EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5837 ISSN 1307-5543 – ejpam.com Published by New York Business Global Almost (m,n)-quasi-ideals and Fuzzy Almost (m,n)-quasi-ideals in Ordered Semigroups P. Khamrot1, P. Chaisuwan, P. Keawton, C. Wangsamphao2, T. Gaketem3,∗ 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 Education, University of Phayao, Phayao, Thailand 3 Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand Abstract. The ordered semigroups are algebraic systems consisting of a non-empty set, an asso- ciative binary operation, and a partial order compatible with this binary operation. This paper aims to define almost (m,n)-quasi-ideals and fuzzy almost (m,n)-qausi-ideals in ordered semi- groups. We prove the union of almost (m,n)-quasi-ideals, including almost (m,n)-qausi-ideals in ordered semigroups. In class, fuzzifications are the same. Finally, we connect relation almost (m,n)-quasi-ideals and fuzzy almost (m,n)-quasi-ideals in ordered semigroups. 2020 Mathematics Subject Classifications: 20M12, 06F05 Key Words and Phrases: Ordered almost (m,n)-quasi-ideal, fuzzy ordered almost (m,n)-quasi- ideal, ordered semigroups 1. Introduction Ordered semigroup is an algebraic structure in a binary operation satisfying associa- tive property and a partial order with the compatibility. The concepts of quasi-ideals of semigroups presented by Steinfied [1] in 1956. The dealing with various problems related to uncertain conditions by fuzzy sets by Zadeh in 1965, [2]. These concepts were applied in many areas, such as medical science, theoretical physics, robotics, computer science, control engineering, information science, measure theory, logic, set theory, and topology. Rosenfeld studied concent of fuzzy subgroups and fuzzy ideals. In 1981 Kuroki studied the typers of fuzzy subsemigroups. In the same year Satko and Grosek [3] discussed concept of an almost-ideal (A-ideal) in a semilattice. And S. Bogdanovic [4] gave the concept of al- most bi-ideals in semigroups. In 2019, S. Suebsung et al. [5] investigated almost ideals and fuzzy almost ideals in ternary semigroups. In 2020, Chinram et al. [6] discussed almost ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5837 Email addresses: thiti.ga@up.ac.th (T. Gaketem) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) P. Khamrot et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5837 2 of 13 interior ideals and weakly almost interior ideals in semigroups and studied the relationship between almost interior ideals and weakly almost interior ideals in semigroups. The research of almost ideals studied in semihypergroups such that in 2021, P. Muang- doo et al. [7] studied almost bi-hyperideals and their fuzzification of semihypergroups. W. Nakkhasen et al. [8] discussed fuzzy, almost interior hyperideals of semihypergroups. in 2022, S. Suebsung et al. [9] introduced almost ideals in ordered semigroups. In the same year T. Gaketem and P. Khamrot [10] explored the concept of almost ideals within the framework of bipolar fuzzy sets, specifically focusing on bipolar fuzzy almost bi-ideals in semigroups. In 2023, R. Chinram et. al [11] studied concept almost (m,n)-quasi-ideals in semigroups and their fuzzifications. In the same year, T. Gaketem and P. Khamrot [12] studied bipolar fuzzy almost interior ideals in semigroups. In 2024, T. Gaketem and P. Khamrot [13] discussed bipolar fuzzy almost ideals in semigroups. In addition, almost ideal’s work also has many studies, such as almost ideals in ordered semigroup [14], al- most ideals in semirings [15], almost ideals in ternary semiring [16], etc. Not long ago in 2025 P. Khamrot et al. [17] studied fuzzy (m,n)-ideals and n-interior ideals in ordered semigroups. In the same year, P. Khamrot et al. [18] extend concepts fuzzy (m,n)-ideals and n-interior ideals to biploar fuzzy sets. In this paper we extend the definition of almost (m,n)-quasi-ideals in semigroups go to ordered semigroups. We discussed the union of almost (m,n)-quasi-ideals, including almost (m,n)-quasi-ideals in ordered semigroups. In class, fuzzifications are the same. Finally, we connect relation almost (m,n)-quasi-ideals and fuzzy almost (m,n)-quasi-ideals in ordered semigroups. 2. Preliminaries Now, we recall the concept of ordered semigroups and fuzzy sets. Additionally, their preliminary results are provided. Definition 1. [19]. Let T be a set with a binary opeation · and a binary opeation relation ≤. Then (T, ·,≤) is called an ordred semigroup if (1) (T, ·) is a semigroup, (2) (T,≤) is a partially ordered set, (3) for all a, b, c ∈ T, we have a ≤ b then ac ≤ bc and ca ≤ cb. For a nonempty subset X and Y of ordered semigroup T, we write (X] := {a ∈ T | 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], P. Khamrot et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5837 3 of 13 (4) (X](Y] ⊆ (XY], (5) ((X](Y]] = (XY], (6) (X ∪Y] = (X] ∪ (Y], (7) (X ∩Y] = (X] ∩ (Y]. Let (T, ·,≤) be an ordered semigroup, (∅ ̸=)K ⊆ T is called a subsemigroup such that K2 ⊆ K. A left (right) ideal of a ordered semigroup (T, ·,≤) is a non-empty set K of T such that SK ⊆ K (KS ⊆ K) and (K]. By an ideal of an ordered semigroup (T, ·,≤), we mean a non-empty set of T which is both a left and a right ideal of T. Definition 2. [20] A subsemigroup K of an ordered semigroup (T, ·,≤) is called an (m,n)- ideal of T if K satisfies the following conditions: (1) KmTKn ⊆ K. (2) K = (K], that is for x ∈ K and y ∈ T, y ≤ x implies y ∈ K. where m,n are non-negative integers. Definition 3. [20] An non-empty subset K of an ordered semigroup (T, ·,≤) is called an (m,n)-quasi-ideal of T if K satisfies the following conditions: (1) (KmT] ∩ (TKn] ⊆ K. (2) K = (K], that is for x ∈ K and y ∈ T, y ≤ x implies y ∈ K. where m,n are non-negative integers. Definition 4. [9] A nonempty subset of K an ordered semigroup T is called a left ordered almost ideal of T if (tK] ∩ K ̸= ∅ for all t ∈ S. Definition 5. [9] A nonempty subset of K an ordered semigroup T is called a right ordered almost ideal of T if (Kt] ∩ K ̸= ∅ for all t ∈ T . For any hi ∈ [0, 1], i ∈ F , define ∨ i∈F hi := sup i∈F {hi} and ∧ i∈F hi := inf i∈F {hi}. We see that for any h, r ∈ [0, 1], we have h ∨ r = max{h, r} and h ∧ r = min{h, r}. A fuzzy set ϑ in a nonempty set T is a function from T into the unit closed interval [0, 1] of real numbers, i.e., ϑ : T → [0, 1]. For any two fuzzy sets ϑ and ξ of a non-empty set T, define the symbol as follows: (1) ϑ ≤ ξ ⇔ ϑ(h) ≤ ξ(h) for all h ∈ T, P. Khamrot et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5837 4 of 13 (2) ϑ = ξ ⇔ ϑ ≤ ξ and ξ ≤ ϑ, (3) (ϑ ∧ ξ)(h) = min{ϑ(h), ξ(h)} = ϑ(h) ∧ ξ(h) for all h ∈ T, (4) (ϑ ∨ ξ)(h) = max{ϑ(h), ξ(h)} = ϑ(h) ∨ ξ(h) for all h ∈ T, (5) the support of ϑ instead by supp(ϑ) = {h ∈ T | ϑ(h) ̸= 0}. For the symbol ϑ ≥ ξ, we mean ξ ≤ ϑ. If K ⊆ T ̸= ∅, then the characteristic function χK of T is a function from T into {0, 1} defined as follows: χK(x) = { 1 if x ∈ K 0 otherwise. for all x ∈ T Lemma 1. If I and L are nonempty subsets of an oredred semigroup T, then the following are true: (1) χI ∧ χL = χI∩L. (2) If I ⊆ L, then χI ⪯ χL. (3) χI ◦ χL = χIL. Definition 6. Let T be an ordered semigroup and Fu be a non-empty subset of T, we define the set Fu by Fu := {(x, y) ∈ T× T | u ≤ xy}. Definition 7. [21] Let ϑ and η be fuzzy sets of an ordered semigroup T. The product of fuzzy subsets ϑ and η of T is defined as follow, for all u ∈ T (ϑ ◦ η)(u) =  ∨ (x,y)∈Fu {ϑ(x) ∧ η(y)} if Fu ̸= ∅, 0 if Fu = ∅. For u ∈ T and t ∈ (0, 1], a fuzzy point xt of a set T is a fuzzy subset of T defined by xt(e) = { t if e = u, 0 otherwise. For k ∈ N, let ϑn := ϑ ◦ ϑ ◦ · · · ◦ ϑ︸ ︷︷ ︸ n-times . Lemma 2. [14] If φ, ν and ξ are fuzzy sets of an ordered semigroup S, then the following are true: (1) If φ ⪯ ν, then φn ⪯ νn P. Khamrot et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5837 5 of 13 (2) If φ ⪯ ν, then φ ◦ ξ ⪯ ν ◦ ξ. (3) If φ ⪯ ν, then φ ∨ ξ ⪯ ν ∨ ξ. (4) If φ ⪯ ν, then φ ∧ ξ ⪯ ν ∧ ξ. (5) If φ ⪯ ν, then supp(φ) ⪯ supp(ν). For a fuzzy set φ of an ordered semigroup S, we define (φ] : S → [0, 1] by (φ] := sup a≤b φ(b) for all a ∈ S. Lemma 3. [14] If φ, ν and ξ are fuzzy sets of an ordered semigroup S, then the following are true: (1) φ ⪯ (φ]. (2) If φ ⪯ ν, then (φ] ⪯ (ξ]. (3) If φ ⪯ ν, then (φ ◦ ξ] ⪯ (ν ◦ ξ] and (ξ ◦ φ] ⪯ (ξ ◦ ν]. Lemma 4. [14] If φ is a fuzzy set of an ordered semigroup S, then the following are equivalent. (1) If a ≤ b, then φ(a) ⪯ φ(b). (2) (φ] = φ. Definition 8. [14] A fuzzy set δ of a semigroup T is said to be a fuzzy ideal of T if δ(uv) ≥ δ(u) ∨ δ(v) for all u, v ∈ T. Definition 9. [19] A fuzzy subsemigroup δ of a ordered semigroup T is said to be a fuzzy (m,n)-ideal of T if (1) δ(u1u2 · · ·umzv1v2 · · · vn) ≥ δ(u1) ∧ δ(u2) ∧ ... ∧ δ(um) ∧ δ(v1) ∧ δ(v2) ∧ ... ∧ δ(vn) for all u1, u2, ..., um, v1, v2, ..., vn, z ∈ T and m,n ∈ N. (2) If u1 ≤ u2, then δ(u1) ≥ δ(u2), for all u1, u2 ∈ T. Definition 10. [9] A nonempty subset of K an ordered semigroup T is called a left ordered almost ideal (right ordered almost ideal) of T if (tK]∩K ̸= ∅ ((K]t]∩K ̸= ∅) for all t ∈ T. 3. Main Results In this section, we define the almost (m,n)-quasi-ideal and fuzzy almost (m,n)-quasi- ideal in ordered semigroup. We prove some basic interesting properties of almost (m,n)- quasi-ideal and fuzzy almost (m,n)-quasi-ideal in ordered semigroup. Definition 11. A non-empty subset B on an ordered semigroup T is called an almost (m,n)-quasi-ideal of T if (Bmt] ∩ (tBn] ∩B ̸= ∅ for all t ∈ T where m,n ∈ {1, 2, ..., n}. P. Khamrot et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5837 6 of 13 Example 1. (1) An almost (1, 1)-ideal of an ordered semigroup T is a right almost quasi- ideal of T. (2) Consider the ordered semigroup Z6 under the usual addition and the partial ordered ≤:= {(a, a) | a ∈ Z6}. We have A = {1, 4, 5} is an almsot (1, 1)-quasi-ideal of Z6. (3) The almost (m,n)-quasi-ideal of an ordered semigroup T is not (m,n)-quasi-ideal of T. Theorem 1. Every (m,n)-quasi-ideal of an ordered semigroup T is an almost (m,n)- quasi-ideal of T. Proof. Assume that B is an (m,n)-quasi-ideal of T and let t ∈ T. Then (Bmt]∩(tBn] ⊆ (BmT] ∩ (TBn]. Thus (BmtBn] ∩B ̸= ∅. We conclude that B is an almost (m,n)-quasi- ideal of T. Theorem 2. Let B1 and B2 be two non-empty subsets of an ordered semigroup T such that B1 ⊆ B2. If B1 is an almost (m,n)-quasi-ideal of T, then B2 is also an almost (m,n)-quasi-ideal of T. Proof. Let B1 be an almost (m,n)-quasi-ideal of T with B1 ⊆ B2 and let t ∈ T Then (Bm 1 t] ∩ (tBn 1 ] ⊆ (Bm 2 t] ∩ (tBn 2 ] Thus, (B m 2 tBn 2 ] ∩B2 ̸= ∅. Hence, B2 is an almost (m,n)-quasi-ideal of T. Corollary 1. Let B1 and B2 be almost (m,n)-quasi-ideals of an ordered semigroup T. Thus B1 ∪B2 is also an almost (m,n)-quasi-ideal of T. Proof. Since B1 and B1 are subsets of B1 ∪B2, by Theorem 2, B1 ∪B2 is an almost (m,n)-quasi-ideal of T. Corollary 2. Let B1 and B2 be nonempty subsets of an ordered semigroup T. If B1 is an almost (m,n)-quasi-ideal of T, then B1 ∪B2 is an almost (m,n)-quasi-ideal of T. Proof. By Corollary 1, and B1 ⊆ B1 ∪B2. Thus, B1 ∪B2 is an almost (m,n)-quasi- ideal of T. Corollary 3. The finite union of almost (m,n)-quasi-ideals of an ordered semigroup T is an almost (m,n)-quasi-ideal of T. Example 2. Consider the ordered semigroup Z6 under the usual addtion and the partial ordered ≤:= {(a, a) | a ∈ Z6}. We have A = {1, 4, 5} and B = {1, 2, 5} are almost (1, 1)- quasi-ideals of Z6. Consider A ∩B = {1, 5} then A ∩B2 ∩ 2A ∩B ∩ A ∩B = ∅. Thus, A ∩B = {1, 5} is not an almost (1, 1)-quasi-ideal of Z6. Definition 12. A fuzzy set ϑ on an ordered semigroup T is called a fuzzy almost (m,n)- quasi-ideal of T if (ϑm ◦ xt] ∧ (xt ◦ ϑn] ∧ ϑ ̸= 0. for any fuzzy point xt ∈ T where m,n ∈ {1, 2, ..., n}. P. Khamrot et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5837 7 of 13 Theorem 3. If ϑ is a fuzzy almost (m,n)-quasi-ideal of an ordered semigroup T and ξ is a fuzzy subset of T such that ϑ ≤ ξ, then ξ is a fuzzy almost (m,n)-quasi-ideal of T. Proof. Suppose that ϑ is a fuzzy almost (m,n)-quasi-ideal of T and ξ is a fuzzy subset of T such that ϑ ≤ ξ. Then for any fuzzy points xt ∈ T, we obtain that (ϑm◦xt]∧(xt◦ϑn]∧ϑ ̸= 0. Thus, (ϑm ◦ xt] ∧ (xt ◦ ϑn] ∧ ϑ ≤ (ξm ◦ xt] ∧ (xt ◦ ξn] ∧ ξ ̸= 0. Hence, (ξm ◦ xt] ∧ (xt ◦ ξn] ∧ ξ ̸= 0. Therefore, ξ is a fuzzy almost (m,n)-quasi-ideal of T. The following result is an obvious of Theorem 3. Theorem 4. Let ϑ and ξ be fuzzy almost (m,n)-quasi-ideal of an ordered semigroup T. Then ϑ ∨ ξ is also a fuzzy almost (m,n)-quasi-ideal of T. Proof. Since ϑ ≤ ϑ ∨ ξ, by Theorem 3, ϑ ∨ ξ is also a fuzzy almost (m,n)-quasi-ideal of T. Theorem 5. If ϑ fuzzy almost (m,n)-quasi-ideal of an ordered semigroup T and ξ is a fuzzy set, then ϑ ∨ ξ is a fuzzy almost (m,n)-quasi-ideal of T. Proof. By Theorem 3, and ϑ ≤ ϑ ∨ ξ. Thus, ϑ ∨ ξ is a fuzzy almost (m,n)-quasi-ideal of T. Corollary 4. Let T be an ordered semigroup. Then the finite maximum of fuzzy almost (m,n)-quasi-ideals of T is a fuzzy almost (m,n)-quasi-ideal of T. Example 3. Consider the ordered semigroup Z6 under the usual addition and the partial ordered ≤:= {(a, a) | a ∈ Z6}. ϑ : Z6 → [0, 1] is defined by ϑ(0) = 0, ϑ(1) = 0.2, ϑ(2) = 0, ϑ(3) = 0, ϑ(4) = 0.5, ϑ(5) = 0.4 and ν : Z6 → [0, 1] is defined by ν(0) = 0, ν(1) = 0.8, ν(2) = 0.4, ν(3) = 0.3, ν(4) = 0, ν(5) = 0.3. We have ϑ and ν are fuzzy almost (1, 1)-quasi-ideals of Z6 but (ϑ ∧ ν)(0) is not a fuzzy almost (1, 1)-quasi-ideal of Z6. Remark The minimum is not fuzzy almost (1, 1)-quasi-ideal of an ordered semigroup by Example 3 Lemma 5. Let A be a subset of T and n ∈ N ∪ {0}. Then (χA) n = χAn Theorem 6. Let B be a nonempty subset of an ordered semigroup T. Then B is an almost (m,n)-quasi-ideal of T if and only if χB is a fuzzy almost (m,n)-quasi-ideal of T. Proof. Suppose thatB is an almost (m,n)-quasi-ideal of T. Then (Bmt]∩(tBn]∩B ̸= ∅ for all t ∈ T and m,n ∈ {1, 2, ..., n}. Thus there exists c ∈ T such that c ∈ (Bmt] ∩ (tBn] and c ∈ B. Let xt ∈ T and t ∈ (0, 1]. Then ((χBm ◦xt] ◦∧(xt ◦χBn ])(c) ̸= 0 and χB(c) ̸= 0 and m,n ∈ {1, 2, ..., n}. Thus, ((χBm ◦ xt] ◦ ∧(xt ◦ χBn ] ∧ χB)(c) = (((χB) m ◦ xt] ◦ ∧(xt ◦ (χB) n])(c) ̸= 0 and m,n ∈ {1, 2, ..., n}. So, (χBm ◦ xt] ◦ ∧(xt ◦ χBn ] ∧ χB) ̸= 0. Hence, χB is a fuzzy almost (m,n)-quasi-ideal of T. P. Khamrot et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5837 8 of 13 Conversely, suppose that χB is a fuzzy almost (m,n)-quasi-ideal of T and let xt ∈ T and t ∈ (0, 1] where m,n ∈ {1, 2, ..., n}. Then ((χBm ◦ xt] ◦ ∧(xt ◦ χBn ] ∧ χB) ̸= 0. Thus, there exists c ∈ B such that ((χBm ◦ xt] ◦ ∧(xt ◦ χBn ] ∧ χB)(c) ̸= 0. It implies that ((χBm ◦ xt] ◦ ∧(xt ◦ χBn ])(c) ̸= 0 and χB(c) ̸= 0 and m,n ∈ {1, 2, ..., n}. Hence c ∈ T such that c ∈ (Bmt] ∩ (tBn] and c ∈ B. So (Bmt] ∩ (tBn] ∩ B ̸= ∅ for all t ∈ T and m,n ∈ {1, 2, ..., n}. We conclude that B is an almost (m,n)-quasi-ideal of T. Theorem 7. Let ϑ be a fuzzy subset of an ordered semigroup T. Then ϑ is a fuzzy almost (m,n)-quasi-ideal of T if and only if supp(ϑ) is an almost (m,n)-quasi-ideal of T. Proof. Assume that ϑ is a fuzzy almost (m,n)-quasi-ideal of T and let xt ∈ T and t ∈ (0, 1]. Then (ϑm◦xt]∧(xt◦ϑn]∧ϑ ̸= 0 for allm,n ∈ {1, 2, ..., n}. Thus, there exists z ∈ T such that ((ϑm◦xt]∧(xt◦ϑn]∧ϑ)(z) ̸= 0. So ϑ(z) ̸= 0 and z = a1a2 · · · amx = xb1b2 · · · bn for some a1a2 · · · am, b1b2 · · · bn ∈ T such that ϑ(a1) ̸= 0, ϑ(a2) ̸= 0, · · · , ϑ(am) ̸= 0 ϑ(b1) ̸= 0, ϑ(b2) ̸= 0, · · ·ϑ(bn) ̸= 0. Thus, a1a2 ·am, b1b2 · · · bn ∈ supp(ϑ). It implies that ((χsupp(ϑ)m◦ xt]∧(xt◦χsupp(ϑ)n ]∧χsupp(ϑ))(z) ̸= 0. Hence, (χsupp(ϑ)m ◦xt]∧(xt◦χsupp(ϑ)n ]∧χsupp(ϑ) ̸= 0. for all m,n ∈ {1, 2, ..., n}. Therefore, χsupp(ϑ) is a fuzzy almost (m,n)-quasi-ideal of T. By Theorem 6, supp(ϑ) is an almost (m,n)-quasi-ideal of T. Conversely, suppose that supp(ϑ) is an almost almost (m,n)-quasi-ideal of T. By Theorem 6, χsupp(ϑ) is a fuzzy almost (m,n)-quasi-ideal of T. Then for any fuzzy point xt ∈ T and m,n ∈ {1, 2, ..., n}, we have (χsupp(ϑ)m ◦ xt] ∧ (xt ◦ χsupp(ϑ)n ] ∧ χsupp(ϑ) ̸= 0. Thus, there exists z ∈ T such that ((χsupp(ϑ)m ◦ xt] ∧ (xt ◦ χsupp(ϑ)n ] ∧ χsupp(ϑ))(z) ̸= 0. So ϑ(z) ̸= 0 and z = a1a2 · · · amx = xb1b2 · · · bn for some a1a2 · · · am, b1b2 · · · bn ∈ T such that ϑ(a1) ̸= 0, ϑ(a2) ̸= 0, · · · , ϑ(am) ̸= 0 ϑ(b1) ̸= 0, ϑ(b2) ̸= 0, · · ·ϑ(bn) ̸= 0. Thus, a1a2 ·am, b1b2 · · · bn ∈ supp(ϑ). So, there exists z ∈ T such that ((ϑm◦xt]∧(xt◦ϑn]∧ϑ)(z) ̸= 0. Hence, (ϑm ◦ xt] ∧ (xt ◦ ϑn] ∧ ϑ ̸= 0 for all m,n ∈ {1, 2, ..., n}. Therefore, ϑ is a fuzzy almost (m,n)-quasi-ideal of T. Next, we investigate connection between minimal and maximal almost (m,n)-quasi- ideals and minimal and maximal fuzzy almost (m,n)-quasi-ideals of ordered semigroups. Definition 13. An almost (m,n)-quasi-ideal B of an ordered semigroup T is called (1) a minimal if for any almost (m,n)-quasi-ideal R of T if whenever R ⊆ B, then R = B, (2) a maximal if for any almost (m,n)-quasi-ideal R of T if whenever B ⊆ R, then R = B. Definition 14. A fuzzy almost (m,n)-quasi-ideal ϑ of an ordered semigroup T is called (1) a minimal if for any fuzzy almost (m,n)-quasi-ideal ξ of T if whenever ξ ≤ ϑ, then supp(ξ) = supp(ϑ), (2) a maximal if for any fuzzy almost (m,n)-quasi-ideal ξ of T if whenever ϑ ≤ ξ, then supp(ξ) = supp(ϑ). P. Khamrot et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5837 9 of 13 Theorem 8. Let B be a nonempty subset of an ordered semigroup T. Then (1) B is a minimal almost (m,n)-quasi-ideal of T if and only if χB is a minimal fuzzy almost (m,n)-quasi-ideal of T. (2) B is a maximal almost (m,n)-quasi-ideal of T if and only if χB is a maximal fuzzy almost (m,n)-quasi-ideal of T. Proof. (1) Assume that B is a minimal almost (m,n)-quasi-ideal of T. Then B is an almost (m,n)-quasi-ideal of T. Thus by Theorem 6, χB is a fuzzy almost (m,n)-quasi-ideal of T. Let ξ be a fuzzy almost (m,n)-quasi-ideal of T such that ξ ≤ χB. Then by Theorem 7, supp(ξ) is an almost (m,n)-quasi-ideal of T such that supp(ξ) ⊆ supp(χB) = B. Since B is minimal we have supp(ξ) = B = supp(χB). Therefore, χB is minimal. Conversely, suppose that χB is a minimal fuzzy almost (m,n)-quasi-ideal of T. Then χB is a fuzzy almost (m,n)-quasi-ideal of T. Thus by Theorem 6, B is an almost (m,n)-quasi-ideal of T. Let R be an almost (m,n)-quasi-ideal of T such that R ⊆ B. Then χR is a fuzzy almost (m,n)-quasi-ideal of T such that χR ≤ χB. Hence, R = supp(χR) = supp(χB) = B. Therefore, B is minimal. (2) Assume that B is a maximal almost (m,n)-quasi-ideal of T. Then B is an almost (m,n)-quasi-ideal of T. Thus by Theorem 6, χB is a fuzzy almost (m,n)-quasi-ideal of T. Let ξ be a fuzzy almost (m,n)-quasi-ideal of T such that χB ≤ ξ. Then by Theorem 7, supp(ξ) is an almost (m,n)-quasi-ideal of T such that B = supp(χB) ⊆ supp(ξ). Since B is maximal we have supp(ξ) = B = supp(χB). Therefore, χB is maximal. Conversely, suppose that χB is a maximal fuzzy almost (m,n)-quasi-ideal of T. Then χB is a fuzzy almost (m,n)-quasi-ideal of T. By Theorem 6, B is an almost (m,n)- quasi-ideal of T. Let R be an almost (m,n)-quasi-ideal of T such that B ⊆ R. Then χR is a fuzzy almost (m,n)-quasi-ideal of T such that χB ≤ χR. Since χB is a maximal we have R = supp(χR) = supp(χB) = B. Therefore, B is maximal. Corollary 5. Let T be an ordered semigroup. Then T has no proper almost (m,n)-quasi- ideal if and only if supp(ϑ) = T for every fuzzy almost (m,n)-quasi-ideal ϑ of T. Next, we give definition of prime (resp., semiprime, strongly prime) almost (m,n)- quasi-ideals and prime (resp., semiprime strongly prime) fuzzy almost (m,n)-quasi-ideals. We study the relationships between prime (resp., semiprime strongly prime) almost (m,n)- quasi-ideals and their fuzzification of ordered semigroups. Definition 15. Let B be an almost (m,n)-quasi-ideal of an ordered semigroup T. Then we called (1) B is a prime if for any two almost (m,n)-quasi-ideals N and H of T such that NH ⊆ B implies that N ⊆ B or H ⊆ B. P. Khamrot et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5837 10 of 13 (2) B is a semiprime if for any almost (m,n)-quasi-ideal N of T such that N2 ⊆ N implies that N ⊆ B. (3) B is a strongly prime if for any almost (m,n)-quasi-ideals N and H of T such that NH ∩ HN ⊆ B implies that N ⊆ B or H ⊆ B. Definition 16. A fuzzy almost (m,n)-quasi-ideal ϑ on an ordered semigroup T. Then we called (1) ϑ is a prime if for any two fuzzy almost (m,n)-quasi-ideals ξ and ν of T such that ξ ◦ ν ≤ ϑ implies that ξ ≤ ϑ or ν ≤ ϑ. (2) ϑ is a semiprime if for any fuzzy almost (m,n)-quasi-ideal ξ of T such that ξ ◦ ξ ≤ ξ implies that ξ ≤ ϑ. (3) ϑ is a strongly prime if for any two fuzzy almost (m,n)-quasi-ideals ξ and ν of T such that (ξ ◦ ν) ∧ (ν ◦ ξ) ≤ ϑ implies that ξ ≤ ϑ or ν ≤ ϑ. It is clearly, every fuzzy strongly prime almost (m,n)-quasi-ideal of a ternary semigroup is a fuzzy prime almost (m,n)-quasi-ideal, and every fuzzy prime almost (m,n)-quasi-ideal of a ternary semigroup is a fuzzy semiprime almost (m,n)-quasi-ideal. Theorem 9. Let B be a nonempty subset of an ordered semigroup T. Then B is a prime almost (m,n)-quasi-ideal of T if and only if χB is a prime fuzzy almost (m,n)-quasi-ideal of T. Proof. Suppose that B is a prime almost (m,n)-quasi-ideal of T. Then B is an almost (m,n)-quasi-ideal of T. Thus by Theorem 6, χB is a fuzzy almost (m,n)-quasi-ideal of T. Let ϑ and ξ be fuzzy almost (m,n)-quasi-ideals such that ϑ◦ξ ≤ χB. Assume that ϑ ≰ χB and ξ ≰ χB. Then there exist h, r ∈ T such that ϑ(h) ̸= 0 and ξ(r) ̸= 0. While χB(h) = 0 and χB(r) = 0. Thus, h ∈ supp(ϑ) and r ∈ supp(ξ), but h, r /∈ B. So supp(ϑ) ⊈ B and supp(ξ) ⊈ B. Since supp(ϑ) and supp(ξ) are almost (m,n)-quasi-ideals of T we have supp(ϑ) supp(ξ) ⊈ B. Thus, there exists m = pq for some p ∈ supp(ϑ) and q ∈ supp(ξ) such that m ∈ B. Hence χB(m) = 0 implies that (ϑ ◦ ξ)(m) = 0. Since ϑ ◦ ξ ≤ χB. we have p ∈ supp(ϑ) and q ∈ supp(ξ). Thus ϑ(p) ̸= 0, and ξ(q) ̸= 0. It implies that (ϑ ◦ ξ)(m) = ∨ (p,q)∈Fm {ϑ(p) ∧ ξ(q)} ≠ 0 It is a contradiction so ϑ ≤ χB or ξ ≤ χB. Therefore χB is a prime fuzzy almost (m,n)- quasi-ideal of T. Conversely, suppose that χB is a prime fuzzy almost (m,n)-quasi-ideal of T. Then χB is a fuzzy almost (m,n)-quasi-ideal of T. Thus by Theorem 6, B is an almost (m,n)- quasi-ideal of T. Let N and H be almost (m,n)-quasi-ideal of T such that NH ⊆ B. Then χB and χH are fuzzy almost (m,n)-quasi-ideals of T. By Lemma 1 χN ◦ χH = χNH ≤ χB. By assumption, χN ≤ χB or χB ≤ χH. Thus N ⊆ B or H ⊆ B. We conclude that B is a prime almost (m,n)-quasi-ideal of T. P. Khamrot et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5837 11 of 13 Theorem 10. Let B be a nonempty subset of an ordered semigroup T. Then B is a semiprime almost (m,n)-quasi-ideal of T if and only if χB is a semiprime fuzzy almost (m,n)-quasi-ideal of T. Proof. Suppose that B is a semiprime almost (m,n)-quasi-ideal of T. Then B is an almost (m,n)-quasi-ideal of T. Thus by Theorem 6, χB is a fuzzy almost (m,n)-quasi- ideal of T. Let ϑ be fuzzy almost (m,n)-quasi-ideal such that ϑ ◦ ϑ ≤ χB. Assume that ϑ ≰ χB. Then there exist h ∈ T such that ϑ(h) ̸= 0. While χB(h) = 0. Thus, h ∈ supp(ϑ), but h /∈ B. So supp(ϑ) ⊈ B. Since supp(ϑ) are almost (m,n)-quasi-ideals of T we have supp(ϑ) ⊈ B. Thus h ∈ supp(ϑ) such that m ∈ B. Hence χB(m) = 0 implies that (ϑ ◦ ϑ)(m) = 0. Since ϑ ◦ ϑ ≤ χB. we have p ∈ supp(ϑ). Thus ϑ(p) ̸= 0. It implies that (ϑ ◦ ϑ)(m) = ∨ (p,b)∈Fm {ϑ(p) ∧ ϑ(b)} ≠ 0 It is a contradiction so ϑ ≤ χB. Therefore χB is a semiprime fuzzy almost (m,n)-quasi- ideal of T. Conversely, suppose that χB is a semiprime fuzzy almost (m,n)-quasi-ideal of T. Then χB is a fuzzy almost (m,n)-quasi-ideal of T. Thus by Theorem 6, B is an almost (m,n)- quasi-ideal of T. Let N be almost (m,n)-quasi-ideal of T such that N2 ⊆ B. Then χN is a fuzzy almost (m,n)-quasi-ideal of T. By Lemma 1 χN ◦ χN = χN2 ≤ χB. By assumption, χN ≤ χB. Thus N ⊆ B. We conclude that B is a semiprime almost (m,n)-quasi-ideal of T. Theorem 11. Let B be a nonempty subset of an ordered semigroup T. Then B is a strongly prime almost (m,n)-quasi-ideal of T if and only if χB is a fuzzy strongly prime almost (m,n)-quasi-ideal of T. Proof. Suppose thatB is a strongly prime almost (m,n)-quasi-ideal of T. ThenB is an almost (m,n)-quasi-ideal of T. Thus by Theorem 6, χB is a fuzzy almost (m,n)-quasi-ideal of T. Let ϑ and ξ be fuzzy almost (m,n)-quasi-ideals of T such that (ϑ ◦ ξ)∧ (ξ ◦ϑ) ≤ χB. Assume that ϑ ≰ χB and ξ ≰ χB. Then there exist h, b ∈ T such that ϑ(h) ̸= 0 and ξ(b) ̸= 0. While χB(h) = 0 and χB(b) = 0. Thus, h ∈ supp(ϑ) and b ∈ supp(ξ), but h, b /∈ B. So, supp(ϑ) ⊈ B and supp(ξ) ⊈ B. Hence, there exists m ∈ [supp(ϑ) supp(ξ)]∩ (supp(ϑ) supp(ξ)) such that m /∈ B. Thus, χB(m) = 0. Since m ∈ supp(ϑ) supp(ξ) and m ∈ supp(ξ) supp(ϑ) we have m = dk and m = gq for some d, q ∈ supp(ϑ), and for some k, g ∈ supp(ξ). we have (ϑ ◦ ξ)(m) = ∨ (d,e)∈Fm {ϑ(d) ∧ ξp(k)} ≠ 0. Similarly (ξ ◦ ϑ)(m) = ∨ (g,q)∈Fm {ξ(g) ∧ ϑ(q)}. P. Khamrot et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5837 12 of 13 So (ϑ ◦ ξ)(m) ∧ (ξ ◦ ϑ)(m) ̸= 0. It is a contradiction so, ϑ ≤ χB or ξ ≤ χB. Therefore, χB is a fuzzy strongly prime almost (m,n)-quasi-ideal of T. Conversely, suppose that χB is a fuzzy strongly prime almost (m,n)-quasi-ideal of T. Then χB is a fuzzy almost (m,n)-quasi-ideal of T. Thus, by Theorem 6, B is an almost (m,n)-quasi-ideal of T. Let N and H be almost (m,n)-quasi-ideals of T such that NH ∩ HN ≤ B. Then χN and χH are fuzzy almost (m,n)-quasi-ideals of T. By Lemma 1 χNH = χN◦χH and χHN = χH◦χN. Thus (χN◦χH)∧(χH◦χN) = χNH∧χHN = χNH∩HN ≤ χB. By assumption, χN ≤ χB and χN ≤ χB. Thus N ⊆ B or H ⊆ B. We conclude that B is a strongly prime almost (m,n)-quasi-ideal of T. 4. Conclusion The aim paper gives the concept of almost (m,n)-quasi-ideals in ordered semigroups. The union of two almost (m,n)-quasi-ideals is also an almost (m,n)-quasi-ideal in ordered semigroups, and the results in class fuzzifications are the same. In Theorems 6, 7, 8, 10, and 11. Finally we prove that if K is a (minimal/ maximal/prime/semiprime/strongly prime) almost (m,n)-quasi-ideals of T if and only if χA is (minimal/ maximal/prime/semiprime strongly prime) fuzzy almost (m,n)-quasi-ideals of T. In future work, we can study other kinds of almost ideals and their fuzzifications in ordered ternary semigroup. Acknowledgements This research was supported by the School of Science University of Phayao. References [1] O. Steinfeid. Uber die quasiidale von albgtuppen. Publ. 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