EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 604-615 ISSN 1307-5543 – ejpam.com Published by New York Business Global Relations Between Derivations and Homomorphisms of Ordered Hyperrings Ruiqi Cai1, Mashaer Alsaeedi2, Maryam Akhoundi3,∗ 1 Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China 2 Department of Mathematics, College of Sciences and Humanities, Prince Sattam bin Abdulaziz University, Al-Kharj, Saudi Arabia 3 Clinical Research Development Unit of Rouhani Hospital, Babol University of Medical Sciences, Babol, Iran Abstract. The present study investigates the relation between derivations and hyperideals on ordered hyperrings with no zero divisors. Also, we identify some results for the ordered hyperrings induced by the homomorphism of the ordered hyperrings by derivations. The present work explores some aspects of derivations in ordered hyperrings. Also, we establish some results in connection with homomorphisms and hyperideals. Furthermore, we describe prime hyperideals associated to a derivation d on an ordered hyperring T and derive several results about homomorphisms and derivations on ordered hyperrings. 2020 Mathematics Subject Classifications: 13N15, 16Y99 Key Words and Phrases: Krasner hyperring, ordered hyperring, derivation, homomorphism, hyperideal 1. Introduction Marty presented the hypergroup ideas in 1934 [1]. Krasner originally considered hyperring, which is a development of ring, in [2]. The study of hyperideals have been made by Heidari and Davvaz in the context of ordered semihypergroups in [3]. The study also demonstrated that the direct product of ordered hyperstructures are ordered hyperstructures. Later on, Davvaz et al. [4] utilized pseudoorders to construct strongly regular relations in ordered semihypergroups and ex- amined the relationships between ordered hyperstructures and ordered structures. Also, see [5, 6]. Al-Tahan and Davvaz [7] use the ordered hyperstructure to communicate with biological inheritance and genetics to do research, and to access applications. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5052 Email addresses: cairic@e.gzhu.edu.cn (R. Cai), m.alsaedi@psau.edu.sa (M. Alsaeedi), Maryam.akhoundi@mubabol.ac.ir (M. Akhoundi) https://www.ejpam.com 604 © 2024 EJPAM All rights reserved. R. Cai, M. Alsaeedi, M. Akhoundi / Eur. J. Pure Appl. Math, 17 (2) (2024), 604-615 605 Derivation in rings was first explored by Posner [8] and later on hyperrings by Asokku- mar [9] and Kamali and Davvaz [10]. Derivation on ordered semihyperring was presented by Rao et al. in [11]. Omidi and Davvaz considered ordered hyperring, which is a devel- opment of ordered ring, in [12]. Also, see [13–16]. The present work explores some aspects of derivations in ordered hyperrings. Also, we establish some results in connection with homomorphisms and hyperideals. 2. Preliminaries We set that OKH: the set of all ordered Krasner hyperrings (E,⊕,⊙,≤), Der(E): the set of all derivations of E. Definition 1. [2] (E,⊕,⊙) is a Krasner hyperring if: (1) (E,⊕) is a canonical hypergroup; (2) (E,⊙) is a semigroup and z ⊙ 0 = 0⊙ z = {0}, ∀z ∈ E; (3) ⊙ is distributive with respect to the hyperaddition ⊕. Definition 2. [12] Let (E,⊕,⊙) be a Krasner hyperring. (E,⊕,⊙,≤) ∈ OKH if (1) (E,≤) is a partially ordered set; (2) (l, l′) ∈≤⇒ l ⊕ t ⪯ l′ ⊕ t,∀l, l′, t ∈ E; (3) (l, l′) ∈≤ and (0, t) ∈≤⇒ (l ⊙ t, l′ ⊙ t) ∈≤ and (t⊙ l, t⊙ l′) ∈≤. Note that for every ∅ ≠ L,L′ ⊆ E, L ⪯ L′ ⇔ ∀l ∈ L,∃l′ ∈ L′ such that (l, l′) ∈≤. Definition 3. [12] Let (E,⊕,⊙,≤) and (E′,⊕′,⊙′,≤′) ∈ OKH. A function Λ : E → E′ is a homomorphism if ∀l, l′ ∈ E, (1) Λ(l ⊕ l′) ⊆ Λ(l)⊕′ Λ(l′); (2) Λ(l ⊙ l′) = Λ(l)⊙′ Λ(l′); (3) (l, l′) ∈≤⇒ (Λ(l),Λ(l′)) ∈≤′. Definition 4. [3] Let (E,⊕,⊙,≤) ∈ OKH. ∅ ≠ X ⊆ E is a hyperideal of E if (1) (X,⊕) is a canonical subhypergroup of (E,⊕); (2) l ⊙ x, x⊙ l ∈ X,∀l ∈ E,∀x ∈ X; R. Cai, M. Alsaeedi, M. Akhoundi / Eur. J. Pure Appl. Math, 17 (2) (2024), 604-615 606 (3) (X] := {l ∈ E | l ≤ x, for some x ∈ X} ⊆ X. Definition 5. [11] Let (E,⊕,⊙,≤) ∈ OKH. d ∈ Der(E) if for all l, l′ ∈ E, (1) d(l ⊕ l′) ⊆ d(l)⊕ d(l′); (2) d(l ⊙ l′) ∈ d(l)⊙ l′ ⊕ l ⊙ d(l′); (3) (l, l′) ∈≤⇒ (d(l), d(l′)) ∈≤. 3. Main Results Let (E,⊕,⊙,≤) ∈ OKH. Then, 0 ̸= z ∈ E is a zero divisor if ∃ 0 ̸= v ∈ E such that z ⊙ v = 0 = v ⊙ z. Theorem 1. Let (E,⊕,⊙,≤) ∈ OKH with no zero divisors and 0 ̸= d ∈ Der(E). If Y is a proper hyperideal of E, then d is nonzero on Y . Proof. Let d(m) = 0, ∀ 0 ̸= m ∈ Y . As Y is a hyperideal of E, m⊙ g ∈ Y , ∀ g ∈ E. Thus, d(m⊙ g) = 0. So, d(m⊙ g) ∈ d(m)⊙ g ⊕m⊙ d(g) = 0⊙ g ⊕m⊙ d(g) = 0⊕m⊙ d(g) = m⊙ d(g). Hence, m⊙ d(g) = d(m⊙ g) = 0. By hypothesis, E has no zero divisors. Thus, d(g) = 0, ∀ g ∈ E a contradiction. Therefore, d is nonzero on Y . Theorem 2. Let (E,⊕,⊙,≤) ∈ OKH and g ∈ g ⊕ g,∀g ∈ E. R. Cai, M. Alsaeedi, M. Akhoundi / Eur. J. Pure Appl. Math, 17 (2) (2024), 604-615 607 IdE(g) = g for any g ∈ E, is a homomorphism iff IdE ∈ Der(E). Proof. Let IdE be a homomorphism and g, g′ ∈ E. Then, IdE(g ⊙ g′) = IdE(g)⊙ IdE(g ′) = g ⊙ g′ ∈ (g ⊙ g′)⊕ (g ⊙ g′) = IdE(g)⊙ g′ ⊕ g ⊙ IdE(g ′). Hence, IdE ∈ Der(E). Conversely, let g, g′ ∈ E. Then IdE(g ⊙ g′) = g ⊙ g′ = IdE(g)⊙ IdE(g ′). So, IdE is a homomorphism. Theorem 3. Let (E,⊕,⊙,≤) ∈ OKH be commutative and and g ∈ g ⊕ g,∀g ∈ E. For a given t ∈ E, we set dt(g) = t⊙ g, ∀g ∈ E. Then dt ∈ Der(E). Proof. Let g, g′ ∈ E. For a given t ∈ E, we have dt(g ⊕ g′) = t⊙ (g ⊕ g′) = t⊙ g ⊕ t⊙ g′ = dt(g)⊕ dt(g ′), and dt(g ⊙ g′) = t⊙ (g ⊙ g′) ∈ t⊙ (g ⊙ g′)⊕ t⊙ (g ⊙ g′) = (t⊙ g)⊙ g′ ⊕ (t⊙ g)⊙ g′ = (t⊙ g)⊙ g′ ⊕ (g ⊙ t)⊙ g′ = (t⊙ g)⊙ g′ ⊕ g ⊙ (t⊙ g′) = dt(g)⊙ g′ ⊕ g ⊙ dt(g ′). Let g, g′ ∈ E and (g, g′) ∈≤. Then R. Cai, M. Alsaeedi, M. Akhoundi / Eur. J. Pure Appl. Math, 17 (2) (2024), 604-615 608 dt(g) = t⊙ g ≤ t⊙ g′ = dt(g ′) by Definition 2, and hence dt ∈ Der(E). Corollary 1. Let (E,⊕,⊙,≤) ∈ OKH be commutative and g ∈ g ⊕ g,∀g ∈ E. Then, the identity function IdE defined by IdE(g) = g for any g ∈ E, is a homomorphism. Proof. We have d1(g) = 1⊙ g = g = IdE(g). By Theorem 3, IdE = d1 ∈ Der(E). Now, by Theorem 2, IdE is a homomorphism. Corollary 2. Let (E,⊕,⊙,≤) ∈ OKH be commutative and g ∈ g ⊕ g,∀g ∈ E. If d = dt, where t ∈ E, satisfies the following condition (g′, g) ∈≤ and d(g) = g ⇒ d(g′) = g′, then Fixd(E) = {x ∈ E | d(x) = x} is a hyperideal of E. Proof. By Theorem 3, d(g) = dt(g) = t⊙ g, for any g ∈ E. Let g, g′ ∈ Fixd(E). Then d(g) = g and d(g′) = g′. We have d(g ⊖ g′) = dt(g ⊖ g′) = t⊙ (g ⊖ g′) = t⊙ g ⊖ t⊙ g′ = dt(g)⊖ dt(g ′) = d(g)⊖ d(g′) = g ⊖ g′. So, g ⊖ g′ ⊆ Fixd(E). R. Cai, M. Alsaeedi, M. Akhoundi / Eur. J. Pure Appl. Math, 17 (2) (2024), 604-615 609 Now, let g ∈ Fixd(E) and q ∈ E. Then, d(g ⊙ q) = dt(g ⊙ q) = t⊙ (g ⊙ q) = (t⊙ g)⊙ q = dt(g)⊙ q = d(g)⊙ q = g ⊙ q. So, g ⊙ q ∈ Fixd(E). Let g ∈ Fixd(E), q ∈ E and q ≤ g. Then, d(q) = dt(q) ≤ dt(g) = d(g) = g. By hypothesis, d(q) = q. So, q ∈ Fixd(E). Hence, Fixd(E) is a hyperideal of E. Example 1. Let E = {0, 1, f, f ′} and ⊕ 0 1 f f ′ 0 0 1 f f ′ 1 1 {0, f} {1, f ′} f f f {1, f ′} {0, f} 1 f ′ f ′ f 1 0 ⊙ 0 1 f f ′ 0 0 0 0 0 1 0 1 f f ′ f 0 f f 0 f ′ 0 f ′ 0 f ′ ≤:= {(z, z) | z ∈ E} ∪ {(0, f), (f ′, 1)}. Then (E,⊕,⊙,≤) ∈ OKH. We set d(z) = df (z) =  0, z = 0, f ′ f, z = 1, f. Then, d = df ∈ Der(E). Indeed: df (0) = f ⊙ 0 = 0, R. Cai, M. Alsaeedi, M. Akhoundi / Eur. J. Pure Appl. Math, 17 (2) (2024), 604-615 610 df (1) = f ⊙ 1 = f , df (f) = f ⊙ f = f , df (f ′) = f ⊙ f ′ = 0. Clearly, Fixd(E) = {0, f} is a hyperideal of E. Theorem 4. Let (E,⊕,⊙,≤) ∈ OKH and d ∈ Der(E) with d(g) = d(1)⊙ g; ∀g ∈ E. If d is a homomorphism, then d is idempotent. Proof. Let g ∈ E. We have d2(g) = d(d(g)) = d(1⊙ d(g)) = d(1)⊙ (1⊙ d(g)) = (d(1)⊙ 1)⊙ d(g) = d(1)⊙ d(g) = d(1⊙ g) = d(g). Hence, d2 = d. Example 2. In Example 1, d(1⊙ 1) = d(1) = f = f ⊙ f = d(1)⊙ d(1), d(1⊙ f) = d(f) = f = f ⊙ f = d(1)⊙ d(f), d(1⊙ f ′) = d(f ′) = 0 = f ⊙ 0 = d(1)⊙ d(f ′), d(f ⊙ f) = d(f) = f = f ⊙ f = d(f)⊙ d(f), d(f ⊙ f ′) = d(0) = 0 = f ⊙ 0 = d(f)⊙ d(f ′). Hence, d is a homomorphism of E. Also, d(g) = d(1)⊙ g; ∀g ∈ E. R. Cai, M. Alsaeedi, M. Akhoundi / Eur. J. Pure Appl. Math, 17 (2) (2024), 604-615 611 Now, by Theorem 4, d is idempotent. Definition 6. Let (E,⊕,⊙,≤) ∈ OKH and d ∈ Der(E) be a homomorphism. A proper hyperideal W of E is said to be a prime hyperideal associated to d if g ⊙ g′ ∈ W ⇒ g ∈ W or d(g′) ∈ W, ∀g, g′ ∈ E. Theorem 5. Let (E,⊕,⊙,≤) ∈ OKH and d ∈ Der(E) be a homomorphism. Then Y is a prime hyperideal of E associated to d iff for any hyperideals G and G′ of E, we have G⊙G′ ⊆ Y ⇒ G ⊆ Y or d(G′) ⊆ Y . Proof. (⇒): Let Y be a prime hyperideal of E associated to d, G⊙G′ ⊆ Y and G ⊈ Y , where G,G′ are hyperideals of E. As G ⊈ Y , ∃g ∈ G such that g /∈ Y . Take any g′ ∈ G′. Then, g ⊙ g′ ∈ G⊙G′ ⊆ Y . Since Y is a prime hyperideal associated to d and g /∈ Y , we get d(g′) ∈ Y . Hence, d(G′) ⊆ Y . (⇐): Suppose that g ⊙ g′ ∈ Y for some g, g′ ∈ E. Then < g ⊙ g′ >⊆ Y . So, < g > ⊙ < g′ >⊆< g ⊙ g′ >⊆ Y . Hence, < g >⊆ Y or d(< g′ >) ⊆ Y . Thus, g ∈ Y or d(g′) ∈ Y . Therefore, Y is a prime hyperideal associated to d. Example 3. In Example 1, Y = {0, f} is a prime hyperideal associated to d. Theorem 6. Let (E,⊕,⊙,≤) ∈ OKH and d ∈ Der(E) be a homomorphism. If W is a prime hyperideal associated to d, then √ W := {t ∈ E | ∃n ∈ N such that tn ∈ W} is a prime hyperideal of E associated to d. Proof. Let z, z′ ∈ √ W . By the proof of Lemma 3.19 in [17], z ⊕ z′ ⊆ √ W and ⊖z ∈ √ W . Also, for any t ∈ E, t⊙ z, z ⊙ t ∈ √ W . Now, let q ∈ ( √ W ]. Then q ≤ z for some z ∈ √ W . As z ∈ √ W , ∃n ∈ N such that zn ∈ W . R. Cai, M. Alsaeedi, M. Akhoundi / Eur. J. Pure Appl. Math, 17 (2) (2024), 604-615 612 Since q ≤ z, we get qn ≤ zn ∈ W . Thus, qn ∈ W . So, q ∈ √ W and hence ( √ W ] ⊆ √ W . Let g ⊙ g′ ∈ √ W and g /∈ √ W for g, g′ ∈ E. Claim: d(g′) ∈ √ W . As g ⊙ g′ ∈ √ W , ∃n ∈ N such that (g ⊙ g′)n ∈ W . So, gn ⊙ g′n ∈ W . As W is a prime hyperideal associated to d and gn /∈ W , d(g′n) ∈ W . Since d is a homomorphism of T , we obtain (d(g′))n = d(g′n) ∈ W . Thus, d(g′) ∈ √ W . So, √ W is a prime hyperideal associated to d. Let Ω be an index set and (Ti,⊕i,⊙i,≤i) ∈ OKH, for all i ∈ Ω. Then,∏ i∈Ω Ti = {(ti)i∈Ω | ti ∈ Ti} ∈ OKH. Indeed: for any (wi)i∈Ω, (w ′ i)i∈Ω ∈ ∏ i∈Ω Ti, (i) (wi)i∈Ω ⊕ (w′ i)i∈Ω = {(ti)i∈Ω | ti ∈ wi ⊕i w ′ i}; (ii) (wi)i∈Ω ⊙ (w′ i)i∈Ω = (wi ⊙i w ′ i)i∈Ω; (iii) (wi)i∈Ω ≤ (w′ i)i∈Ω ⇔ wi ≤i w ′ i, ∀i ∈ Ω. Define the map πi : ∏ i∈Ω Ti → Ti by πi((wi)i∈Ω) = wi, for each (wi)i∈Ω ∈ ∏ i∈Ω Ti and i ∈ Ω and define the map ρi : Ti → ∏ i∈Ω Ti by (ρi(t))(j) =  t, if i = j 0j , otherwise for each t ∈ Ti. Theorem 7. Let Ω be an index set and (Ti,⊕i,⊙i,≤i) ∈ OKH, for all i ∈ Ω. If d ∈ Der( ∏ i∈Ω Ti), then di = πidρi ∈ Der(Ti), for all i ∈ Ω. R. Cai, M. Alsaeedi, M. Akhoundi / Eur. J. Pure Appl. Math, 17 (2) (2024), 604-615 613 Proof. Let d ∈ Der( ∏ i∈Ω Ti) and z, z′ ∈ Ti, for all i ∈ Ω. Then, di(z ⊕i z ′) = πidρi(z ⊕i z ′) = πid(ρi(z ⊕i z ′)) = πi(d(ρi(z)⊕i ρi(z ′))) ⊆ πi(d(ρi(z))⊕i d(ρi(z ′))) = πidρi(z)⊕i πidρi(z ′) = di(z)⊕i di(z ′), and di(z ⊙i z ′) = πidρi(z ⊙i z ′) = πid(ρi(z ⊙i z ′)) = πi(d(ρi(z)⊙i ρi(z ′))) ∈ πi((d(ρi(z))⊙i ρi(z ′))⊕i (ρi(z)⊙i d(ρi(z ′))) = (πidρi(z)⊙i πiρi(z ′))⊕i (πiρi(z)⊙i πidρi(z ′)) = (πidρi(z)⊙i z ′)⊕i (z ⊙i πidρi(z ′)) = (di(z)⊙i z ′)⊕i (z ⊙i di(z ′)). Since d ∈ Der( ∏ i∈Ω Ti), it follows that d is isotone. Also, since πi and ρi are isotone, we get πidρi is isotone. Therefore, di ∈ Der(Ti), for all i ∈ Ω. Let Ω be an index set, (Ti,⊕i,⊙i,≤i) ∈ OKH and di ∈ Der(Ti), for all i ∈ Ω. Define∏ i∈Ω di : ∏ i∈Ω Ti → ∏ i∈Ω Ti by ( ∏ i∈Ω di)((wi)i∈Ω) = (di(wi))i∈Ω, for each (wi)i∈Ω ∈ ∏ i∈Ω Ti. Corollary 3. Let Ω be an index set and (Ti,⊕i,⊙i,≤i) ∈ OKH, for all i ∈ Ω. If d ∈ Der( ∏ i∈Ω Ti), then d = ∏ i∈Ω πidρi iff d ∈ ∏ i∈Ω Der(Ti). Proof. (⇒): Let d ∈ Der( ∏ i∈Ω Ti) and d = ∏ i∈Ω πidρi. By Theorem 7, we have πidρi ∈ Der(Ti), ∀i ∈ Ω. Thus, d ∈ ∏ i∈Ω Der(Ti). REFERENCES 614 (⇐): Let d ∈ ∏ i∈Ω Der(Ti) and w ∈ Ti. Then (πi( ∏ i∈Ω di)ρi)(w) = di(w), where di ∈ Der(Ti). So, (πi( ∏ i∈Ω di)ρi) = di. Thus, d = ∏ i∈Ω di for some di ∈ Der(Ti). Hence, d = ∏ i∈Ω πidρi. 4. Conclusions This study was conducted to investigate the significant relationship between homo- morphisms and derivations in ordered hyperrings. 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