EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5596 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fuzzy (m,n)-Ideals and n-Interior Ideals in Ordered Semigroups P. Khamrot1, A. Iampan2, T. Gaketem2,∗ 1 Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna of Phitsanulok, Phitsanulok, Thailand 2 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand Abstract. In 2019, Ahsan et al. developed the concept of fuzzy (m,n)-ideals in semigroups. Later, in 2022 Tiprachot studied n-interior ideals in ordered semigroup, which is a generalization of fuzzy ideals and interior ideals in semigroups. The aim of this paper is to study the concept of fuzzy (m,n)-ideals and n-interior ideals in ordered semigroup and investigate the properties of fuzzy (m,n)-ideals and n-interior ideals. 2020 Mathematics Subject Classifications: 20M12, 06F05 Key Words and Phrases: Fuzzy (m,n)-ideals, fuzzy n-interior ideals, regular ordered semigroups 1. Introduction The theory of ordered semigroups originated as a generalization of the semigroup theory. The concept of (m,n)-T. Changphas gave ideals in ordered semigroups in [3] which was obtained by generalizing the idea of (m,n)-ideals in semigroups. As a theory of fuzzy set it is tool for dealing with possibilities of uncertainty, connected with the imprecision of states, perceptions, and preferences, and was studied by Zadeh in 1965 [14]. It has been applied to many areas, such as medical science, robotics, com- puter science, information science, control engineering, measure theory, logic, set theory, topology and others. The study of fuzzy sets in semigroups was introduced by Kuroki in 1981. In 2020 Kehayopulu and Tsingelis [5] extended the concept of fuzzy semigroups to the fuzzy ordered semigroups and studied some properties of fuzzy left (right) ideals and fuzzy filters in ordered semigroups. In 2019, Ahsan et al. [9] extended the notion of (m,n)-ideals in semigroups to the notion of fuzzy (m,n)-ideals in semigroups and they characterized (m,n)-regular semigroups by using fuzzy (m,n)-ideals. Tiprachot et al. [12] ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5596 Email addresses: pk g@rmutl.ac.th (P. Khamrot), aiyared.ia@up.ac.th (A. Iampan), thiti.ga@up.ac.th (T. Gaketem) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) P. Khamrot, A. Iampan, T. Gaketem / Eur. J. Pure Appl. Math, 18 (1) (2025), 5596 2 of 12 discussed the notion of n-interior ideals as a generalization of interior ideals and character- ized many classes of ordered semigroups in terms of (m,n)-ideals and n-interior ideals. In 2023, Tiprachot et al. [13] extend n-interior ideals and (m,n)-ideals to hybrid in ordered semigroups. Recently T. Gaketem and P. Khamrot [6] studied concepts interval valued fuzzy (m,n)-ideals in semigroups. In the same year A. Mahoob et al. [8] gave the concepts of structures of fuzzy (m,n)-quasi ideals in ordered semigroups. Other work of ordered semigroup studied more like fuzzy quasi-ideal [4], fuzzy filters [1, 2, 7] and fuzzy prime [10]. The purpose of this paper is to extend the study of fuzzy (m,n)-ideals in semigroups to ordred semigroups. We prove the properties of fuzzy (m,n)-ideals in ordered semigroup and investigate the properties of minimal fuzzy (m,n) ideals, and fuzzy prime (semiprime) (m,n) ideals in semigroups. The relationship between (m,n) ideals and fuzzy (m,n) ideals in ordered semigroups. Finally, we discuss the properties of fuzzy n-interior ideals and the relationship between n-interior ideals and fuzzy n-interior ideals in ordered semigroups. 2. Preliminaries In this section, we review some basic concepts that are necessary to understand our next section. Let (S, ·) be a semigroup ordered semigroup and S,≤) is a partially ordered set. Then (S, ·,≤) is an ordered semigroup if for all a, b, c ∈ S, we have a ≤ b then ac ≤ bc and ca ≤ cb. For a nonempty subset X and Y of ordered semigroup S, we write (X ] := {a ∈ S | a ≤ b for some b ∈ X} and XY := {xy | x ∈ X and y ∈ Y}. It is observed that (1) X ⊆ (X ], (2) if X ⊆ Y, then (X ] ⊆ (Y], (3) ((X ]] = (X ], (4) (X ](Y] ⊆ (XY], (5) ((X ](Y]] = (XY], (6) (X ∪ Y] = (X ] ∪ (Y], (7) (X ∩ Y] = (X ] ∩ (Y]. Let (S, ·,≤) be an ordered semigroup, (∅ ≠)K ⊆ K is called a subsemigroup such that K2 ⊆ K. A left (right) ideal of a ordered semigroup (S, ·,≤) is a non-empty set K of K such that SK ⊆ K (KS ⊆ K) and (K]. By an ideal of an ordered semigroup (S, ·,≤), we mean a non-empty set of S which is both a left and a right ideal of S. Definition 1. [3] A subsemigroup K of an ordered semigroup (S, ·,≤) is called an (m,n)- ideal of S if K satisfies the following conditions: P. Khamrot, A. Iampan, T. Gaketem / Eur. J. Pure Appl. Math, 18 (1) (2025), 5596 3 of 12 (1) KmSKn ⊆ K. (2) K = (K], that is for x ∈ K and y ∈ S, y ≤ x implies y ∈ K. where m,n are non-negative integers. We see that for any δ1, δ2 ∈ [0, 1], we have δ1 ∨ δ2 = max{δ1, δ2} and δ1 ∧ δ2 = min{δ1, δ2}. A fuzzy set δ of a non-empty set T is function from T into unit closed interval [0, 1] of real numbers, i.e., δ : T → [0, 1]. For any two fuzzy sets δ and ϑ of a non-empty set T , define ≥,=,∧ and ∨ as follows: (1) δ ≥ ϑ ⇔ δ(e) ≥ ϑ(e) for all e ∈ T , (2) δ = ϑ ⇔ δ ≥ ϑ and ϑ ≥ δ, (3) (δ ∧ ϑ)(e) = min{δ(e), ϑ(e)} = δ(e) ∧ ϑ(e) for all e ∈ T , (4) (δ ∨ ϑ)(e) = max{δ(e), ϑ(e)} = δ(e) ∨ ϑ(e) for all e ∈ T . For the symbol δ ≤ ϑ, we mean ϑ ≥ δ. The following definitions are types of fuzzy substructures of a semigroup. Definition 2. [11] A fuzzy set δ of a semigroup S is said to be a fuzzy ideal of S if δ(uv) ≥ δ(u) ∨ δ(v) for all u, v ∈ S. Definition 3. [9] A fuzzy subsemigroup of δ if a semigroup S is said to be a fuzzy (m,n)- ideal of S if for all u1, u2, ..., um, v1, v2, ..., vn, z ∈ S and m,n ∈ N, we have δS(u1u2 · · ·umzv1v2 · · · vn) ≥ δS(u1)∧ ϑF(u2)∧ ...∧ δS(um)∧ δS(v1)∧ δS(v2)∧ ...∧ δS(vn). For any element k in an ordered semigroup S, define the set Fk by Fk := {(y, z) ∈ S × S | k ≤ yz}. For two fuzzy sets δ and ϑ on a semigroup S, define the product δ ◦ ϑ as follows: For all k ∈ S, (δ ◦ ϑ)(k) =  ∨ (y,z)∈Fk {δ(y) ∧ ϑ(z)} if Fk ̸= ∅, 0 if Fk = ∅. Definition 4. Let I be a non-empty set of an ordered semigroup S. A characteristic function are respectively defined by λI : S → [0, 1], k 7→ λI(u) := { 1 k ∈ I, 0 k /∈ I, The following definitions are types of fuzzy subsemigroups on ordered semigroups. P. Khamrot, A. Iampan, T. Gaketem / Eur. J. Pure Appl. Math, 18 (1) (2025), 5596 4 of 12 Definition 5. [11] A fuzzy set ξ of an ordered semigroups S is said to be A fuzzy left (right) ideal of S if u ≤ v implies δ(u) ≥ δ(y) for all u, v ∈ S and δ(uv) ≥ δ(v) (δ(uv) ≥ δ(u)) for all u, v ∈ S. Lemma 1. Let K be a nonempty subset of an ordered semigroup S. Then K is a sub- semigroup of S if and only if the characteristic function λK is a fuzzy subsemigroup of S. 3. Fuzzy (m,n)-ideals In this section, we outline the concept of fuzzy (m,n)-ideals and explore their properties within ordered semigroups. Definition 6. A fuzzy subsemigroup δ of an ordered semigroup S is called a fuzzy (m,n)- ideal of S if (1) δ(u1u2 · · ·umkv1v2 · · · vn) ≥ δ(u1)∧ δ(u2)∧ · · · ∧ δ(um)∧ δ(v1)∧ δ(v2)∧ · · · ∧ δ(vn) for all u1, u2, . . . , um, k, v1, v2, . . . vn of S and m,n ∈ N. (2) u ≤ v implies δ(u) ≥ δ(v) for all u, v ∈ S. Example 1. Consider the ordered semigroup S = {w, x, y, z} with the following Cayley table: · w x y z w w w w w x w w z w y w w w w z w w w w and ≤: {(w,w), (x, x), (y, y), (z, z)}. Define a function δ : S → [0, 1] by δ(w) = 0.4, δ(x) = 0.4, δ(y) = 0, δ(z) = 0. Then δ is a fuzzy (m,n)-ideal of S. Theorem 1. Let {δi | i ∈ J } be a family of fuzzy (m,n)-ideals of an ordered semigroup S with δ(u) ≥ δ(v) whenever u ≤ v. Then ∧ i∈F ϑi is a fuzzy (m,n)-ideal of S. Proof. Let u, v ∈ S. Then,∧ i∈J δi(uv) ≥ ∧ i∈J {δi(u) ∧ δ(v)} = ∧ i∈J δi(u) ∧ ∧ i∈J δi(v). Thus, ∧ i∈J δi is a fuzzy subsemigroup of S. Let u1, u2, . . . , um, k, v1, v2, . . . vn ∈ S. Then,∧ i∈J δi(u1u2 · · ·umkv1v2 · · · vn) P. Khamrot, A. Iampan, T. Gaketem / Eur. J. Pure Appl. Math, 18 (1) (2025), 5596 5 of 12 ≥ ∧ i∈J {δi(u1) ∧ δi(u2) · · · ∧ δi(un) ∧ δi(v1) ∧ δi(v2) . . . δi(vn)} = ∧ i∈J δi(u1) ∧ ∧ i∈J δi(u2) · · · ∧ ∧ i∈J δi(un) ∧ ∧ i∈J δi(v1) ∧ ∧ i∈J δi(v2) . . . ∧ i∈J δi(vn). Thus, ∧ i∈J δi is a fuzzy (m,n)-ideal of S. Theorem 2. Let K be a nonempty subset of an ordered semigroup S and m,n are positive integers. Then K is an (m,n)-ideal of S if and only if the characteristic function λK is a fuzzy (m,n)-ideal of S. Proof. Suppose that K is an (m,n)-ideal of S. Then, K is a subsemigroup of S. By Lemma 1, λK is a fuzzy subsemigroup of S. Let u1, u2, . . . um, k, v1, v2, . . . , vn ∈ S. Then the following cases: Case 1 If ui, vj ∈ K for all i ∈ {1, 2, . . . ,m} and j ∈ {1, 2, . . . , n}, then u1u2 · · ·umkv1v2 · · · vn ∈ KmSKn. Thus, λK(u1u2 · · ·umkv1v2 · · · vn) = 1, λK(ui) = 1 for all i ∈ {1, 2, . . . ,m} and λK(rj) = 1 for all j ∈ {1, 2, . . . , n}. So, we have λK(u1u2 · · ·umkv1v2 · · · vn) ≥ λK(u1)∧λK(u2)∧· · ·∧λK(um)∧· · ·∧λK(v1)∧λK(r2)∧· · ·∧λK(rn). Case 2 If ei /∈ K or rj /∈ K for some i ∈ {1, 2, . . . ,m} and j ∈ {1, 2, . . . , n}, then λK(u1u2 · · ·umkv1v2 · · · vn) ≥ λK(u1)∧λK(u2)∧· · ·∧λK(um)∧· · ·∧λK(v1)∧λK(r2)∧· · ·∧λK(rn). Let u, v ∈ S such that u ≤ v and u ∈ K. Then λK(u) = 1. Thus, λK(u) ≥ λK(v). Therefore, λK is a fuzzy (m,n)-ideal of S. Conversely, suppose that λK is a fuzzy (m,n)-ideal of S. Then λK is a fuzzy subsemi- group of S. By Lemma 1, K is a subsemigroup of S. Let u1, u2, . . . um, k, v1, v2, . . . , vn ∈ KmSKn. Then λK(ui) = 1 and λK(vj) = 1 for some i ∈ {1, 2, . . .m} and j ∈ {1, 2, . . . , n}. By assumption, λK(u1u2 · · ·umkv1v2 · · · vn) ≥ λK(u1) ∧ λK(u2) ∧ · · · ∧ λK(um) ∧ · · · ∧ λK(v1) ∧ λK(v2) ∧ · · · ∧ λK(vn). Thus, λK(u1u2 · · ·umkv1v2 · · · vn) = 1. It impiles that, e1e2 · · · emkv1v2 · · · vn ∈ K. Hence, KmSKn ⊆ K. Let u ∈ K such that v ≤ u and v ∈ S. Then λK(u) ≥ λK(v) ≥ 1. Thus, v ∈ K. Therefore, K is an (m,n)-ideal of S. Let δ be a fuzzy set and t ∈ [0, 1]. Define the set Ut := {e ∈ S | δ(e) ≥ t} is called an t-level subset of fuzzy set of δ. Lemma 2. A fuzzy set δ is a fuzzy subsemigroup of a semigroup S if and only if the level set Ut is a subsemigroup of S for all t ∈ [0, 1]. Proof. Let δ be a fuzzy subsemigroup of S and u, v ∈ Ut. Then δ(u1) ≥ t, δ(v) ≥ t. By assumption, δ(uv) ≥ δ(u)∧ δ(ev). Thus, δ(uv) ≥ δ(u)∧ δ(v) ≥ t. It impiles that, uv ∈ Ut. Hence, Ut is a subsemigroup of S. P. Khamrot, A. Iampan, T. Gaketem / Eur. J. Pure Appl. Math, 18 (1) (2025), 5596 6 of 12 Conversely, suppose that Ut is a subsemigroup of S and u, v ∈ S. If u, v ∈ Ut, then δ(u) ≥ t, δ(v) ≥ t. Thus, δ(uv) ≥ δ(u) ∧ δ(v). If u /∈ Ut or v /∈ Ut, then δ(uv) ≥ δ(u) ∧ δ(v). Hence, δ be a fuzzy subsemigroup of S. Theorem 3. A fuzzy set δ is a fuzzy (m,n)-ideal of an ordered semigroup S if and only if the level set Ut is an (m,n)-ideal of S for all t ∈ [0, 1]. Proof. Let δ be a fuzzy (m,n)-ideal of S. Then δ is a fuzzy subsemigroup of S. By Lemma 2, Ut is a subsemigroup of S. Let u1, u2, . . . um, k, v1, v2, . . . , vn ∈ Ut. Then δ(ui) ≥ t, δ(vj) ≥ t for some i ∈ {1, 2, . . . ,m} and j ∈ {1, 2, . . . ,m}. By assumption, δ(u1u2 · · ·umkv1v2 · · · vn) ≥ δ(u1) ∧ δ(u2) ∧ · · · ∧ δ(um) ∧ · · · ∧ δ(v1) ∧ δ(v2) ∧ · · · ∧ δ(vn). Thus, δ(u1u2 · · ·umkv1v2 · · · vn) ≥ t. It impiles that, u1u2 · · ·umkv1v2 · · · vn ∈ Ut. Let u, v ∈ S such that u ≤ v and v ∈ Ut. Then δ(u) ≥ δ(v) ≥ t. Thus, u ∈ Ut. Hence, Ut is an (m,n)-ideal of S. Conversely, suppose that Ut is an (m,n)-ideal of S. Then Ut is a subsemigroup of S. By Lemma 2, δ is a fuzzy subsemigroup of a semigroup S. Let u, v ∈ S such that u ≤ v. We choose δ(v) = t. Thus, v ∈ Ut. By assumption, u ∈ Ut. Then δ(u) ≥ t = δ(v). If δ is not a fuzzy (m,n)-ideal of S, then there exists ui, k, vj ∈ S such that δ(u1u2 · · ·umkv1v2 · · · vn) < δ(u1) ∧ δ(u2) ∧ · · · ∧ δ(um) ∧ · · · ∧ δ(v1) ∧ δ(v2) ∧ · · · ∧ δ(vn). By assumption, we have u1u2 · · ·umkv1v2 · · · vn ∈ Ut. Thus, δ(u1u2 · · ·umkv1v2 · · · vn) ≥ δ(u1) ∧ δ(u2) ∧ · · · ∧ δ(um) ∧ · · · ∧ δ(v1) ∧ δ(v2) ∧ · · · ∧ δ(vn). It is a contradiction. Hence, δ be a fuzzy (m,n)-ideal of S. Definition 7. An (m,n)-ideal K of a ordered semigroup S is called (1) a minimal if for every (m,n)-ideal of J of S such that J ⊆ K, we have J = K. (2) a maximal if for every (m,n)-ideal of J of S such that K ⊆ J , we have J = K. Definition 8. A fuzzy (m,n)-ideal δ of an ordered semigroup S is (1) a minimal if for all fuzzy (m,n)-ideal ξ of S such that ξ ≤ δ, then ξ = δ. (2) a maximal if for all fuzzy (m,n)-ideal ξ of S such that δ ≤ ξ, then ξ = δ. Theorem 4. A non-empty subset K of an ordered semigroup S. Then the following statements ture (1) K is a minimal (m,n)-ideal if and only if λK is a minimal fuzzy (m,n)-ideal. (2) K is a maximal (m,n)-ideal if and only if λK is a maximal fuzzy (m,n)-ideal. Proof. P. Khamrot, A. Iampan, T. Gaketem / Eur. J. Pure Appl. Math, 18 (1) (2025), 5596 7 of 12 (1) Let K be a minimal (m,n)-ideal of S. Then K is an (m,n)-ideal of S. Thus, by Theorem 2, λK is a fuzzy (m,n)-ideal of S. Let J be an (m,n)-ideal of S such that J ⊆ K. Then by Theorem 2, λJ is a fuzzy (m,n)-ideal of S and λJ ≤ λK. Since K is a minimal (m,n)-ideal of S we have J = K. Thus, λJ = λK. Hence, λK is minimal fuzzy (m,n)-ideal of S. Conversely, λK is minimal fuzzy (m,n)-ideal of S. Then λK is a fuzzy (m,n)-ideal of S. Thus, by Theorem 2, K is an (m,n)-ideal of S. Let λJ be a fuzzy (m,n)-ideal of S such that λJ ≤ λK. Then by Theorem 2, J is an (m,n)-ideal of S such that J ⊆ K. Since λK is minimal fuzzy (m,n)-ideal of S we have λJ = λK. Thus, J = K. Hence, K is a minimal (m,n)-ideal of S. (2) If follows from (1). Next, we give the relationship between prime, semiprime (m,n)-ideals and prime, semiprime fuzzy (m,n)-ideals. Definition 9. Let K be an (m,n)-ideal of an ordered semigroup S is called (1) prime if uv ∈ K implies u ∈ K or v ∈ K for all e, h ∈ S, (2) semiprime if u2 ∈ K implies u ∈ K for all u ∈ S. Definition 10. Let δ be a fuzzy (m,n)-ideal of a ordered semigroup is called (1) prime if δ(uv) ≤ δ(u) ∨ δ(v) for all u, v ∈ S, (2) semiprime if δ(u2) ≤ δ(u) for all u ∈ S. Remark 1. Every prime (m,n)-ideal is semiprime (m,n)-ideal in an ordered semigroup. Theorem 5. Let K be a non-empty subset of an ordered semigroup S. Then K is a prime (m,n)-ideal of S if and only if λK is a prime fuzzy (m,n)-ideal of S. Proof. Suppose that K is a prime (m,n)-ideal of S. Then K is an (m,n)-ideal of S. Thus, by Theorem 2 λK is a fuzzy (m,n)-ideal of S. Let u, v ∈ S. Case 1: If uv ∈ K, then u ∈ K or v ∈ K. Thus, λK(uv) = 1 = λK(u) and λK(uv) = −1 = λK(u) or λK(v) = 1 = λK(uv). Hence, λK(uv) ≤ λK(u) ∨ λK(v). Case 2: If uv /∈ K, then λK(uv) = 0. Thus, λK(uv) ≤ λK(u) ∨ λK(v). Therefore, λK is a prime fuzzy (m,n)-ideal of S. Conversely, suppose that λK is a prime fuzzy (m,n)-ideal of S. Then λK is a fuzzy (m,n)-ideal of S. Thus, by Theorem 2, K is an (m,n)-ideal of S. Let u, v ∈ S with uv ∈ K. Then, λK(uv) = 1. If u /∈ K and v /∈ K, then λK(u) = 0 = λK(v). By assumption, λK(uv) ≤ λK(u) ∨ λK(v) . Thus,λK(uv) = 0. It is a contradiction, so u ∈ K or v ∈ K. Hence, K is a prime (m,n)-ideal of S. Theorem 6. Let K be a non-empty subset of an ordered semigroup S. Then K is a semiprime (m,n)-ideal of K if and only if λK is a semiprime fuzzy (m,n)-ideal of S. Proof. It follows from Theorem 5. P. Khamrot, A. Iampan, T. Gaketem / Eur. J. Pure Appl. Math, 18 (1) (2025), 5596 8 of 12 4. Fuzzy n-interior ideals Before, we will review the definition of n-interior ideals in ordered semigroups. Definition 11. [12] A subsemigroup K of an ordered semigroup S is said to be an n- interior ideal of S if SKnS ⊆ K where n is an integer and K = (K], that is for x ∈ K and y ∈ S, y ≤ x implies y ∈ K. Definition 12. A fuzzy subsemigroup δ in an ordered semigroup S is called fuzzy n- interior ideal of S if (1) δ(ukni v) ≥ δ(k1) ∧ δ(k2) ∧ · · · ∧ δ(kn) (2) u ≤ v imples δ(u) ≥ δ(v) for all u, kni , v ∈ S and where i ∈ {1, 2, . . . , n}. Example 2. Consider the ordered semigroup S = {u, v, w, x, y, z} with the following Cay- ley table: · u v w x y z u u u w u u u v u u u u u v w u u u u u v z u u u u u x y u x x u u x z u x x x y z and ≤: {(u, u), (v, v), (w,w), (x, x), (y, y), (z, z)}. Define a function δ : S → [0, 1] by δ(u) = 0.6, δ(v) = 0.2, δ(w) = 0.4, δ(x) = 0.5 δ(y) = 0.3 δ(z) = 0.1. Then δ is a fuzzy n-interior ideal of S. Theorem 7. Let {δi | i ∈ J } be a family of fuzzy n-interior ideals of an ordered semigroup S with δ(u) ≥ δ(v) whenever u ≤ v. Then ∧ i∈F ϑi is a fuzzy n-interior ideal of S. Proof. Let u, v ∈ S. Then,∧ i∈J δi(uv) ≥ ∧ i∈J {δi(u) ∧ δi(v)} = ∧ i∈J δi(u) ∧ ∧ i∈J δi(v). Thus, ∧ i∈J δi is a fuzzy subsemigroup of S. Let u, kni , v ∈ S for all i ∈ {1, 2, . . . , n}. Then,∧ i∈J δi(uk n i v) ≥ ∧ i∈J {δi(k1) ∧ δi(k2) · · · ∧ δi(kn)} = ∧ i∈J δi(k1) ∧ ∧ i∈J δi(k2) · · · ∧ ∧ i∈J δi(kn). Thus, ∧ i∈J δi is a fuzzy n-interior ideal of S. P. Khamrot, A. Iampan, T. Gaketem / Eur. J. Pure Appl. Math, 18 (1) (2025), 5596 9 of 12 Theorem 8. Let K be an ideal of a semigroup S and m,n are positive integers. Then K is an n-interior ideal of S if and only if the characteristic function λK is a fuzzy n-interior ideal of S. Proof. Suppose that K is an n-interior ideal of S. Then K is a subsemigroup of S. Thus, by Theorem 1, λK is a BF subsemigroup of E . Let h, ri, k ∈ E where i ∈ {1, 2, . . . , n}. If ri ∈ K for all i ∈ {1, 2, . . . , n}, then hrni k ∈ K. Thus, λK(ri) = λK(hr n i k) = 1 for all i ∈ {1, 2, . . . , n}. Hence, λK(hr n i k) ≥ λK(r1) ∧ λK(r2) ∧ · · · ∧ λK(rn). If ri /∈ K for some i ∈ {1, 2, . . . , n}, then λK(ri) = 0 for some i ∈ {1, 2, . . . , n}. Thus, λK(hr n i k) ≥ λK(r1) ∧ λK(r2) ∧ · · · ∧ λK(rn). Therefore, λK is a fuzzy n-interior ideal of S. Conversely, suppose that λK is a fuzzy n-interior ideal of S. Then λK is a fuzzy subsemigroup of S. Thus, by Theorem 1, K is a subsemigroup of S. Let rni ∈ SKnS where n is an integer and for all i ∈ {1, 2, . . . , n}. Then λK(r n i ) = 1 for all i ∈ {1, 2, . . . , n}. By assumption, λK(hr n i k) ≥ λK(r1) ∧ λK(r2) ∧ · · · ∧ λK(rn). Thus, λK(hr n i k) = 1 for all i ∈ {1, 2, . . . , n}. Hence, rni ∈ K for all i ∈ {1, 2, . . . , n}. Therefore, K is an n-interior ideal of S. Theorem 9. A fuzzy set δ is a fuzzy n-interior ideal of a semigroup S if and only if the level set Ut is an n-interior ideal of S for all t ∈ [0, 1]. Proof. Let δ be a fuzzy n-interior ideal of S. Then δ is a fuzzy subsemigroup of S. By Lemma 2, Ut is a subsemigroup of S. Let r1, r2, . . . rm, k, h ∈ U (s,t) ϑ . Then δ(ri) ≥ t for some i ∈ {1, 2, . . . , n}. By assumption, δ(hrni k) ≥ δ(r1) ∧ δ(r2) ∧ · · · ∧ δ(rn). Thus, δP (hrni k) ≥ t. It impiles that, rni ∈ Ut. Hence, Ut is an n-interior ideal of S. Conversely, suppose that Ut is an n-interior ideal of S. Then Ut is a subsemigroup of S. By Lemma 2, δ is a fuzzy subsemigroup of S. If δ is not a fuzzy n- interior ideal of E, then there exists ri, k, h ∈ S such that δ(hrni k) < δ(r1)∧ δ(r2)∧ · · · ∧ δ(rn). By assumption, we have hrni k ∈ Ut. Thus, δ(hr n i k) ≥ δ(r1) ∧ δ(r2) ∧ · · · ∧ δ(rn). It is a contradiction. Hence, δ is a fuzzy n-interior ideal of S. Definition 13. An n-interior ideal K of a semigroup S is called a minimal if for every n-interior ideal of J of S such that J ⊆ K, we have J = K. Definition 14. A fuzzy n-interior ideal δ of a semigroup S is a minimal if for all fuzzy n-interior ideal ξ of S such that ξ ≤ δ, then ξ = δ. Theorem 10. A non-empty subset K of a semigroup S is a minimal n-interior ideal if and only if λK is a minimal fuzzy n-interior ideal S. Proof. Let K be a minimal n-interior ideal of S. Then K is an n-interior ideal of S. Thus, by Theorem 8, λK is a fuzzy n-interior ideal of E. Let J be an n-interior ideal of E such that J ⊆ K. Then by Theorem 8, λJ is a fuzzy n-interior ideal of S and λJ ≤ λK. Since K is a minimal n-interior ideal of S we have J = K. Thus, λJ = λK. Hence, λK is minimal fuzzy n-interior ideal of S. P. Khamrot, A. Iampan, T. Gaketem / Eur. J. Pure Appl. Math, 18 (1) (2025), 5596 10 of 12 Conversely, λK is minimal fuzzy n-interior ideal of S. Then λK is a fuzzy n-interior ideal of S. Thus, by Theorem 8, K is an n-interior ideal of S. Let λJ be a fuzzy n-interior ideal of S such that λJ ≤ λK. Then by Theorem 8, J is an n-interior ideal of S such that J ⊆ K. Since λK is minimal fuzzy n-interior ideal of S we have λJ = λK. Thus, J = K. Hence, K is a minimal n-interior ideal of S. Definition 15. An n-interior ideal K of a semigroup S is called a maximalif for every n-interior ideal of J of E such that K ⊆ J , we have J = K. Definition 16. A fuzzy n-interior ideal δ of a semigroup S is a maximal if for all fuzzy n-interior ideal ξ of E such that δ ≤ ξ, then ξ = δ. Theorem 11. A non-empty subset K of a semigroup S is a maximal n-interior ideal if and only if λK is a maximal fuzzy n-interior ideal S. Proof. Let K be a maximal n-interior ideal of S. Then K is an n-interior ideal of S. Thus, by Theorem 8, λK is a fuzzy n-interior ideal of E. Let J be an n-interior ideal of E such that K ⊆ J . Then by Theorem 8, λJ is a fuzzy n-interior ideal of S and λK ≤ λJ . Since K is a maximal n-interior ideal of S we have J = K. Thus, λJ = λK. Hence, λK is maximal fuzzy n-interior ideal of S. Conversely, λK is maximal fuzzy n-interior ideal of S. Then λK is a fuzzy n-interior ideal of S. Thus, by Theorem 8, K is an n-interior ideal of S. Let λJ be a fuzzy n-interior ideal of S such that λK ≤ λJ . Then by Theorem 8, J is an n-interior ideal of S such that K ⊆ J . Since λK is maximal fuzzy n-interior ideal of S we have λJ = λK. Thus, J = K. Hence, K is a maximal n-interior ideal of S. Next, we give the relationship between prime, semiprime n-interior ideals and prime, semiprime fuzzy n-interior ideals. Definition 17. Let K be an n-interior ideal of a semigroup S is called (1) prime if eh ∈ K implies e ∈ K or h ∈ K for all e, h ∈ S, (2) semiprime if e2 ∈ K implies e ∈ K for all e ∈ S. Definition 18. Let δ be a fuzzy n-interior ideal of a semigroup S is called (1) prime if δ(eh) ≤ δ(e) ∨ δ(h) for all e, h ∈ S, (2) semiprime if δ(e2) ≤ δ(e) for all e ∈ S. Remark 2. Every prime n-interior ideal is semiprime n-interior ideal in a semigroup. Theorem 12. Let K be a non-empty subset of a semigroup S. Then the following state- ment holds: (1) K is a prime n-interior ideal of S if and only if λK is a prime fuzzy n-interior ideal of S. P. Khamrot, A. Iampan, T. Gaketem / Eur. J. Pure Appl. Math, 18 (1) (2025), 5596 11 of 12 (2) K is a semiprime n-interior ideal of S if and only if λK is a semiprime fuzzy n-interior ideal of S. Proof. (1) Suppose that K is a prime n-interior ideal of S. Then K is an n-interior ideal of S. Thus, by Theorem 8 λK is a fuzzy n-interior ideal of S. Let e, h ∈ S. Case 1: If eh ∈ K, then e ∈ K or h ∈ K. Thus, λK(eh) = 1 = λK(e) and λK(eh) = 1. Hence, λK(eh) ≤ λK(e) ∨ λK(h). Case 2: If eh /∈ K, then λK(eh) = 0. Thus, λK(eh) ≤ λK(e) ∨ λK(h). Therefore, λK is a prime fuzzy n-interior ideal of S. Conversely, suppose that λK is a prime fuzzy n-interior ideal of S. Then λK is a fuzzy n-interior ideal of S. Thus, by Theorem 8, K is an n-interior ideal of S. Let e, h ∈ S with eh ∈ K. Then, λK(eh) = 1. If e /∈ K and h /∈ K, then λK(e) = 0 = λK(h). By assumption, λK(eh) ≤ λK(e) ∨ λK(h). Thus,λK(eh) = 0.. It is a contradiction, so e ∈ K or h ∈ K. Hence, K is a prime n-interior ideal of S. (2) It follows from (1). 5. Conclusion In this paper, we introduce the concept of bipolar fuzzy (m,n)-ideals in semigroups and investigate their properties. Additionally, we establish the relationship between (m,n)- ideals and fuzzy (m,n)-ideals. Furthermore, we define bipolar fuzzy n-interior ideals in semigroup and prove the relationship between n-interior ideals and fuzzy n-interior ideals. In the future, we plan to explore bipolar (m,n)-ideals and n-interior ideals in ordered semigroups or within the algebraic context. Acknowledgements This research was supported by the University of Phayao and the Thailand Science Research and Innovation Fund (Fundamental Fund 2025, Grant No. 5027/2567). References [1] M. Al-Tahan, B. Davvaz, A. Mahboob, and N. M. Khan. On a generalization of fuzzy filters for ordered semigroups. 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