EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5646 ISSN 1307-5543 – ejpam.com Published by New York Business Global Generalization of bi-antiideals in semigroups Madeleine Al Tahan1, Sarka Hoskova-Mayerova2,∗, Saba Al-Kaseasbeh3 1 Department of Mathematics and Statistics, Abu Dhabi University, United Arab Emirates 2 Department of Mathematics and Physics, University of Defence, Czech Republic 3 Department of Mathematis, Tafila Technical University, Jordan Abstract. Algebraic structure consisting of a set together with an associative internal binary operation on it, so called semigroup has applications in different fields of science. For a better understanding of these applications, semigroups are characterized through their subsets. Fuzzy sets deal with uncertainties, and because many real-life problems have an associated algebraic structure, fuzzification of these structures makes sense and is useful. This paper investigates the generalization of bi-antiideals in semigroups and their fuzzification to enhance understanding of algebraic structures with uncertainties. Building upon prior research, we define and explore (m,n)- bi-antiideals as an extension of bi-antiideals, studying their properties through theoretical analysis and illustrative examples. We introduce fuzzy (m,n)-bi-antiideals by leveraging fuzzy set theory to model uncertainties, establishing a connection with (m,n)-bi-antiideals via level sets. 2020 Mathematics Subject Classifications: 06F05, 08A72 Key Words and Phrases: Antiideal, bi-antiideal, (m,n)-bi-antiideal, Fuzzy (m,n)-bi-antiideal, level set 1. Introduction The initial paper on semigroups emerged in 1905 as a concise work by L.E. Dickson. However, the true inception of the theory occurred in 1928 when A.K. Suschkewitsch [20] published a paper of paramount significance. In contemporary language, he demonstrated that within any finite semigroup, there exists a “kernel” (referred to as a simple ideal), and he comprehensively characterized the structure of finite simple semigroups. Semigroups provide a foundational framework for understanding how elements combine under certain operations, and their applications span across multiple branches of mathematics and vari- ous interdisciplinary fields such as Coding theory, Automata, etc. For more details about semigroup terminology and history, we refer to [6]. The history of fuzzy sets can be traced back to the mid-20th century when Lotfi Zadeh [21] introduced the concept of fuzzy logic in 1965. Zadeh’s groundbreaking idea ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5646 Email addresses: altahan.madeleine@gmail.com (M. Al Tahan), sarka.mayerova@unob.cz (S. Hoskova-Mayerova), saba.alkaseasbeh@gmail.com (S. Al-Kaseasbeh) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Al Tahan, S. Hoskova-Mayerova, S. Al-Kaseasbeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5646 2 of 11 challenged the traditional binary approach of classical set theory by allowing elements to possess degrees of membership in sets, rather than being strictly classified as either inside or outside a set. This innovative notion found its roots in the observation that many real-world concepts are not easily definable in precise terms. Fuzzy sets quickly garnered attention across various disciplines, including algebraic structures. The combination of the two concepts led to the launch of fuzzy algebraic structures. The latter was established by Rosenfeld [17] in 1971 when he introduced fuzzy groups. There are many research items in the literature characterizing semigroups through its (fuzzy) subsets. For example, (fuzzy) filters of a semigroup were studied in [3, 4, 10], and (fuzzy) ideals of a semigroup were studied in [12–15]. For further details, we refer to the work cited in [7, 9, 11, 16]. Inspired by the literature, our present work sets out on an exploration of specific subsets within semigroups and fuzzifies them. The remaining part is constructed as follows. After an Introduction, in Section 2 we present some results about (fuzzy) antiideals and (fuzzy) bi-antiideals of semigroups that are used in the subsequent sections. In Section 3 we generalizes bi-antiideals to (m,n)-bi-antiideals, discuss some of their properties, and give some non-trivial examples. In Section 4 we fuzzify (m,n)-bi- antiideals by introducing fuzzy (m,n)-bi-antiideals of a semigroup. Moreover, we link the two new notions by means of level sets. 2. (Fuzzy) Left(right) antiideals and bi-antiideals of a semigroup In this section, we present some definitions and results that are used throughout the paper. Antiideals were introduced by Schwarz [19] and were studied and generalized by Iseksi [8, 9]. Other antiideals of semigroups were introduced. For example, Al-Tahan and Sarka [5] introduced interior antiideals of a semigroup and investigated their properties and Al-Kaseasbeh et al. [18] studied antiideals of a semiring. A non-empty set X with an associative binary operation is called a semigroup and a non-empty subset A of X is a subsemigroup of X if it is a semigroup. If X has an identity, then it is called a monoid. As simple examples, the set of non-negative even integers under standard addition is a semigroup and the set of positive real numbers under standard multiplication is a semigroup. Definition 1. [8] Let (X, ·) be a semigroup and A ̸= ∅ ⊆ X. Then (i) A is a left antiideal of X if XA ∩A = ∅; (ii) A is a right antiideal of X if AX ∩A = ∅; (iii) A is an antiideal of X if it is both a left and right antiideal of X. Example 1. Let (K, ·) be the semigroup of integers greater than 1 under standard multi- plication of integers and A = {2, 3}. Then A is an antiideal of K. This is clear as KA ∩A = AK ∩A = {4, 6, 8, 9, . . .} ∩ {2, 3} = ∅. M. Al Tahan, S. Hoskova-Mayerova, S. Al-Kaseasbeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5646 3 of 11 We present an example of an infinite antiideal. Example 2. Let M = {1, 2, 3, 4, . . .} and define the semigroup (M,⋆) as follows. x ⋆ y = { 1 if x is an odd number; y otherwise. Then A = {3, 5, 7, 9, . . .} is an antiideal of M . This is clear as AM ∩A = {1} ∩A = ∅. Al-Tahan et al. [2] introduced bi-antiideals of a semigroup. We present some of their results. Definition 2. [2] Let (X, ·) be a semigroup and A ̸= ∅ ⊆ X. Then A is a bi-antiideal of X if AXA ∩A = ∅. Example 3. Let (K, ·) be the semigroup defined in Example 1 and B = {2, 3, 4}. Then B is a bi-antiideal of K. Moreover, it is not a left(right) antiideal of K. This is clear as BK ∩B = {4} ≠ ∅. Proposition 1. [4] Evey left(right) antiideal of a semigroup X is a bi-antiideal of X. Fuzzy sets were introduced by Zadeh [21] in 1965 to accommodate uncertainties that classical sets fail to deal with. In a fuzzy set, the element’s membership is a real number in the unit interval. Definition 3. [21] Let X be a universal set, I = [0, 1], and µ : X → I. Then a fuzzy set of X is given as: A = {(x, µ(x)) : x ∈ X}. Here µ(x) denotes the membership’s grade of the element x in X. Definition 4. [7] For the fuzzy sets µ1, µ2 of X, the fuzzy sets µ1 ∧ µ2, µ1 ∨ µ2 of X are defined as follows. (µ1 ∧ µ2)(x) = min{µ1(x), µ2(x)} for all x ∈ X. (µ1 ∨ µ2)(x) = max{µ1(x), µ2(x)} for all x ∈ X. Definition 5. [7] Let X1, X2 be non-empty sets and µ1, µ2 be fuzzy sets of X1, X2 respec- tively. Then the fuzzy set µ = µ1 × µ2 of X1 ×X2 is defined as follows. µ((x1, x2)) = min{µ1(x1), µ2(x2)} for all x1 ∈ X1, x2 ∈ X2. Definition 6. [2] Let (X, ·) be a semigroup and µ : X → [0, 1] be a non-zero fuzzy set of X. Then (i) µ is a fuzzy left antiideal of X if µ(ra) ∧ µ(a) = 0 for all r, a ∈ X; (ii) µ is a fuzzy right antiideal of X if µ(ar) ∧ µ(a) = 0 for all r, a ∈ X; M. Al Tahan, S. Hoskova-Mayerova, S. Al-Kaseasbeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5646 4 of 11 (iii) µ is a fuzzy antiideal of X if µ is a fuzzy left antiideal of X and a fuzzy right antiideal of X; (iv) µ is a fuzzy bi-antiideal of X if µ(xry) ∧ µ(x) ∧ µ(y) = 0 for all r, x, y ∈ X. Definition 7. [7] Let (X, ·) be a semigroup, µ be a non-zero fuzzy set of X, and t ∈ [0, 1]. Then the level set µt is defined as follows. µt = {x ∈ X : µ(x) ≥ t}. Example 4. Let (K, ·) be the semigroup defined in Example 1 and define the fuzzy sets µ1, µ2 on K as follows. µ1(k) = { 0.54 if k = 2; 0 otherwise. and µ2(k) =  0.65 if k = 4; 0.6 if k = 3; 0.55 if k = 2; 0 otherwise. Then µ1 is a fuzzy antiideal of K and µ2 is a fuzzy bi-antiideal of K. Theorem 1. [2] Let X be a semigroup, t ∈]0, 1], and µ a non-zero fuzzy set of X. Then the following statements hold. (i) µ is a fuzzy left(right) antiideal of X if and only of µt ̸= ∅ is a left(right) antiideal of X. (ii) µ is a fuzzy bi-antiideal of X if and only of µt ̸= ∅ is a bi-antiideal of X. 3. (m,n)-bi-antiideals of a semigroup In this section and inspired by (m,n)-antiideals [1, 8] and by bi-antiideals [2], we introduce (m,n)-bi-antiideals of a semigroup as a generalization of bi-antiideals and study their properties. The results of this section are considered as a generalization of some results in [2]. Definition 8. Let (X, ·) be a semigroup, m,n be positive integers, and A ̸= ∅ ⊆ X. Then A is an (m,n)-bi-antiideal of X if AmXAn ∩A = ∅. Remark 1. A monoid can have (m,n)-bi-antiideals. (See Example 5.) Example 5. Let (P0,+) be the monoid of non-negative integers under standard addition of integers. Then {1, 2} is a (2, 1)-bi-antiideal of P0. This is clear as ({1, 2}+ {1, 2}+ P0 + {1, 2}) ∩ {1, 2} = {x ∈ P0 : x ≥ 3} ∩ {1, 2} = ∅. Proposition 2. Let (X, ·) be a semigroup and A ̸= ∅ ⊆ X be a bi-antiideal of X. Then A is an (m,n)-bi-antiideal of X. M. Al Tahan, S. Hoskova-Mayerova, S. Al-Kaseasbeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5646 5 of 11 Proof. The proof results form having AmXAn ∩A = A(Am−1XAn−1)A ∩A ⊆ AXA ∩A = ∅. Remark 2. The converse of Proposition 2 may not hold. (See Example 6.) Example 6. Let M2(P0) be the semigroup of all two by two matrices with non-negative integral entries under multiplication of matrices and A = { ( 2 0 0 0 ) , ( 4 0 0 0 ) }. Then A is a (2, 1)-bi-antiideal of M2(P0). Furthermore, it is not a bi-antiideal of M2(P0). Proposition 3. Let (X, ·) be a semigroup and A ̸= ∅ ⊆ X be an (m,n)-bi-antiideal of X. If k ≥ m, andl ≥ n, then A is a (k, l)-bi-antiideal of X. Proof. The proof results form having AkXAl ∩A = Am(Ak−mXAl−n)An ∩A ⊆ AmXAn ∩A = ∅. Example 7. Let N be the senigroup of natural numbers under standard multiplication and A = {2, 3, 6, 12}. Then A is a (3, 1)-bi-antiideal of N that is not a (2, 1)-bi-antiideal of N. This is clear as 2(3)(1)(2) ∈ A2NA ∩A. Proposition 4. Let (X, ·) be a semigroup and A ̸= ∅ ⊆ X be an (m,n)-bi-antiideal of X. Then A is not a subsemigroup of X. Proof. Let A be an (m,n)-bi-antiideal ofX that is subsemigroup of A. Then Am+n+1 = AmAAn ̸= ∅ ⊆ AmXAn ∩A = ∅. Al-Tahan et al. [2] proved that every left(right) antiideal of a semigroup X is a bi- antiideal of X. Example 8 shows that the converse may not hold. Example 8. Let M2(P0) be the semigroup of all two by two matrices with non-negative integer entries and A = { ( 2 0 0 0 ) }. Then A is a bi-antiideal of M2(P0). Furthermore, it is not a left(right) antiideal of P0. Proposition 5. Let (X, ·) be a semigroup and A ̸= ∅ ⊆ X be an (m,n)-bi-antiideal of X. Then every non-empty subset of A is an (m,n)-bi-antiideal of X. Proof. The proof is straightforward. Corollary 1. Let (X, ·) be a semigroup and Ai ̸= ∅ ⊆ X for i ∈ N. If Ai is an (m,n)- bi-antiideal of X for some i ∈ N, then every non-empty intersection of Ai is an (m,n)-bi- antiideal of X. M. Al Tahan, S. Hoskova-Mayerova, S. Al-Kaseasbeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5646 6 of 11 Theorem 2. Let X1, X2 be semigroups, f : X1 → X2 be an onto semigroup homomor- phism, and A1 ̸= ∅ ⊆ X1 an (m,n)-bi-antiideal of X1. Then f(A1) is an (m,n)-bi-antiideal of X2. Proof. Let y ∈ f(A1) mX2f(A1) n ∩ f(A1). Then there exist x1, . . . , xm, z1, . . . , zm ∈ A1, x ∈ X1, r = f(x) ∈ X2 with y = f(x1) . . . f(xm)f(r)f(y1) . . . f(yn) ∈ f(A1). Having f a semigroup homomorphism implies that y = f(x1 . . . xmry1 . . . yn) ∈ f(A1) and hence, x1 . . . xmry1 . . . yn ∈ Am 1 X1A n 1 ∩A1 = ∅. Theorem 3. Let X1, X2 be semigroups, f : X1 → X2 be a semigroup homomorphism, and A2 ̸= ∅ ⊆ X2 an (m,n)-bi-antiideal of X2. Then f−1(A2) ̸= ∅ is an (m,n)-bi-antiideal of X1. Proof. Let x ∈ f−1(A2) mX1f −1(A2) n ∩ f−1(A2). Then there exist xi, zj ∈ f−1(A2) with i ∈ {1, . . . ,m}, j ∈ {1, . . . , n}, r ∈ X1 satisfying x = x1 . . . xmry1 . . . yn ∈ f−1(A2) and hence f(x) = f(x1 . . . xmry1 . . . yn) ∈ A2. Having f a semigroup homomorphism implies that y = f(x1) . . . f(xm)f(r)f(y1) . . . f(yn) ∈ A2 and hence, y ∈ Am 2 X2A n 2 ∩A2 = ∅. Example 9. Let M2(P0), M2(2P0) be the semigroups of all two by two matrices with non- negative integer entries and with non-negative even integral entries under multiplication of matrices respectively defined in Example 6 and f : M2(P0) → M2(2P0) be defined as follows. For every matrix M ∈ M2(P0), f(M) = 2M . From Example 8, we have A = { ( 2 0 0 0 ) , ( 4 0 0 0 ) } is a (2, 1)-bi-antiideal of M2(P0). Having f an onto semigroup homorphism implies that f(A) = { ( 4 0 0 0 ) , ( 8 0 0 0 ) } is a (2, 1)-bi-antiideal of M2(2P0). 4. Fuzzy (m,n)-bi-antiideals of a semigroup In this section, we introduce new fuzzy algebraic structures and study their proper- ties. More precisely and inspired by fuzzy interior antiideals introduced in [2] and fuzzy antiideals of a semiring [18], we define fuzzy (m,n)-bi-antiideals of a semigroup. Definition 9. Let (X, ·) be a semigroup, m,n be positive integers, and µ : X → [0, 1] be a non-zero fuzzy set of X. Then µ is a fuzzy (m,n)-bi-antiideal of X if for all xi, yj , r ∈ X, µ(x1 . . . xmry1 . . . yn) ∧ µ(x1) ∧ . . . ∧ µ(xm) ∧ µ(y1) ∧ . . . ∧ µ(yn) = 0. M. Al Tahan, S. Hoskova-Mayerova, S. Al-Kaseasbeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5646 7 of 11 Example 10. Let P0 be the semigroup of non-negative integers under standard addition and µ be the fuzzy set on P0 defined as follows. µ(k) =  0.65 if k=1; 0.54 if k=2; 0 otherwise. Then µ is a fuzzy (2, 1)-bi-antiideal of P0. Moreover, it is not a fuzzy bi-antiideal of P0. This is clear as 0.54 = µ(2) ∧ µ(1) = µ(1 + 0 + 1) ∧ µ(1). Next, we study fuzzy (m,n)-bi-antiideals of a semigroup under some operations of fuzzy sets such as the intersection, union, and product of fuzzy sets. Theorem 4. Let (X, ·) be a semigroup and µi be a non-zero fuzzy set of X for i = 1, . . . , k. If µi is a fuzzy (m,n)-bi-antiideal of X for some i ∈ {1, . . . , k}, then so is µ = µ1 ∧ µ2 ∧ . . . ∧ µk. Proof. Let x1, . . . , xm, r, y1, . . . , yn ∈ X. Without loss of generality, let µ1 be a fuzzy (m,n)-bi-antiideal of X. Then µ(x1 . . . xmry1 . . . yn) ∧ µ(x1) ∧ . . . µ(xm) ∧ µ(y1) ∧ . . . ∧ µ(yn) ≤ µ1(x1 . . . xmry1 . . . yn) ∧ µ1(x1) ∧ . . . µ1(xm) ∧ µ1(y1) ∧ . . . ∧ µ1(yn) = 0. Remark 3. The union of fuzzy (m,n)-bi-antiideals is not necessarily a fuzzy (m,n)-bi- antiideal. (See Example 11.) Example 11. Let M2(P0) be the semigroup defined in Example 6 and µ1, µ2 be defined as follows. µ1(B) = 0.6 if B = ( 2 0 0 0 ) ; 0 otherwise. and µ2(B) =  0.8 if B = ( 4 0 0 0 ) ; 0.7 if B = ( 16 0 0 0 ) ; 0 otherwise. Then µ1, µ2 are fuzzy (2, 1)-bi-antiideals of M2(P0). The fuzzy set µ = µ1 ∨ µ2 of M2(P0) is given by: µ(B) =  0.6 if B = M1 = ( 2 0 0 0 ) ; 0.8 if B = M2 = ( 4 0 0 0 ) ; 0.7 if B = M3 = ( 16 0 0 0 ) ; 0 otherwise. M. Al Tahan, S. Hoskova-Mayerova, S. Al-Kaseasbeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5646 8 of 11 Having M3 = M1M1 ( 1 0 0 0 ) M2 and 0.7 = µ(M3) implies that µ(M1M1 ( 1 0 0 0 ) M2) ∧ µ(M1) ∧ µ(M2) ̸= 0. Theorem 5. Let X1, X2 be semigroups and µ1, µ2 be non-zero fuzzy sets of X1, X2 re- spectively. If µ1 or µ2 is a fuzzy (m,n)-bi-antiideal of X1, X2, then µ = µ1 ×µ2 is a fuzzy (m,n)-bi-antiideal of X1 ×X2. Proof. Let x1, . . . , xm, r, y1, . . . , yn ∈ X1, z1, . . . , zm, r′, w1, . . . , wn ∈ X2. Without loss of generality, let µ1 be a fuzzy (m,n)-bi-antiideal of X1. Then µ((x1, z1) . . . (xm, zm)(r, r′)(y1, w1) . . . (yn, wn)∧µ((x1, z1))∧ . . . µ((xm, zm))∧µ((y1, w1))∧ . . .∧µ((yn, wn)) ≤ µ1(x1 . . . xmry1 . . . yn)∧µ1(x1)∧ . . . µ1(xm)∧µ1(y1)∧ . . .∧µ1(yn) = 0. Theorem 6. Let (Xi, ·) be a semigroup for i = 1, 2, . . . , k and µi be a non-zero fuzzy set of Xi for i = 1, . . . , k. If µi is a fuzzy (m,n)-bi-antiideal of Xi for some i ∈ {1, . . . , k}, then µ = µ1 × µ2 × . . .× µk is a fuzzy (m,n)-bi-antiideal of X1 × . . .×Xk. Proof. The proof is similar to that of Theorem 5. Next, we link fuzzy (m,n)-bi-antiideals of a semigroup X to (m,n)-bi-antiideals of X. Theorem 7. Let (X, ·) be a semigroup, µ be a non-zero fuzzy set of X, and t ∈ [0, 1]. Then µ is a fuzzy (m,n)-bi-antiideal of X if and only if µt is either the empty set or an (m,n)-bi-antiideal of X. Proof. Let µ be a fuzzy (m,n)-bi-antiideal of X, and α ∈ µm t Xµn t ∩ µt ̸= ∅. Then there exist x1, . . . , xm, y1, . . . , yn ∈ µt, r ∈ X with α = x1 . . . xmry1 . . . yn. Having α, x1, . . . , xm, y1, . . . , yn ∈ µt implies that 0 = µ(x1 . . . xmry1 . . . yn) ∧ µ(x1) ∧ . . . ∧ µ(xm) ∧ µ(y1) ∧ . . . ∧ µ(yn) ≥ t. Conversely, let µ(x1 . . . xmry1 . . . yn)∧µ(x1)∧ . . .∧µ(xm)∧µ(y1)∧ . . .∧µ(yn) = t > 0. Then x1 . . . xmry1 . . . yn, x1, . . . , xm, y1, . . . , yn ∈ µt ̸= ∅ and hence, x1 . . . xmry1 . . . yn ∈ µm t Xµn t ∩ µt = ∅. Theorem 8. Let (X, ·) be a semigroup. Then every (m,n)-bi-antiideal of X can be rep- resented as a level set of a fuzzy (m,n)-bi-antiideal of X. Proof. Let A be an (m,n)-bi-antiideal of X and define the fuzzy set µ of X as follows. µ(x) = { 0.94 if x ∈ A; 0 otherwise. One can easily see that µ0.94 = A and that µ is a fuzzy (m,n)-bi-antiideal of X. M. Al Tahan, S. Hoskova-Mayerova, S. Al-Kaseasbeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5646 9 of 11 Corollary 2. Every fuzzy bi-antiideal of a semigroup X is a fuzzy (m,n)-bi-antiideal of X. Proof. The proof follows from Theorem 1, Proposition 2 and Theorem 8. Corollary 3. Every fuzzy left(right) antiideal is a fuzzy (m,n)-bi-antiideal. Proof. The proof follows from Proposition 1, Proposition 2, and Theorem 8. Theorem 9. Let (X, ·) be a semigroup and A ̸= ∅ ⊆ X. Then A is an (m,n)-bi-antiideal of X if and only if µA is a fuzzy (m,n)-bi-antiideal of X. Here, for all x ∈ X, µA(x) = { 1 if x ∈ A; 0 otherwise. Proof. Let A be an (m,n)-bi-antiideal of X and x1, . . . , xm, r, y1, . . . , yn ∈ X. If there exist i ∈ {1, . . . ,m} or j ∈ {1, . . . , n} with xi /∈ A or yj /∈ A, then µA(x1 . . . xmry1 . . . yn)∧ µA(x1)∧ . . .∧µA(xm)∧µA(y1)∧ . . .∧µA(yn) = 0. Otherwise and having AmXAn∩A = ∅ implies that x1 . . . xmry1 . . . yn /∈ A and hence, µA(x1 . . . xmry1 . . . yn)∧µA(x1)∧µA(xm)∧ µA(y1) ∧ µA(yn) = 0. Conversely, let µA be a fuzzy (m,n)-bi-antiideal of X and α ∈ AmXAn ∩ A. Then there exist x1, . . . , xm, y1, . . . , yn ∈ A, r ∈ X with α = x1 . . . xmry1 . . . yn ∈ A. The latter implies that µA(x1 . . . xmry1 . . . yn) ∧ µA(x1) ∧ µA(xm) ∧ µA(y1) ∧ µA(yn) = 1 ̸= 0. 5. Conclusion In this paper, we have extended the theory of semigroups by introducing and charac- terizing (m,n)-bi-antiideals and their fuzzy counterparts. Our exploration began with a review of the foundational concepts of antiideals and bi-antiideals, followed by the gener- alization to (m,n)-bi-antiideals. We demonstrated the properties of these new structures through various propositions and examples. Furthermore, we incorporated fuzzy set theory to handle uncertainties in semigroups, defining and analyzing fuzzy (m,n)-bi-antiideals. We established connections between fuzzy (m,n)-bi-antiideals and their classical coun- terparts using level sets, providing a comprehensive framework for understanding these concepts. Our findings contribute to the broader understanding of semigroups and their appli- cations in different fields of mathematics and science. Future research could focus on exploring additional properties of (m,n)-bi-antiideals, identifying all antiideals in specific semigroups, extending the theory to other algebraic structures, and finding practical ap- plications for these theoretical concepts. By advancing the study of semigroups and their fuzzy generalizations, we hope to inspire further investigations and applications in both theoretical and applied mathematics. M. Al Tahan, S. Hoskova-Mayerova, S. Al-Kaseasbeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5646 10 of 11 Acknowledgements This research was supported by Abu Dhabi University with grant number: 19300884. This research (APC) was also supported by the grant VAROPS granted by the Ministry of Defence of the Czech Republic. The authors declare no conflict of interest. References [1] M. Al-Tahan and I. Cristea. A classical and a fuzzy approach to study the (m, n)-antiideals of a semigroup. Journal of Multiple-Valued Logic & Soft Computing, 44:291–302, 2025. [2] M. 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