a3 {u1, u2} 0.6 b3 {u1, u3, u4} 0.8 c3 {u1, u4} 0.9 Therefore, (ϱ̃γ , Y ) is an HINA upon (ξ̃λ, L). Theorem 1. (ϱ̃γ , Y ) is an HINA upon an HF (ξ̃λ, L) if and only if nonempty set ϱ̃γ [α, t] is an ideal of Y upon the field ξ̃λ[α, t],∀t ∈ [0, 1], α ∈ P (U). Proof. Let t ∈ [0, 1] be such that ϱ̃γ [α, t] ̸= ∅ and q, ς ∈ ϱ̃γ [α, t], s ∈ ξ̃λ[α, t]. Then q, ς ∈ Y, s ∈ L and ϱ̃(q) ⊇ α, γ(q) ≤ t, ϱ̃(ς) ⊇ α, γ(ς) ≤ t, ξ̃(s) ⊇ α, λ(s) ≤ t, so that q−ς ∈ Y and sq ∈ Y . Also, ϱ̃(q−ς) ⊇ ϱ̃(q)∩ϱ̃(ς) ⊇ α∩α = α and γ(q−ς) ≤ ∨ {γ(q), γ(ς)} ≤ ∨ {t, t} = t. P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 5 of 16 Also, ϱ̃(sq) ⊇ ξ̃(s) ∩ ϱ̃(q) ⊇ α ∩ α = α and γ(sq) ≤ ∨ {λ(s), γ(q)} ≤ ∨ {t, t} = t. Thus, q − ς ∈ ϱ̃γ [α, t] and sq ∈ ϱ̃γ [α, t]. Hence, ϱ̃γ [α, t] is a subspace of the linear space Y over a field ξ̃λ[α, t]. Suppose that ξ̃λ[α, t] ̸= ∅. We know that ξ̃λ[α, t] is a subfield of L. Let q ∈ Y, i ∈ ϱ̃γ [α, t]. Then i ∈ Y and ϱ̃(i) ⊇ α, γ(i) ≤ t. Thus, i, q ∈ Y, iq ∈ Y and ϱ̃(iq) ⊇ ϱ̃(i) ⊇ α, γ(iq) ≤ γ(i) ≤ t. Thus, iq ∈ ϱ̃γ [α, t]. Let q, ς ∈ Y, i ∈ ϱ̃γ [α, t]. Then i ∈ Y and ϱ̃(i) ⊇ α, γ(i) ≤ t. Thus, i, q, ς ∈ Y, ς(q + i) − ςq ∈ Y and ϱ̃(ς(q + i) − ςq) ⊇ ϱ̃(i) ⊇ α, γ(ς(q + i) − ςq) ≤ γ(i) ≤ t. Thus, ς(q + i) − ςq ∈ ϱ̃γ [α, t]. Hence, ϱ̃γ [α, t] is an ideal of Y over a field ξ̃λ[α, t]. Conversely, suppose that ϱ̃γ [α, t] ̸= ∅ is an ideal of Y . Since (ϱ̃γ , Y ) is an HNA upon (ξ̃λ, L), the first three conditions of HINA holds directly. If possible, suppose that there exists q, ς ∈ Y such that ϱ̃(qς) ⊆ ϱ̃(q), γ(qς) ≥ γ(q). Put u1 = 1 2{ϱ̃(qς) + ϱ̃(q)}, v1 = 1 2{γ(qς) + γ(q)}. Then ϱ̃(qς) ⊂ u1 ⊂ ϱ̃(q) and γ(qς) > v1 > γ(q). Hence, ϱ̃(qς) ⊂ u1, ϱ̃(q) ⊃ u1 and γ(qς) > v1, γ(q) < v1. Since q, ς ∈ Y , we have qς ∈ Y . Thus, ϱ̃(qς) ⊂ u1, ϱ̃(q) ⊃ u1 and γ(qς) > v1, γ(q) < v1. Therefore, qς /∈ ϱ̃γ [u1, t] ∩ ϱ̃γ [v1, t], q ∈ ϱ̃γ [u1, t] ∩ ϱ̃γ [v1, t], ∀ς ∈ Y , which is a contradiction. Hence, ϱ̃(qς) ⊇ ϱ̃(q), γ(qς) ≤ γ(q). If possible, presume that there exists i, q, ς ∈ Y such that ϱ̃(ς(q + i) − ςq) ⊆ ϱ̃(i), γ(ς(q + i)− ςq) ≥ γ(i). Put u2 = 1 2{ϱ̃(ς(q+ i)− ςq) + ϱ̃(i)}, v2 = 1 2{γ(ς(q+ i)− ςq) + γ(i)}. Then ϱ̃(ς(q + i) − ςq) ⊂ u2 ⊂ ϱ̃(i) and γ(ς(q + i) − ςq) > v2 > γ(i). Hence, ϱ̃(ς(q + i) − ςq) ⊂ u2, ϱ̃(i) ⊃ u2 and γ(ς(q + i) − ςq) > v2, γ(i) < v2. Since i, q, ς ∈ Y , ς(q + i) − ςq ∈ Y . Thus, ϱ̃(ς(q + i) − ςq) ⊂ u2, ϱ̃(i) ⊃ u2 and γ(ς(q + i) − ςq) > v2, γ(i) < v2. Therefore, ς(q + i) − ςq /∈ ϱ̃γ [u2, t] ∩ ϱ̃γ [v2, t], i ∈ ϱ̃γ [u2, t] ∩ ϱ̃γ [v2, t], which is a contradiction. Thus, ϱ̃(ς(q + i)− ςq) ⊇ ϱ̃(i), γ(ς(q + i)− ςq) ≤ γ(i). Hence, (ϱ̃γ , Y ) is an HINA upon (ξ̃λ, L). Theorem 2. Let ϱ̃γ and h̃µ be two HINAs of Y upon an HF (ξ̃λ, L). Then ϱ̃γ ⋒ h̃µ is an HINA of Y upon an HF (ξ̃λ, L). Proof. We know that ϱ̃γ ⋒ h̃µ is an HNA of Y upon an HF ξ̃ of L. Let q, ς, i ∈ Y . Then (ϱ̃γ ⋒ h̃µ)(qς) = (ϱ̃∩̃h̃)(qς) = ϱ̃(qς)∩̃h̃(qς) ⊇ (ϱ̃(q)∩̃(h̃(q)) = (ϱ̃ ∩ h̃)(q), (γ ∨ µ)(qς) = ∨{γ(qς), µ(qς)} ≤ ∨{γ(q), µ(q)} = (γ ∨ µ)(q), (ϱ̃γ ⋒ h̃µ)(ς(q + i)− ςq) = (ϱ̃∩̃h̃)(ς(q + i)− ςq) = ϱ̃(ς(q + i)− ςq)∩̃h̃(ς(q + i)− ςq) ⊇ (ϱ̃(i)∩̃(h̃(i)) P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 6 of 16 = (ϱ̃ ∩ h̃)(i), and (γ ∨ µ)(ς(q + i)− ςq) = ∨{γ(ς(q + i)− ςq), µ(ς(q + i)− ςq)} ≤ ∨{γ(i), µ(i)} = (γ ∨ µ)(i). Definition 7. Let Y and Y ′ be two NAs and ω : Y → Y ′ be a mapping. Then: (i) If h̃µ is an HS of Y ′ , then the preimage of h̃µ under ω is the HS in Y upon U demarcated by ω−1(h̃µ)(q) = (ω−1(h̃)(q), ω−1(µ)(q)) = (h̃(ω)(q), µ(ω)(q)), ∀q ∈ Y . (ii) If ϱ̃γ is an HS of Y , then the image of ϱ̃γ under ω is the HS in Y ′ upon U demarcated by ω(ϱ̃)(ς) =  ⋃ q∈ω−1(ς) ϱ̃(q), if ω−1(ς) ̸= ∅ 0, otherwise ω(γ)(ς) =  ∧ q∈ω−1(ς) γ(q), if ω−1(ς) ̸= ∅ 1, otherwise for every ς ∈ Y . Theorem 3. Let Y and Y ′ be two near algebras upon a field L and ω : Y → Y ′ be an onto near algebra homomorphism. If (ϱ̃γ , Y ) is an HINA upon an HF (ξ̃λ, L), then (ω(ϱ̃γ), Y ′ ) is an HINA upon (ξ̃λ, L). Proof. Let q, ς ∈ Y ′ . Then {r | r ∈ ω−1(q + ς)} ⊇ {d+ t | d ∈ ω−1(q) and t ∈ ω−1(ς)} and {r | r ∈ ω−1(qς)} ⊇ {dt | d ∈ ω−1(q) and t ∈ ω−1(ς)}. If ω−1(q) ̸= ∅ and ω−1(ς) ̸= ∅, then ω−1(qς) ̸= ∅. (i) For all q, ς ∈ Y ′∃ d, t ∈ Y such that q = ω(d), ς = ω(t), ω(ϱ̃)(q + ς) = ⋃ r∈ω−1(q+ς) ϱ̃(r) ⊇ ⋃ d∈ω−1(q),t∈ω−1(ς) ϱ̃(d+ t) ⊇ ⋃ d∈ω−1(q),t∈ω−1(ς) ϱ̃(d) ∩ ϱ̃(t) = ( ⋃ d∈ω−1(q) ϱ̃(d)) ∩ ( ⋃ t∈ω−1(ς) ϱ̃(t)) P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 7 of 16 = (ω(ϱ̃)(q)) ∩ (ω(ϱ̃)(ς)) and ω(γ)(q + ς) = ∧ r∈ω−1(q+ς) γ(r) ≤ ∧ d∈ω−1(q),t∈ω−1(ς) γ(d+ t) ≤ ∧ d∈ω−1(q),t∈ω−1(ς) (∨{γ(d), γ(d)}) = ∨{ ∧ d∈ω−1(q) γ(d), ∧ t∈ω−1(ς) γ(t)} = ∨{ω(γ)(q), ω(γ)(ς)}. (ii) Let s ∈ L. Then ω(ϱ̃)(sq) = ⋃ r∈ω−1(sq) ϱ̃(r) ⊇ ⋃ d∈ω−1(q) ϱ̃(sd) ⊇ ⋃ d∈ω−1(q) ξ̃(s) ∩ ϱ̃(d) = ξ̃(s) ∩ ( ⋃ d∈ω−1(q) ϱ̃(q)) = ξ̃(s) ∩ ω(ϱ̃)(q) and ω(γ)(sq) = ∧ r∈ω−1(sq) γ(r) ≤ ∧ d∈ω−1(q) γ(sd) ≤ ∧ d∈ω−1(q) ( ∨ {λ(s), γ(d)}) = ∨ {λ(s), ∧ d∈ω−1(q) γ(d)} = ∨ {λ(s), ω(γ)(q)}. (iii) ξ̃(1) ⊇ ω(ϱ̃)(q) and λ(1) ≤ ω(γ)(q),∀q ∈ Y ′ . P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 8 of 16 (iv) ω(ϱ̃)(qς) = ⋃ r∈ω−1(qς) ϱ̃(r) ⊇ ⋃ d∈ω−1(q),t∈ω−1(ς) ϱ̃(dt) ⊇ ⋃ d∈ω−1(q) ϱ̃(d) = (ω(ϱ̃)(q)) and ω(γ)(qς) = ∧ r∈ω−1(qς) γ(r) ≤ ∧ d∈ω−1(q),t∈ω−1(ς) γ(dt) ≤ ∧ d∈ω−1(q) γ(d) = ω(γ)(q). (v) Let q, ς, i ∈ Y ′ . Then there exist d, t,m ∈ Y such that q = ω(d), ς = ω(t), and i = ω(m). Thus, ω(ϱ̃)(ς(q + i)− ςq) = ⋃ r∈ω−1(ς(q+i)−ςq) ϱ̃(r) ⊇ ⋃ d∈ω−1(q),t∈ω−1(ς),m∈ω−1(i) ϱ̃(t(d+m)− td) ⊇ ⋃ m∈ω−1(i) ϱ̃(i) = (ω(ϱ̃)(i)) and ω(γ)(ς(q + i)− ςq) = ∧ r∈ω−1(ς(q+i)−ςq) γ(r) ≤ ∧ d∈ω−1(q),t∈ω−1(ς),m∈ω−1(i) γ(t(d+m)− td) ≤ ∧ m∈ω−1(i) γ(i) P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 9 of 16 = ω(γ)(i). Hence, (ω(ϱ̃γ), Y ′ ) is an HINA upon (ξ̃λ, L). Theorem 4. Let Y and Y ′ be two near algebras upon a field L and ω : Y → Y ′ be an onto near algebra homomorphism. If h̃µ is an HINA in Y ′ upon (ξ̃λ, L), then ω−1(h̃µ) = (ω−1(h̃), ω−1(µ)) is an HINA in Y upon (ξ̃λ, L). Proof. Let q, ς, i ∈ Y . Then (i) ω−1(h̃)(qς) = h̃(ω(qς)) = h̃(ω(q)ω(ς)) ⊇ h̃(ω(q)) = ω−1(h̃)(q) and ω−1(µ)(qς) = µ(ω(qς)) = µ(ω(q)ω(ς)) ≤ µ(ω(q)) = ω−1(µ)(q). (ii) ω−1(h̃)(ς(q + i) − ςq) = h̃(ω(ς(q + i) − ςq)) = h̃(ω(ς)(ω(q) + ω(i)) − ω(q)ω(ς)) ⊇ h̃(ω(i)) = ω−1(h̃)(i) and ω−1(µ)(ς(q + i) − ςq) = µ(ω(ς(q + i) − ςq)) = µ(ω(ς)(ω(q) + ω(i))− ω(q)ω(ς)) ≤ µ(ω(i)) = ω−1(µ)(i). Therefore, ω−1(h̃µ) = (ω−1(h̃), ω−1(µ)) is an HINA in Y upon (ξ̃λ, L). Theorem 5. Let Y and Y ′ be two near algebras upon a field L and ω : Y → Y ′ be an onto near algebra homomorphism. If h̃µ is an HINA in Y ′ upon (ξ̃λ, L), then ω−1(h̃µ) = (ω−1(h̃), ω−1(µ)) is an HINA in Y upon (ξ̃λ, L). Proof. Let q, ς, i ∈ Y . Then (i) ω−1(h̃)(qς) = h̃(ω(qς)) = h̃(ω(q)ω(ς)) ⊇ h̃(ω(q)) = ω−1(h̃)(q) and ω−1(µ)(qς) = µ(ω(qς)) = µ(ω(q)ω(ς)) ≤ µ(ω(q)) = ω−1(µ)(q). (ii) ω−1(h̃)(ς(q + i) − ςq) = h̃(ω(ς(q + i) − ςq)) = h̃(ω(ς)(ω(q) + ω(i)) − ω(q)ω(ς)) ⊇ h̃(ω(i)) = ω−1(h̃)(i) and ω−1(µ)(ς(q + i) − ςq) = µ(ω(ς(q + i) − ςq)) = µ(ω(ς)(ω(q) + ω(i))− ω(q)ω(ς)) ≤ µ(ω(i)) = ω−1(µ)(i). Therefore, ω−1(h̃µ) = (ω−1(h̃), ω−1(µ)) is an HINA in Y upon (ξ̃λ, L). Definition 8. Let ϱ̃γ and h̃µ be two hybrid structures of near algebras Y and Y ′ over L, respectively. Then the Cartesian product of ϱ̃γ and h̃µ is denoted by ϱ̃γ × h̃µ, is defined to be a hybrid structure ϱ̃γ × h̃µ : Y × Y ′ → P (U) × I; (q, q ′ ) 7→ ((ϱ̃ × h̃)(q, q ′ ), (γ × µ)(q, q ′ )), ∀(q, q′ ) ∈ Y × Y ′ , where ϱ̃ × h̃ : Y × Y ′ → P (U); (q, q ′ ) 7→ ϱ̃(q) ∩ h̃(q ′ ) and γ × µ : Y × Y ′ → I; (q, q ′ ) 7→ ∨ {γ(q), µ(q′ )}. Theorem 6. Let ϱ̃γ and h̃µ be two HINAs of Y and Y ′ upon an HF (ξ̃λ, L). Then ϱ̃γ× h̃µ is an HINA of Y × Y ′ upon (ξ̃λ, L). Proof. Let (q, q ′ ), (ς, ς ′ ), (i, i ′ ) ∈ Y × Y ′ and s ∈ L. Then (i) (ϱ̃× h̃)[(q, q ′ ) + (ς, ς ′ )] = (ϱ̃× h̃)(q + ς, q ′ + ς ′ ) = ϱ̃(q + ς) ∩ h̃(q ′ + ς ′ ) ⊇ [ϱ̃(q) ∩ ϱ̃(ς)] ∩ [h̃(q ′ ) ∩ h̃(ς ′ )] = [ϱ̃(q) ∩ h̃(q ′ )] ∩ [ϱ̃(ς) ∩ h̃(ς ′ )] P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 10 of 16 = (ϱ̃× h̃)(q, q ′ ) ∩ (ϱ̃× h̃)(ς, ς ′ ) and (γ × µ)[(q, q ′ ) + (ς, ς ′ )] = (γ × µ)(q + ς, q ′ + ς ′ ) = ∨ {γ(q + ς), µ(q ′ + ς ′ )} ≤ ∨ { ∨ {γ(q), γ(ς)}, ∨ {µ(q′ ), µ(ς ′ )}} ≤ ∨ { ∨ {γ(q), γ(q′ )}, ∨ {µ(ς), µ(ς ′)}} = ∨ {(γ × µ)(q, q ′ ), (γ × µ)(ς, ς ′ )}. (ii) (ϱ̃× h̃)(s(q, q ′ )) = (ϱ̃× h̃)(sq, sq ′ ) = ϱ̃(sq) ∩ h̃(sq ′ ) ⊇ [ξ̃(s) ∩ ϱ̃(q)] ∩ [ξ̃(s) ∩ h̃(q ′ )] = ξ̃(s) ∩ (ϱ̃(q) ∩ h̃(q ′ )) = ξ̃(s) ∩ (ϱ̃× h̃)(q, q ′ ) and (γ × µ)[s(q, q ′ )] = (γ × µ)(sq, sq ′ ) = ∨ {γ(sq), µ(sq′ )}} ≤ ∨ { ∨ {λ(s), γ(q)}, ∨ {λ(s), µ(q′ )}} ≤ ∨ {λ(s), ∨ {γ(q), µ(q′ )}} = ∨ {λ(s), (γ × µ)(q, q ′ )}. (iii) Let 1 be the unity in L. Since (ϱ̃γ , Y ) and (h̃µ, Y ′ ) are HINAs over HF (ξ̃λ, L), we have ξ̃(1) ⊇ ϱ̃(q), ∀q ∈ Y, λ(1) ≤ γ(q) and ξ̃(1) ⊇ h̃(q ′ ),∀q′ ∈ Y ′ , λ(1) ≤ µ(q ′ ). Then ξ̃(1) ⊇ ϱ̃(q) ∩ h̃(q ′ ) = (ϱ̃× h̃)(q, q ′ ) and λ(1) ≤ ∨ {γ(q), µ(q′ )} = (γ × µ)(q, q ′ ). (iv) (ϱ̃× h̃)[(q, q ′ )(ς, ς ′ )] = (ϱ̃× h̃)(qς, q ′ ς ′ ) = ϱ̃(qς) ∩ h̃(q ′ ς ′ ) ⊇ ϱ̃(q) ∩ h̃(q ′ ) = (ϱ̃× h̃)(q, q ′ ) and (γ × µ)[(q, q ′ )(ς, ς ′ )] = (γ × µ)(qς, q ′ ς ′ ) P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 11 of 16 = ∨ {γ(qς), µ(q′ ς ′ )} ≤ ∨ {γ(q), µ(q′ )} = (γ × µ)(q, q ′ ). (v) (ϱ̃× h̃)[(ς, ς ′ )((q, q ′ ) + (i, i ′ ))− (ς, ς ′ )(q, q ′ ] = (ϱ̃× h̃)[(ς, ς ′ )(q + i, q ′ + i ′ )− (ςq, ς ′ q ′ )] = (ϱ̃× h̃)[(ς(q + i), ς ′ (q ′ + i ′ ))− (ςq, ς ′ q ′ )] = (ϱ̃× h̃)[(ς(q + i)− ςq, ς ′ (q ′ + i ′ ))− ς ′ q ′ )] = ϱ̃(ς(q + i)− ςq) ∩ h̃(ς ′ (q ′ + i ′ ))− ς ′ q ′ ) ⊇ ϱ̃(i) ∩ h̃(i ′ ) = (ϱ̃× h̃)(i, i ′ ) and (γ × µ)[(ς, ς ′ )((q, q ′ ) + (i, i ′ ))− (ς, ς ′ )(q, q ′ ] = (γ × µ)[(ς, ς ′ )(q + i, q ′ + i ′ )− (ςq, ς ′ q ′ )] = (γ × µ)[(ς(q + i), ς ′ (q ′ + i ′ ))− (ςq, ς ′ q ′ )] = (γ × µ)[(ς(q + i)− ςq, ς ′ (q ′ + i ′ ))− ς ′ q ′ )] = γ(ς(q + i)− ςq) ∩ µ(ς ′ (q ′ + i ′ ))− ς ′ q ′ ) ≤ γ(i) ∩ µ(i ′ ) = (γ × µ)(i, i ′ ). Hence, ϱ̃γ × h̃µ is an HINA of Y × Y ′ upon (ξ̃λ, L). Definition 9. Let ϱ̃γ be an HINA of Y upon U and ς ∈ Y . Then the hybrid coset (or coset) of ϱ̃γ is denoted by ς + ϱ̃γ and is defined by (ς + ϱ̃)(q) = ϱ̃(q − ς) and (ς + γ)(q) = γ(q − ς),∀q ∈ Y . Theorem 7. Let ϱ̃γ be an HINA of Y upon U and q, ς ∈ Y . Then q + ϱ̃γ = ς + ϱ̃γ if and only if ϱ̃(q − ς) = ϱ̃(0) and γ(q − ς) = γ(0). Proof. Let q, ς ∈ Y . Suppose that q + ϱ̃γ = ς + ϱ̃γ . Then ϱ̃(q − ς) = (ς + ϱ̃)(q) = (q + ϱ̃)(q) = ϱ̃(q − q) = ϱ̃(0) and γ(q − ς) = (ς + γ)(q) = (q + γ)(q) = γ(q − q) = γ(0). Conversely, suppose that ϱ̃(q − ς) = ϱ̃(0) and γ(q − ς) = γ(0). For every κ ∈ Y , we have (q + ϱ̃)(κ) = ϱ̃(κ− q) = ϱ̃(κ− ς + ς − q) = ϱ̃[(κ− ς) + (ς − q)] ⊇ ϱ̃(κ− ς) ∩ ϱ̃(ς − q) = ϱ̃(κ− ς) ∩ ϱ̃(q − ς) = ϱ̃(κ− ς) ∩ ϱ̃(0) P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 12 of 16 = ϱ̃(κ− ς) = (ς + ϱ̃)(κ) and (q + γ)(κ) = γ(κ− q) = γ(κ− ς + ς − q) = γ[(κ− ς) + (ς − q)] ≤ ∨ {γ(κ− ς), γ(ς − q)} = ∨ {γ(κ− ς), γ(q − ς)} = ∨ {γ(κ− ς), γ(0)} = γ(κ− ς) = (ς + γ)(κ). Thus, q + ϱ̃ ⊇ ς + ϱ̃ and q + γ = ς + γ. Now, (ς + ϱ̃)(κ) = ϱ̃(κ− ς) = ϱ̃(κ− q + q − ς) = ϱ̃[(κ− q) + (q − ς)] ⊇ ϱ̃(κ− q) ∩ ϱ̃(q − ς) = ϱ̃(κ− q) ∩ ϱ̃(0) = ϱ̃(κ− q) = (q + ϱ̃)(κ) and (ς + γ)(κ) = γ(κ− ς) = γ(κ− q + q − ς) = γ[(κ− q) + (q − ς)] ≤ ∨ {γ(κ− q), γ(q − ς)} = ∨ {γ(κ− q), γ(0)} = γ(κ− q) = (q + γ)(κ). Thus, ς + ϱ̃ ⊇ q + ϱ̃ and ς + γ = q + γ. Hence, q + ϱ̃ = ς + ϱ̃ and q + γ = ς + γ. Theorem 8. Let ϱ̃γ be an HINA of Y upon U . Then the following two statements hold: (i) If q+ ϱ̃ = p+ ϱ̃, ς + ϱ̃ = m+ ϱ̃, then (q+ ς) + ϱ̃ = (p+m) + ϱ̃, qς + ϱ̃ = pm+ ϱ̃, and if q + γ = p+ γ, ς + γ = m+ γ, then (q + ς) + γ = (p+m) + γ, qς + γ = pm+ γ P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 13 of 16 (ii) If q + ϱ̃ = p+ ϱ̃, then sq + ϱ̃ = sp+ ϱ̃, and if q + γ = p+ γ, then sq + γ = sp+ γ for all q, ς, p,m ∈ Y and s ∈ L. Proof. (i) Suppose that q+ ϱ̃ = p+ ϱ̃, ς + ϱ̃ = m+ ϱ̃ and q+ γ = p+ γ, ς + γ = m+ γ. Then ϱ̃(q − p) = ϱ̃(0), ϱ̃(ς −m) = ϱ̃(0) and γ(q − p) = γ(0), γ(ς −m) = γ(0). Consider, ϱ̃[(q + ς)− (p+m)] = ϱ̃[(q − p) + (ς −m)] ⊇ ϱ̃(q − p) ∩ ϱ̃(ς −m) = ϱ̃(0) ∩ ϱ̃(0) = ϱ̃(0) and γ[(q + ς) − (p +m)] = γ[(q − p) + (ς −m)] ≤ ∨ {γ(q − p), γ(ς −m)} = ∨ {γ(0), γ(0)} = γ(0). But ϱ̃(0) ⊇ ϱ̃[(q + ς) − (p + m)] and γ(0) ≤ γ[(q + ς) − (p + m)]. Therefore, ϱ̃[(q+ ς)− (p+m)] = ϱ̃(0) and γ[(q+ ς)− (p+m)] = γ(0). Thus, (q+ ς)+ ϱ̃ = (p+m)+ ϱ̃ and (q + ς) + γ = (p+m) + γ. Again, ϱ̃[qς − pm] = ϱ̃[pm− qς] = ϱ̃(pm− qm+ qm− qς) = ϱ̃[(p− q)m+ q(ς + (−ς +m))− qς)] ⊇ ϱ̃((p− q)m) ∩ ϱ̃(q(ς + (−ς +m))− qς) = ϱ̃(p− q) ∩ ϱ̃(ς −m) = ϱ̃(0) ∩ ϱ̃(0) = ϱ̃(0) and γ[qς − pm] = γ[pm− qς] = γ(pm− qm+ qm− qς) = γ[(p− q)m+ q(ς + (−ς +m))− qς] ≤ ∨ {γ((p− q)m), γ(q(ς + (−ς +m))− qς)} ≤ ∨ {γ(p− q), γ(ς −m)} = ∨ {γ(0), γ(0)} = γ(0). But ϱ̃(0) ⊇ ϱ̃[(qς) − (pm)] and γ(0) ≤ γ[(qς) − (pm)]. Therefore, ϱ̃[qς − pm] = ϱ̃(0) and γ[qς − pm] = γ(0). Thus, qς + ϱ̃ = pm+ ϱ̃ and qς + γ = pm+ γ. (ii) Suppose that q + ϱ̃ = p + ϱ̃ and q + γ = p + γ. Then ϱ̃(q − p) = ϱ̃(0) and γ(q − p) = γ(0). Now, ϱ̃(sq − sp) = ϱ̃(s(q − p)) ⊇ ϱ̃(q − p) = ϱ̃(0) and γ(sq − sp) = γ(s(q − p)) ≤ γ(q − p) = γ(0). But ϱ̃(0) ⊇ ϱ̃(sq − sp) and γ(0) ≤ γ(sq − sp). Therefore, ϱ̃(sq − sp) = ϱ̃(0) and γ(sq − sp) = γ(0). Thus, sq + ϱ̃ = sp+ ϱ̃ and sq + γ = sp+ γ. Definition 10. Let ϱ̃γ be an HINA of Y upon U . Then the set of all cosets of ϱ̃γ is Y/ϱ̃γ = {ς + ϱ̃γ | ς ∈ Y }, where Y/ϱ̃ = {ς + ϱ̃ | ς ∈ Y } and Y/γ = {ς + γ | ς ∈ Y }. P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 14 of 16 Theorem 9. Let ϱ̃γ be an HINA of Y upon U . Then Y/ϱ̃γ is a near algebra with respect to the operations defined by (q + ϱ̃) + (ς + ϱ̃) = (q + ς) + ϱ̃ (q + ϱ̃)(ς + ϱ̃) = qς + ϱ̃ s(q + ϱ̃) = sq + ϱ̃ (q + γ) + (ς + γ) = (q + ς) + γ (q + γ)(ς + γ) = qς + γ s(q + γ) = sq + γ for all q, ς ∈ Y and s ∈ L. Proof. A direct verification shows that Y/ϱ̃γ is a linear space. Let q+ϱ̃, ς+ϱ̃, j+ϱ̃ ∈ Y/ϱ̃ and q + γ, ς + γ, j + γ ∈ Y/γ, where q, ς, j ∈ Y . Then [(q + ϱ̃)(ς + ϱ̃)](j + ϱ̃) = (qς + ϱ̃)(j + ϱ̃) = (qς)j + ϱ̃ = (q + ϱ̃)[(ς + ϱ̃)(j + ϱ̃)] and [(q + γ)(ς + γ)](j + γ) = (qς + γ)(j + γ) = (qς)j + γ = (q + γ)[(ς + γ)(j + γ)]. This shows that Y/ϱ̃γ is a semigroup under multiplication. Consider, [(q + ϱ̃) + (ς + ϱ̃)](j + ϱ̃) = ((q + ς) + ϱ̃)(j + ϱ̃) = (q + ς)j + ϱ̃ = (qj + ςj) + ϱ̃ = (qj + ϱ̃) + (ςj + ϱ̃) = (q + ϱ̃)(j + ϱ̃) + (ς + ϱ̃)(j + ϱ̃) and [(q + γ) + (ς + γ)](j + γ) = ((q + ς) + γ)(j + γ) = (q + ς)j + γ = (qj + ςj) + γ = (qj + γ) + (ςj + γ) = (q + γ)(j + γ) + (ς + γ)(j + γ). Let s ∈ L. Then P. N. Swamy et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6273 15 of 16 (s(q + ϱ̃))(ς + ϱ̃) = (sq + ϱ̃)(ς + ϱ̃) = (sq)ς + ϱ̃ = s(qς) + ϱ̃ = s(qς + ϱ̃) = s((q + ϱ̃)(ς + ϱ̃)) and (s(q + γ))(ς + γ) = (sq + γ)(ς + γ) = (sq)ς + γ = s(qς) + γ = s(qς + γ) = s((q + γ)(ς + γ)). Hence, Y/ϱ̃γ is a near algebra over L. 4. Conclusion In this work, we combined soft sets and fuzzy sets in near algebras, thereby developing a novel structure of hybrid ideals. This manuscript provided an in-depth exploration of hybrid ideals within a near algebra, elucidating their unique properties through a system- atic investigation. Theoretical findings were substantiated with illustrative examples. We also introduced the notions of HINA homomorphism, the Cartesian product of HINA, and the coset of HINA. Future research can be extended to investigate the hybrid gamma near algebra, the hybrid ideal of a gamma near algebra, and explore the concept of hybrid near algebra on anti-fuzzy sets. Acknowledgements This research was supported by University of Phayao and Thailand Science Research and Innovation Fund (Fundamental Fund 2025, Grant No. 5027/2567). References [1] G. Pilz. Near-ring: The Theory and Its Applications. North Holland Publishers, Amsterdam, 1983. [2] H. Brown. Near algebras. 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