EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6141 ISSN 1307-5543 – ejpam.com Published by New York Business Global Hypergroups Obtained from a Latin Square Mohammad Ali Dehghanizadeh1,∗, Saeed Mirvakili2 1 Department of Mathematics, National University of Skills(NUS), Tehran, Iran 2 Department of Mathematical Sciences, Yazd University, Yazd, Iran ”This paper is dedicated to Professor Ali Akbar Mohammadi Hassanabadi on his 80th birth- day.” Abstract. In this study, we explore the construction of a sequence of hypergroupoids derived from a quasigroup (Latin square). We demonstrate that cyclic groups yield sequences of commutative hypergroups. Additionally, under specific conditions, we establish the formation of Hv-groups and hypergroups. Various examples are provided to illustrate and support the theoretical concepts presented in this paper, offering insights into their structure and applicability. 2020 Mathematics Subject Classifications: 20N20, 05B15 Key Words and Phrases: Latin Square, Quasigroup, Quasihypergroup, hypergroup, Hv-group 1. Introduction The linkage between algebraic structures and combined design problems in combina- torial models has been a field of great interdisciplinary potential, both in abstract algebra theory and in practical applications ranging from coding theory, cryptography, and ex- perimental design [1, 2]. One of the fundamental conductivities is the relation between quasigroups and Latin squares. In recent years, the topic has also been extended to hy- perstructures, and here also to quasihypergroups and hypergroups, providing new avenues for algebraic generalization. The paper by Iranmanesh and Ashrafi [3] opens new fields of algebraic generalization involving the connections between quasihypergroups and Latin squares with their hyperstructural homologues. The study of Latin squares dates back to ancient mathematical traditions, with early examples in Arabic manuscripts and medieval combinatorial puzzles. However, formal- ization started in the 18th century, beginning with Leonhard Euler’s famous ”36 officers problem” that launched a systematic investigation of their properties. A Latin square of order n is a n × n grid of points, whose values are described by [1, 2, 4]. This set of n independent symbols (which can all occur simultaneously in rows and columns) became ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6141 Email addresses: mdehghanizadeh@nus.ac.ir (M. A. Dehghanizadeh), saeedmirvakili@yazd.ac.ir (S. Mirvakili) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. A. Dehghanizadeh, S. Mirvakili / Eur. J. Pure Appl. Math, 18 (2) (2025), 6141 2 of 10 a key object of design theory. By the 20th century Latin squares had rigorous applica- tions in statistical experimental design (through R. A. Fisher’s work), error-correcting codes, and group theory where they became known as multiplication tables of quasigroups (non-associative algebraic structures satisfying the Latin square property)[1, 2, 4]. The desire to use the problem of Latin squares for applications outside of algebra en- abled a number of generalizations, for example, reduced Latin squares, orthogonal arrays, and partial Latin squares. All three variants relaxed or extended classical constraints in some way. They proved helpful in solving problems in graph theory, finite geometry, and cryptography. At the same time, quasigroups (algebraic systems with multiplication tables of Latin squares) developed as the basis of non-associative algebras and mathematically relevant solutions of equations. Quasigroups are not associative but contain cancellativ- ity in their behavior. They therefore offer the bridge between discrete mathematics and algebraic abstraction. On the other hand, the mid-20th century also brought about the development of hy- perstructures, a generalization of classical algebraic operations by changing binary oper- ations (contrary to traditional algebraic operations) to set-valued operations. The fields were opened by the work of F. Marty (1934) in the invention of hypergroups–structures in which the product of two variables is a nonempty set—in the direction of work from Corsini, Davvaz and Vougiouklis [5–8]. hyperstructures and their variants—such as quasi- hypergroups and Hv-structures—have been used for modelling uncertainty theory, granu- lar computing, and non-deterministic systems. Since hyperstructures are compatible with multivaluedness, they offer more powerful algebraic representations for complex relational systems. The construction of a sequence of hypergroupoids derived from a quasigroup (Latin square) is investigated here. We prove that cyclic groups are capable of yielding sequences of commutative hypergroups. In addition, under some special conditions, we show that Hv-groups and hypergroups are formed. Experiments are given in this paper to show and support the theoretical notions presented, and to show how they are formulated and how they may be extended to practice. 2. Main Results A quasigroup (Q, ⋆) is a non-empty set Q with a binary operation ⋆, obeying the Latin square property[4, 9]. This states that, for each a and b in Q, there exist unique elements x and y in Q such that both a ⋆ x = b, y ⋆ a = b hold. In other words, each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup’s multiplication table, or Cayley table. This prop- erty ensures that the Cayley table of a finite quasigroup, and, in particular, a finite group, is a Latin square. M. A. Dehghanizadeh, S. Mirvakili / Eur. J. Pure Appl. Math, 18 (2) (2025), 6141 3 of 10 We recall basic definitions from [8]. A hypergroupoid or hyperstructure is a non-empty set H with a hyperoperation ◦ defined on H, that is, a mapping of H ×H into the family of non-empty subsets of H. If (x, y) ∈ H × H, its image under ◦ is defined by x ◦ y. If A,B are non-empty subsets of H then A ◦B is given by A ◦B = ⋃ {x ◦ y|x ∈ A, y ∈ B}. The notation a ◦A is used for {a} ◦A, and A ◦ a for A ◦ {a}. Generally, the singleton {a} is identified with its member a. The relational notation A ≈ B (read A meets B) is used to assert that the sets A and B have an element in common, that is, A ∩B ̸= ∅. A hypergroupoid (H, ◦) is called a semihypergroup if for all x, y, z ∈ H we have x ◦ (y ◦ z) = (x ◦ y) ◦ z. Moreover, a hypergroupoid (H, ◦) is called a Hv-semigroup if for all x, y, z ∈ H we have x ◦ (y ◦ z) ∩ (x ◦ y) ◦ z ̸= ∅ or x ◦ (y ◦ z) ≈ (x ◦ y) ◦ z. A hypergroupoid (H, ◦) is called a commutative hypergroupoid if x ◦ y = y ◦ x, for all x, y ∈ H. Moreover it is called weak commutative if x ◦ y ∩ y ◦ x ̸= ∅ or x ◦ y ≈ y ◦ x, for all x, y ∈ H. A hypergroupoid (H, ◦) is called a quasihypergroup if for all x ∈ H we have x ◦H = H ◦ x = H. We defin two hypercompositions on (H, ◦), right extension / and left extension \, each an inverse to ·, are defined by: a/b = {x | a ∈ x ◦ b} and b\a = {x | a ∈ b ◦ x}. Hence, x ≈ a/b if and only if a ≈ x ◦ b, and x ≈ b\a if and only if a ≈ b ◦ x. Definition 1. Let X be a n-set and let A = [aij ] be a n×n matrix with aij ⊆ X for all of 1 ≤ i, j ≤ n. A is called a generalized Latin square on n-set X if the following condition is satisfied: n⋃ i=1 aij = X = n⋃ j=1 aij . Example 1. Let n = 4, X = {a, b, c, d} and aii = X and aij = {a}, where i ̸= j. Then A = [aij ] is a generalized Latin square. In fact A = X a a a a X a a a a X a a a a X Definition 2. Let X be a n-set and let A = [aij ] be a n× n matrix with aij ⊆ X for all of 1 ≤ i, j ≤ n. A is called a Latin k-hypersquare on n-set X if the following condition is satisfied: (1) |aij | = k. (2) Equations a · x = c and y · b = d has exactly k solutions in X, Where the hyperoper- ation · on X is defined as x · y = axy. M. A. Dehghanizadeh, S. Mirvakili / Eur. J. Pure Appl. Math, 18 (2) (2025), 6141 4 of 10 Example 2. The generalized Latin square A in Example 1 is not a Latin k-hypersquare. But the generalized Latin square B is a is not a Latin 2-hypersquare, where B = e, c e, a a, b b, c e, a a, b b, c e, c b, c e, c e, a a, b a, b b, c e, c e, a Let (H, ◦) be a semihypergroup. The relation β∗ is the transitive closure of the relation β = ∪n≥1βn, where β1 is the diagonal relation and, for every integer n > 1, βn is the relation defined as follows: xβny ⇔ ∃(z1, . . . , zn) ∈ Hn : {x, y} ⊆ n∏ i=1 zi. β∗ is the smallest strongly regular equivalence onH.Moreover, the canonical projection ψ : H → H/β∗ is a homomorphism and if H is a hypergroup, the kernel of ψ is called heart of H and is denoted with ωH . Let (H, ◦) be a hypergroupoid. Let U denote the set of all finite products of elements of H. Then relation β can be defined on H as follows: xβy ⇔ ∃u ∈ Usuch that{x, y} ⊆ u. If (H, ◦) is a hypergroupoid then The relation β∗ is the transitive closure of the relation β. Theorem 1. (Theorem 81 in [5]) If H is a hypergroup then β = β∗. As a consequence of Theorem 1, in every hypergroup the relation β is transitive. But in semihypergroups this not true and in Hv-groups this is an open problem. Definition 3. Let (H, ◦) be a hypergroupoid. The hyperoperation ◦ is called total associa- tive, if x ◦ (y ◦ z) = H = (x ◦ y) ◦ z, ∀x, y, z ∈ H. Example 3. Let (H, ◦T ) be a total hypergroup, i. e., for all x, y ∈ H, x ◦T y = H. Then ◦ is a total associative hyperoperation. Example 4. Let H ̸= ∅ and |H| > 2. For every x, y ∈ H, define x◦y = { H − {y}, x ̸= y H, x = y. Then for every x, y, z ∈ H, we have y ◦ z = H − {z} and so |H − {z} = 2. So x ◦ (y ◦ z) = x ◦ (H − {z}) = ∪w∈H−{z}x ◦ w = H. In the similar way we have H = (x ◦ y) ◦ z. Therefore the hyperoperation ◦ is total associative M. A. Dehghanizadeh, S. Mirvakili / Eur. J. Pure Appl. Math, 18 (2) (2025), 6141 5 of 10 Now, we construct some hyperoperations from a quasigroup as follows: Definition 4. Let (H, ⋆) be a finite quasigroup. Let H = {a0, a1, . . . , an−1} and ai ⋆ aj = aij , for every i, j = 0, 1, . . . , n − 1. For k ∈ {0, 1, . . . , n − 1} set ai ⋆ C k aj = aih, when j − k ≡ h mod n. It easily to see that (H, ⋆k) is a quasigroup. Now for every x, y ∈ H define the hyperoperation ◦Ck as follows x ◦Ck y = {x ⋆Cm y|m = 0, 1, . . . , k − 1}. Definition 5. Let (H, ⋆) be a finite quasigroup. Let H = {a0, a1, . . . , an−1} and ai ⋆ aj = aij , for every i, j = 0, 1, . . . , n−1. For k ∈ {0, 1, . . . , n−1} set ai⋆ R k aj = ahj , when j−k ≡ h mod n. It easily to see that (H, ⋆Rk ) is a quasihypergroup. Now for every x, y ∈ H define the hyperoperation ◦Rk as follows x ◦Rk y = {x ⋆Rm y|m = 0, 1, . . . , k − 1}. Proposition 1. Let (H, ⋆) be a finite quasigroup and ◦Rk and ◦Ck be the hyperoperations in Definition 4 and 5. Then (H, ◦Cn−1) and (H, ◦Rn−1) are total hypergroup. Proof. LetH = {a0, a1, . . . , an−1}. Since (H, ⋆) is a quasigroup then for every x, y ∈ H, |x ◦Cn−1 y| = n and x ◦Rn−1 y ⊆ H. So for every x, y ∈ H, x ◦Rn−1 y = H. Proposition 2. Let (H, ⋆) be a finite quasigroup and ◦Rk and ◦Ck be the hyperoperations in Definition 4 and 5. Then for k ≥ |H| 2 , (H, ◦Rk ) and (H, ◦Ck ) are weak commutative hypergroupoids. Proof. For every x, y ∈ H, |x ◦Ck y| = k = |y ◦Ck x|. If x ◦Ck y ∩ y ◦Ck x = ∅ then |x◦Ck y∪y ◦Ck x| = k+k+2 > |H| and this is a contradiction. Therefore x◦Ck y∩y ◦Ck x ̸= ∅. In Theorem 2, the given boundary for k is the best boundary. See the next Example. Example 5. Let (H, ⋆) be a quasigroup. ⋆0 e a b c e e a b c a a b c e b c e a b c b c e a ⋆0 e a b c e e a b c a a b c e b c e a b c b c e a ⋆1 e a b c e c e a b a e a b c b b c e a c a b c e ⋆2 e a b c e b c e a a c e a b b a b c e c e a b c ⋆3 e a b c e a b c e a b c e a b e a b c c c e a b M. A. Dehghanizadeh, S. Mirvakili / Eur. J. Pure Appl. Math, 18 (2) (2025), 6141 6 of 10 ◦C0 e a b c e e a b c a a b c e b c e a b c b c e a ◦C1 e a b c e e, c e, a a, b b, c a e, a a, b b, c e, c b b, c e, c e, a a, b c a, b b, c e, c e, a ◦C2 e a b c e e, b, c e, a, c e, a, b a, b, c a e, a, c e, a, b a, b, c e, b, c b a, b, c e, b, c e, a, c e, a, b c e, a, b a, b, c e, b, c e, a, c (H, ◦C1 ) is not weak commutative. Proposition 3. Let (H, ⋆) be a finite quasigroup and ◦Ck be the hyperoperation in Defini- tion 4. If ⋆ is commutative then ◦Ck is weak commutative, for all k. Proof. For every x, y ∈ H, x ⋆ y = y ⋆ x. We have x ⋆ y ∈ x ◦Ck y and y ⋆ x ∈ y ◦Ck x. Therefore x ◦Ck y ∩ y ◦Ck x ̸= ∅ and proof is complete. Proposition 4. Let (H, ⋆) be a finite quasigroup and ◦Ck be the hyperoperation in Defini- tion 4. If (H, ⋆) is a cyclic group, then for every k, ◦Ck is commutative; Proof. Let (H, ⋆) is a cyclic group then (H, ⋆) ∼= (Z,+) or (H, ⋆) ∼= (Zn,+). If (H, ⋆) ∼= (Zn,+) then for every x, y ∈ Zn, x ◦Ck y = {x+ y, x+ y − 1, . . . , x+ y − k} = {y + x, y + x− 1, . . . , y + x− k} = x ◦Ck y. Therefore (H, ◦Ck ) is a commutative hypergroupoid. Lemma 1. (H, ◦Ck ) and (H, ◦Rk ) are quasihypergroups. Proof. For every a, b ∈ H, a ⋆ x = b and y ⋆ a = b have solutions in H. Since for every k, a ⋆x ⊆ a ◦Ck x and y ⋆ a ⊆ y ◦Ck a therefore b ∈ a ◦Ck x and b ∈ y ◦Ck a have solutions in H. Lemma 2. We have ⋆ = ◦C0 ⊂ ◦C1 ⊂ . . . ⊂ ◦Cn−2 ⊂ ◦Cn−1 = ◦T , and ⋆ = ◦R0 ⊂ ◦R1 ⊂ . . . ⊂ ◦Rn−2 ⊂ ◦Rn−1 = ◦T . When ⋆ is the operation in Definition 5 and ◦T is the hyperoperation in Example 3. Proof. It is straightforward. Example 6. Let (H, ⋆) be a cycle group of order 3 by the following Cayley table ⋆ a b c a a b c b b c a c c a b M. A. Dehghanizadeh, S. Mirvakili / Eur. J. Pure Appl. Math, 18 (2) (2025), 6141 7 of 10 then we obtain ⋆0 a b c a a b c b b c a c c a b ⋆1 a b c a c a b b a b c c b c a ⋆3 a b c a b c a b c a b c a b c and therefore ◦C0 a b c a a b c b b c a c c a b ◦C1 a b c a a, c a, b b, c b a, b b, c a, c c b, c a, c, a a, b ◦C2 a b c a H H H b H H H c H H H . (H, ◦Ck ) are commutative hypergroups. Example 7. Let (H, ⋆) be a finite quasigroup by the following Cayley table ⋆ a b c a b c a b a b c c c b a then we obtain ⋆0 a b c a b c a b a b c c c a b ⋆1 a b c a a b c b c a b c b c a ⋆2 a b c a c a b b b c a c a b c and therefore ◦C0 a b c a b c a b a b c c c a b ◦C1 a b c a a, b b, c a, c b a, c a, b b, c c b, c c, a b, a ◦C2 a b c a H H H b H H H c H H H . (H, ◦C0 ) is neither commutative nor weak commutative (because a ◦C0 c ̸≈ c ◦C0 a). (H, ◦C1 ) is not commutative but it is weak commutative (because x ◦C1 y ≈ y ◦C1 x for all x, y ∈ H). Theorem 2. For every x, y ∈ H, (1) |x ◦Ck y| = k + 1; (2) |{u|x ∈ u ◦Ck y}| = k + 1; (3) |{u|x ∈ y ◦Ck u}| = k + 1. M. A. Dehghanizadeh, S. Mirvakili / Eur. J. Pure Appl. Math, 18 (2) (2025), 6141 8 of 10 Proof. It obtains from definition ◦Ck . Corollary 1. for all k = 1, 2, . . . , n, the quasihypergroups (H, ◦Ck−1) are Latin k-hypersquare. In the next Theorem we show that for all k = |H| 2 , |H| 2 +1, . . . , n, the quasihypergroups (H, ◦Ck ) are hypergroups, Theorem 3. If k ≥ |H| 2 then ◦Ck is a total associative hyperoperation. Proof. Let H = {a1, a2, . . . , an} and k ≥ n 2 . Suppose that x, y, z ∈ H. Then by part (1) of Theorem 2, |y ◦Ck z| = k + 1 > n 2 . So we have y ◦Ck z = {ai0 , ai2 , . . . , aik}. Now if h ∈ H then by part (3) of Theorem 2, there exists j = 0, 2, . . . , k such that u ∈ x ◦Ck aij . Hence x ◦Ck (y ◦Ck z) = H. In the similar way we obtain (x ◦Ck y) ◦Ck z = H. Therefore ◦Ck is a total associative hyperoperation. Theorem 4. If k ≥ |H| 2 then (H, ◦Ck ) is a hypergroup. Proof. By Theorem 3 (H, ◦Ck ) is a semihypergroup and by Lemma 1 (H, ◦Ck ) is a quasihypergroup. Therefore (H, ◦Ck ) is a hypergroup Example 8. In Theorem 4, the given boundary for k is the best boundary. For example if (H, ⋆) is the quasigroup in Example 7 then (H, ◦C1 ) is not hypergroup. Theorem 5. Let (H, ⋆) be a cyclic group of order n, i. e., (H, ⋆) ∼= (Zn,+). Then (H, ◦Ck ) is a commutative hypergroup, for all k = 0, 1, . . . , n− 1. Example 9. Let (H, ⋆) ∼= (Z2 × Z2,+). ⋆ e a b c e e a b c a a e c b b b c e a c c b a e then we obtain ⋆0 e a b c e e a b c a a e c b b b c e a c c b a e ⋆1 e a b c e c e a b a b a e c b a b c e c e c b a ⋆2 e a b c e b c e a a c b a e b e a b c c a e c b ⋆3 e a b c e a b c e a e c b a b c e a b c b a e c and therefore ◦C0 e a b c e e a b c a a e c b b b c e a c c b a e ◦C1 e a b c e e, c e, a a, b b, c a a, b e, a e, c b, c b a, b b, c e, c e, a c e, c c, b a, b e, a ◦C2 e a b c e e, b, c e, a, c e, a, b a, b, c a e, a, b e, a, c e, b, c a, b, c b e, a, b a, b, c e, b, c e, a, c c e, a, c e, b, c a, b, c e, a, b M. A. Dehghanizadeh, S. Mirvakili / Eur. J. Pure Appl. Math, 18 (2) (2025), 6141 9 of 10 We have (H, ⋆) is commutative but (H, ◦C1 ) and (H, ◦C2 ) are weak commutative(are not commutative) Theorem 6. Let (H, ⋆) be a group of order n. Then (H, ◦Ck ) is an Hv-group, for all k = 0, 1, . . . , n− 1. Proof. For every x, y, z ∈ X, x ⋆ (y ⋆ z) = (x ⋆ y) ⋆ z, x ⋆ (y ⋆ z) ∈ x ◦Ck (y ◦Ck z) and (x ⋆ y) ⋆ z) ∈ (x ◦Ck y) ◦Ck z. Therefore x ◦Ck (y ◦Ck z) ∩ (x ◦Ck y) ◦Ck z ̸= ∅. Theorem 7. Let (H, ⋆) be a quasigroup. (1) If k = 0, then β∗ = {(x, x)|x ∈ H}; (2) If k > 0, then β∗ = H ×H. Proof. If k = 0 then ∏n i=1 zi is singleton and so β = {(x, x)|x ∈ H}. Now, Let k > 0 and x, y ∈ H. Consider (a1 ⋆0 a1, . . . , a1 ⋆0 an) = (b1, . . . , bn) so (a1 ⋆1 a1, . . . , a1 ⋆1 an) = (b2, . . . , bn, b1). Therefore {bi, bi+1} ⊆ a1 ◦C2 ai ⊆ a1 ◦Ck ai. Since H = {b1, . . . , bn} then for every x, y ∈ H there exist 1 ≤ i, j ≤ n such that i < j and bi = x and bj = y. x = biβbi+1β . . . βbj = y. and therefore xβ∗y. Corollary 2. For quasihypergroups (H, ◦Ck ), k = 0, 1, . . . , n− 1, we have (1) If k = 0, then the fundamental quasigroup H β∗ is isomorphic to H; (2) If k > 0, then the fundamental quasigroup H β∗ is trivial group. Definition 6. A hypergroupoid (H, ◦) is called a transposition hypergroup if it satisfies the axiom, (Transposition) b \ a ≈ c/d =⇒ a ◦ d ≈ b ◦ c for all a, b, c, d ∈ H. If the hypergroupoid (H, ◦) is commutative then hyperoperation \ = / and a commutative transposition hypergroup is called a join space. Theorem 8. If k = |H| 2 , . . . , n then the quasihypergroup (H, ◦Ck ) is a transposition hyper- groupoid. Proof. Let a, b, c, d ∈ H and b \ a ≈ c/d. Since |a ◦Ck d| = k + 1 = |b ◦Ck c| then a ◦Ck d ∩ b ◦Ck c ̸= ∅. Corollary 3. Let (H, ⋆) be a cyclic group of order n. Then (H, ◦Ck ) is a join space. Proof. It obtains from Theorems 5 and 8. Remark 1. For every result true for ◦Ck then it also holds for ◦Rk . M. A. Dehghanizadeh, S. Mirvakili / Eur. J. Pure Appl. Math, 18 (2) (2025), 6141 10 of 10 3. conclusion In this paper, we initiated the construction of sequences of hypergroupoids derived from quasigroups (Latin squares). Our findings demonstrate that cyclic groups can gener- ate sequences of commutative hypergroups, providing a foundational framework for further exploration. Furthermore, under specific conditions, the formation of Hv-groups and hy- pergroups was established, enriching the theoretical understanding of these structures. To bridge theory with application, we conducted experiments that not only corroborate the presented theoretical concepts but also illustrate their formulation and potential practical extensions. These contributions pave the way for further research in hyperstructure theory and its real-world applicability. Future work may delve into extending the current constructions from Γ-structures to Γ- hyperstructures, further enriching the theoretical framework. Moreover, this method can be applied to explore fuzzy algebraic structures, expanding its applicability. 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