Construction of k-Hyperideals by P -Hyperoperations H. Hedayati, R. Ameri∗ Department of Mathematics, Faculty of Basic Science, University of Mazandaran, Babolsar, Iran e-mail : {h.hedayati, ameri}@umz.ac.ir Abstract In this note we present a method to construction new k-hyperideals from given k-ideals of a semiring R by using of the P -hyperoperations. Then we investigate the relationship between them. In particular, we describe all k- hyperideals of the semihyperring of the nonnegative integers. Keywords: (semi)hyperring, k-(hyper)ideal, P -hyperoperation, weak distributive 1 1 Introduction Hyperstructures theory was born in 1934 when Marty [12] defined hypergroups as a generalization of groups. Also Wall in 1937 defined the notion of cyclic hyper- group. This theory has been studied in the following decades and nowadays by many mathematicians. A short review of the theory of hypergroups appears in [2]. A re- cent books [2], [3] and [15] contain a wealth of applications. There are applications 1* Correspondence Author 75 to the following subjects: geometry, hypergraphs, binary relations, combinatorics, codes, cryptography, probability, groups, rational algebraic functions and etc. One of the several contexts which they arise is hyperring. First M. Krasner studied hy- perrings, which is a triple (R, +, .), where (R, +) is a canonical hypergroup and (R, .) is a semigroup, such that for all a, b, c ∈ R, a(b + c) = ab + ac, (b + c)a = ba + ca ([10]). The notion of k-ideals in ordinary semirings was introduced by D. R. Latore in 1965 ([11]). Also M. K. Sen and others worked on one-sided k-ideals and maximal k-ideals of semirings ([14], [16]). The authors in [6] introduced the notion of k-hyperideals in the sense of Krasner and obtained some related results about this notion. We now follow [6] to introduce a method to construct new k-hyperideals from given k-ideals. In section 2 of this paper, we gather all the preliminaries of (semi)hyperrings and k-(hyper)ideals which will be used in the next sections. In section 3, we represent some methods for construction semihyperrings from semirings by P -hyperoperations and then we investigate the relationship between their k-hyperideals and k-ideals. As an important result of this section, all k-hyperideals of the nonnegative integers N∗ as a semihyperring, constructed by P -hyperoperations, are described. In section 4, we characterize the k-hyperideals of product of semihyperrings which are made by P -hyperoperations and a family of semirings. 2 Preliminaries A map ◦ : H × H −→ P∗(H) is called hyperoperation or join operation. A hypergroupoid is a set H with together a (binary) hyperoperation ◦. A hypergroupoid (H, ◦), which is associative, that is x ◦ (y ◦ z) = (x ◦ y) ◦ z, ∀x, y, z ∈ H is called a semihypergroup . A hypergroup is a semihypergroup such that ∀x ∈ H we have x◦H = H = H ◦x, which is called reproduction axiom (see [2]). Let H be a hypergroup and K be a nonempty subset of H. Then K is said to be 76 a subhypergroup of H if itself is a hypergroup under hyperoperation ”◦” restricted to K. Hence it is clear that a subset K of H is a subhypergroup if and only if aK = Ka = K, under the hyperoperation on H. Definition 2.1. A hyperalgebra (R, +, .) is called a semihyperring if and only if (i) (R, +) is a semihypergroup; (ii) (R, .) is a semigroup; (iii) ∀a, b, c ∈ R, a.(a + b) = a.b + a.c and (b + c).a = b.a + c.a. Remark. In Definition 2.1, if we replace (iii) by ∀a, b, c ∈ R, a.(a + b) ⊆ a.b + a.c and (b + c).a ⊆ b.c + c.a, we say that R is a weak distributive semihyperring. A semihyperring R is called with zero element, if there exists an unique element 0 ∈ R such that 0 + x = x = x + 0 and 0x = 0 = x0 for all x ∈ R. A semihyperring R is called additive commutative, if x + y = y + x, ∀x, y ∈ R. A semihyperring (R, +, .) is called a hyperring provided (R, +) is a canonical hypergroup. Definition 2.2. A hyperring (R, +, .) is called (i)commutative if a.b = b.a for all a, b ∈ R; (ii)with identity, if there exists an element, say 1 ∈ R, such that 1.x = x.1 = x for all x ∈ R. Let (R, +, .) be a hyperring, a nonempty subset S of R is called a subhyperring of R if (S, +, .) is itself a hyperring. Definition 2.3. A subhyperring I of a hyperring R is said to be a (resp. right) left hyperideal of R provided that ( resp. x.r ∈ I ) r.x ∈ I for all r ∈ R and for all x ∈ I. We say that I is a hyperideal if I is both a left and right hyperideal. Definition 2.4.[11] Let (R, +, .) be a semiring. A nonempty subset I of R is called a left k-ideal of R, if I is a left ideal of R and for a ∈ I and x ∈ R we have a + x ∈ I or x + a ∈ I =⇒ x ∈ I. 77 Similarly a right k-ideal is defined. A two sided k-ideal or simply a k-ideal is both a left and right k-ideal. We denote I as k-ideal (resp. ideal) of R by I Ck R (resp. I C R). In the sequel, by R we mean a semihyperring, unless otherwise specified. Definition 2.5.[6] Let (R, +, .) be a ( weak distributive ) semihyperring. A nonempty subset I of R is called (i) a left ( resp. right) hyperideal of R if and only if (a) (I, +) is a semihypergroup of (R, +); and (b) rx ∈ I (resp. xr ∈ I), for all r ∈ R and for all x ∈ I. (ii) a hyperideal of R if it is both left and right hyperideal of R. The hyperideal I of R is denoted by I Ch R. (iii) a left k-hyperideal of R, if I is a left hyperideal of R and for a ∈ I and x ∈ R we have a + x ≈ I or x + a ≈ I =⇒ x ∈ I, where by A ≈ B we mean A ∩ B 6= ∅. (iv) Similarly a right k-hyperideal is defined. A two sided k-hyperideal or simply a k-hyperideal is both a left and right k-hyperideal. We denote I as k-hyperideal of R by I Ck.h R. 3 Construction of k-hyperideals by P -hyperoperations In this section we apply three kinds of P -hyperoperations (which were introduced for Hv-structures in [15]) to construct semihyperrings from semirings. Then we investigate the relationship between their k-hyperideals and k-ideals . Definition 3.1. Let (R, +, .) be semiring and ∅ 6= P ⊆ R. We define two hyperop- erations as follows x ⊕c y = {x + t + y | t ∈ P}, 78 x � y = x.y = xy, which ⊕c is called centre P -hyperoperation. Proposition 3.2. Let (R, +, .) be semiring and P ⊆ R be a nonempty such that P R ⊆ P and RP ⊆ P , then (R, ⊕c, �) is a weak distributive semihyperring. Proof . First, we show (R, ⊕c) is a semihypergroup. For this we prove that (x ⊕c y) ⊕c z = x ⊕c (y ⊕c z). For x, y, z ∈ R we have a ∈ (x ⊕c y) ⊕c z =⇒ ∃a1 ∈ x ⊕c y, a ∈ a1 ⊕c z =⇒ ∃t1, t2 ∈ P, a = a1 + t1 + z, a1 = x + t2 + y =⇒ a = x + t2 + y + t1 + z =⇒ a = x + t2 + b, b = y + t1 + z ∈ y ⊕c z =⇒ a ∈ x ⊕c b, b ∈ y ⊕c z =⇒ a ∈ x ⊕c (y ⊕c z) =⇒ (x ⊕c y) ⊕c z ⊆ x ⊕c (y ⊕c z). Similarly, we obtain that (x ⊕c y) ⊕c z ⊇ x ⊕c (y ⊕c z). Clearly (R, �) is a semigroup, since (R, .) is a semigroup and x � y = xy. We now prove weak distributivity, that is x � (y ⊕c z) ⊆ (x � y) ⊕c (x � z) = xy ⊕c xz. For this we have a ∈ x � (y ⊕c z) =⇒ ∃a1 ∈ y ⊕c z, a = x � a1 = xa1 =⇒ ∃t ∈ P, a = xa1, a1 = y + t + z =⇒ a = x(y + t + z) = xy + xt + xz ∈ xy ⊕c xz ( RP ⊆ P ) =⇒ x � (y ⊕c z) ⊆ xy ⊕c xz. 79 Similarly we conclude that (y ⊕c z) � x ⊆ yx ⊕c zx.� Definition 3.3. Let (R, +, .) be a semiring and ∅ 6= P ⊆ R. We define the following hyperoperations x ⊕r y = {x + y + t | t ∈ P}, x ⊕l y = {t + x + y | t ∈ P}, x � y = xy, which ⊕r and ⊕l are called right P -hyperoperation and left P -hyperoperation respec- tively. Proposition 3.4. Let (R, +, .) be a semiring and P ⊆ R be a nonempty such that P R ⊆ P and RP ⊆ P and x + P = P + x, for all x ∈ R. Then (R, ⊕r, �) and (R, ⊕l, �) are weak distributive semihyperrings. Proof. First, we prove that (x ⊕r y) ⊕r z = x ⊕r (y ⊕r z). For this we have a ∈ (x ⊕r y) ⊕r z =⇒ ∃a1 ∈ x ⊕r y, a ∈ a1 ⊕r z =⇒ ∃t1, t2 ∈ P, a1 = x + y + t1, a = a1 + z + t2 =⇒ ∃t1, t2 ∈ P, a = x + y + t1 + z + t2 (1) also we have b ∈ x ⊕r (y ⊕r z) =⇒ ∃b1 ∈ y ⊕r z, b ∈ x ⊕r b1 =⇒ ∃w1, w2 ∈ P, b1 = y + z + w1, b = x + b1 + w2 =⇒ ∃w1, w2 ∈ P, b = x + y + z + w1 + w2 (2) From (1) we have a = x + y + t1 + z + t2 = x + y + z + w1 + t2, ∃w1 ∈ P (z + P = P + z) =⇒ a ∈ x ⊕r (y ⊕r z) (by (2)) =⇒ (x ⊕r y) ⊕r z ⊆ x ⊕r (y ⊕r z). 80 Similarly we can prove that (x ⊕r y) ⊕r z ⊇ x ⊕r (y ⊕r z). Clearly (R, �) is semigroup, since (R, .) is a semigroup. In a similar way to the Proposition 3.2 we can prove weak distributivity. Therefore (R, ⊕r, �) is a weak distributive semihyperring. Analogously we can prove that (R, ⊕l, �) is a weak distributive semihyperring. � Remark. In Propositions 3.2 and 3.4, if we replace the conditions RP ⊆ P and P R ⊆ P by rP = P = P r for all r ∈ R, then (R, ⊕c, �) and (R, ⊕r, �) and (R, ⊕l, �) become semihyperring. Theorem 3.5. Let (R, +, .) be a semiring with zero and P be the same as Propo- sition 3.2 such that 0 ∈ P . Then there is a one-to-one correspondence between the k-ideals of (R, +, .) containing P and k-hyperideals of (R, ⊕c, �). Proof. Let I be a k-ideal of (R, +, .) containing P . First we prove that I /h (R, ⊕c, �). Suppose that x, y ∈ I, we prove x ⊕c y ⊆ I. For this we have z ∈ x ⊕c y =⇒ ∃t ∈ P ⊆ I, z = x + t + y =⇒ z = x + t + y ∈ I ( since x, t, y ∈ I ) =⇒ x ⊕c y ⊆ I. Also if r ∈ R and x ∈ I, then r � x = rx ∈ I, since I / (R, +, .). Thus I is a hyperideal of (R, ⊕c, �). We now prove that I /k.h (R, ⊕c, �). For r ∈ R and x ∈ I we have r ⊕c x ≈ I =⇒ ∃z ∈ r ⊕c x ≈ I =⇒ ∃t ∈ P, z = r + t + x, z ∈ I =⇒ r + t + x ∈ I, t + x ∈ I =⇒ r ∈ I ( since I Ck (R, +, .) ) =⇒ I /k.h (R, ⊕c, �). 81 Conversely, suppose that I /k.h (R, ⊕c, �). We prove that I is a k-ideal of (R, +, .) containing P . For this we have x, y ∈ I =⇒ x ⊕c y ⊆ I ( I Ch (R, ⊕c, �) ) =⇒ ∀t ∈ P, x + t + y ∈ I =⇒ x + y ∈ I ( 0 ∈ P ) . On the other hand r ∈ R, x ∈ I =⇒ r � x ∈ I ( I Ch (R, ⊕c, �) ) =⇒ rx ∈ I. Also we have r + x ∈ I, x ∈ I =⇒ r + 0 + x ∈ I, x ∈ I (0 ∈ P ) =⇒ r ⊕c x ≈ I, x ∈ I =⇒ r ∈ I ( I Ck.h (R, ⊕c, �) =⇒ I Ck (R, +, .). We have 0 ⊕c 0 ⊆ I, then {0 + t + 0 | t ∈ P} ⊆ I, therefore P ⊆ I. � Theorem 3.6. Let (R, +, .) be a semiring with zero and P be the same as Propo- sition 3.4 such that 0 ∈ P . Then there is a one-to-one correspondence between k-ideals of (R, +, .) containing P and k-hyperideals of ( (R, ⊕l, �) ) (R, ⊕r, �). Proof. The proof is similar to the proof of Theorem 3.5 by some manipulation. � Examples. (i) Let N be the set of natural numbers and 2N = {2, 4, 6, 8, ...}. Clearly (N, +, .) is a semiring and 2N is a k-ideal of (N, +, .). Now if P = {4, 8, 12, 16, ...} ⊆ 2N, then it is easy to verify that (N, ⊕c, �) is a weak distributive semihyperring, where for all m, n ∈ N we have m ⊕c n = {m + k + n | k ∈ P} and m � n = mn. Thus 2N is a k-hyperideal of (N, ⊕c, �). 82 (ii) Let N∗ = N ∪ {0} and N∗[x] = {f (x) = n∑ i=1 aix i | ai ∈ N∗}. Clearly (N∗[x], +, .) is a semiring and < x >= {f (x) ∈ N∗[x] | a0 = 0} is a k-ideal of (N∗[x], +, .) generated by x. Set P =< xm > for m ∈ N. Obviously, 0 ∈ P ⊆< x >. Then by Propositions 3.2 and 3.5, (N∗[x], ⊕c, �) is a weak distributive semihyperring and < x > is a k-hyperideal of (N∗[x], ⊕c, �). In the next theorem we describe all k-hyperideals of semihyperring of the nat- ural numbers constructed by P -hyperoperation. For this we consider the semiring (N, +, .) of natural numbers by usual ordinary operations. Theorem 3.7. Let 0 ∈ P ⊆ N∗ and P N∗ ⊆ P and N∗P ⊆ P and P ⊆ I. Then I is a k-hyperideal of (N∗, ⊕c, �) if and only if there exists a ∈ N∗ such that I = {na | n ∈ N∗}. Proof. By Theorem 3.5, I Ck.h (N∗, ⊕c, �) if and only if I Ck (N∗, +, .). Also by Proposition 4.1 [14], I Ck (N∗, +, .) if and only if there exists a ∈ N∗ such that I = {na | n ∈ N∗}. � 4 Product of k-hyperideals In the sequel by ∏ i∈I Ri, we mean the cartesian product of the family {Ri}i∈I . It means ∏ i∈I Ri = {(xi)i∈I | xi ∈ Ri}. Proposition 4.1. Let {Ri}i∈I be a family of semirings and Pi ⊆ Ri be nonempty such that RiPi ⊆ Pi and PiRi ⊆ Pi, for all i ∈ I. For (xi)i∈I , (yi)i∈I ∈ ∏ i∈I Ri. Define (xi)i∈I ⊕c (yi)i∈I = {(xi + ti + yi)i∈I | ti ∈ Pi}, (xi)i∈I � (yi)i∈I = (xiyi)i∈I . Then ( ∏ i∈I Ri, ⊕c, �) is a weak distributive semihyperring . 83 Proof. First we show that ( ∏ i∈I Ri, ⊕c) is a semihypergroup. For this we prove that (xi)i∈I ⊕c [(yi)i∈I ⊕c (zi)i∈I ] = [(xi)i∈I ⊕c (yi)i∈I ] ⊕c (zi)i∈I . We have A ∈ (xi)i∈I ⊕c [(yi)i∈I ⊕c (zi)i∈I ] =⇒ ∃ti ∈ Pi, A ∈ (xi)i∈I ⊕c (yi + ti + zi)i∈I =⇒ ∃t′i ∈ Pi, A = (xi + t ′ i + yi + ti + zi)i∈I =⇒ A ∈ (xi + t′i + yi)i∈I ⊕c (zi)i∈I =⇒ A ∈ [(xi)i∈I ⊕c (yi)i∈I ] ⊕c (zi)i∈I =⇒ (xi)i∈I ⊕c [(yi)i∈I ⊕c (zi)i∈I ] ⊆ [(xi)i∈I ⊕c (yi)i∈I ] ⊕c (zi)i∈I . In a similar way, we can prove the reverse inclusion. Therefore, ( ∏ i∈I Ri, ⊕c) is a semihypergroup. Clearly ( ∏ i∈I Ri, �) is a semigroup. It is enough we prove weak distributivity. For this we should prove that (xi)i∈I � [(yi)i∈I ⊕c (zi)i∈I ] ⊆ (xiyi)i∈I ⊕c (xizi)i∈I . We have A ∈ (xi)i∈I � [(yi)i∈I ⊕c (zi)i∈I ] =⇒ ∃ti ∈ Pi, A ∈ (xi)i∈I � (yi + ti + zi)i∈I =⇒ A = (xi(yi + ti + zi))i∈I = (xiyi + xiti + xizi)i∈I ∈ (xiyi)i∈I ⊕c (xizi)i∈I ( RiPi ⊆ Pi ). This completes the proof. � Proposition 4.2. If {Ri}i∈I is a family of semirings and for all i ∈ I, Pi ⊆ Ri is nonempty such that RiPi ⊆ Pi and PiRi ⊆ Pi and xi + Pi = Pi + xi, for all xi ∈ Ri, then ( ∏ i∈I Ri, ⊕r, �) and ( ∏ i∈I Ri, ⊕l, �) are weak distributive semihyperring where (xi)i∈I ⊕r (yi)i∈I = {(xi + yi + ti)i∈I | ti ∈ Pi}, (xi)i∈I ⊕l (yi)i∈I = {(ti + xi + yi)i∈I | ti ∈ Pi}, 84 (xi)i∈I � (yi)i∈I = (xiyi)i∈I . Proof. First we prove that ( ∏ i∈I Ri, ⊕r) is a semihypergroup. For this we prove that (xi)i∈I ⊕r [(yi)i∈I ⊕r (zi)i∈I ] = [(xi)i∈I ⊕r (yi)i∈I ] ⊕r (zi)i∈I . We have A ∈ (xi)i∈I ⊕r [(yi)i∈I ⊕r (zi)i∈I ] =⇒ ∃ti ∈ Pi, A ∈ (xi)i∈I ⊕r (yi + zi + ti)i∈I =⇒ ∃t′i ∈ Pi, A = (xi + yi + zi + ti + t ′ i)i∈I =⇒ ∃wi ∈ Pi, A = (xi + yi + wi + zi + t ′ i)i∈I ( since zi + Pi = Pi + zi ) ∈ (xi + yi + wi)i∈I ⊕r (zi)i∈I ⊆ [(xi)i∈I ⊕r (yi)i∈I ] ⊕r (zi)i∈I =⇒ (xi)i∈I ⊕r [(yi)i∈I ⊕r (zi)i∈I ] ⊆ [(xi)i∈I ⊕r (yi)i∈I ] ⊕r (zi)i∈I . Similarly, we can prove that the reverse inclusion. Clearly ( ∏ i∈I Ri, �) is a semigroup. Also the weak distributivity is obtained sim- ilar to the proof of Proposition 4.1. Therefore ( ∏ i∈I Ri, ⊕r, �) is a semihyperring. Analogously we can prove that ( ∏ i∈I Ri, ⊕l, �) is a weak distributive semihyperring. This completes the proof. � Remark. In Propositions 4.1 and 4.2, if we replace the conditions RiPi ⊆ Pi and PiRi ⊆ Pi by the condition riPi = Pi = Piri, for all ri ∈ Ri and for all i ∈ I, then ( ∏ i∈I Ri, ⊕c, �), ( ∏ i∈I Ri, ⊕r, �) and ( ∏ i∈I Ri, ⊕l, �) will be semihyperrings. Proposition 4.3. If {Rj}j∈J is a family of semirings and for all j ∈ J, Pj ⊆ Rj is nonempty such that RjPj ⊆ Pj and PjRj ⊆ Pj. Then I is a k-hyperideal of ( ∏ j∈J Rj, ⊕c, �) if and only if I = ∏ j∈J Ij such that Ij /k.h (Rj, ⊕cj , �j), where xj ⊕cj yj = {xj + tj + yj | tj ∈ Pj}, xj �j yj = xjyj. 85 Proof. (=⇒) For all j ∈ J define Ij = {x ∈ Rj | (xi)i∈J ∈ I, ∃xi ∈ Ri, x = xj}. We have x, y ∈ I =⇒ ∃xi, yi ∈ Ri, (xi)i∈J , (yi)i∈J ∈ I, x = xj, y = yj =⇒ (xi)i∈J ⊕c (yi)i∈J ⊆ I (I Ch ( ∏ j∈J Rj, ⊕c, �)) =⇒ ∀ti ∈ Pi, (xi + ti + yi)i∈J ∈ I (∀i ∈ J) =⇒ ∀tj ∈ Pj, x + tj + y ∈ Ij =⇒ x ⊕cj y ⊆ Ij. Now suppose that rj ∈ Rj, x ∈ Ij =⇒ ∃ri ∈ Ri, (ri)i∈J ∈ ∏ i∈J Ri and ∃xi ∈ Ri, (xi)i∈J ∈ I, x = xj =⇒ (ri)i∈J � (xi)i∈J ∈ I ( I Ch ( ∏ i∈J Ri, ⊕c, �) ) =⇒ (rixi)i∈J ∈ I =⇒ rjxj ∈ Ij ( by definition of Ij ) . Therefore Ij Ch Rj. We now show that Ij /k.h Rj for all j ∈ J. We have rj ∈ Rj, xj ∈ Ij, rj ⊕cj xj ≈ Ij =⇒ ∃tj ∈ Pj, rj + tj + xj ∈ Ij =⇒ (rj)j∈J ⊕c (xj)j∈J ≈ I, where (rj)j∈J ∈ ∏ j∈J Rj, (xj)j∈J ∈ ∏ j∈J Ij. Then since I Ck.h ( ∏ j∈J Rj, ⊕c, �) we have (rj)j∈J ∈ I =⇒ rj ∈ Ij, ∀j ∈ J =⇒ Ij /k.h Rj. (⇐=) Suppose that I = ∏ j∈J Ij such that Ij /k.h (Rj, ⊕cj , �j). First we prove I /h ( ∏ j∈J Rj, ⊕c, �). Let (xj)j∈J , (yj)j∈J ∈ I, then (xj)j∈J ⊕c (yj)j∈J = {(xj + tj + yj)j∈J | tj ∈ Pj} ⊆ ∏ j∈J Ij; 86 also we have Ij /h (Rj, ⊕cj , �j) =⇒ ∀tj ∈ Pj, xj + tj + yj ∈ Ij =⇒ (xj)j∈J ⊕c (yj)j∈J ⊆ I. Now if (rj)j∈J ∈ ∏ j∈J Rj and (xj)j∈J ∈ I, then (rj)j∈J � (xj)j∈J = (rjxj)j∈J ∈ ∏ j∈J Ij, since rjxj ∈ Ij by hypothesis. We now prove that I /k.h ( ∏ j∈J Rj, ⊕c, �). For this we have (rj)j∈J ∈ ∏ j∈J Rj, (xj)j∈J ∈ I, (rj)j∈J ⊕c (x1, x2) ≈ I =⇒ ∃tj ∈ Pj, (rj + tj + xj)j∈J ∈ I = ∏ j∈J Ij =⇒ ∃tj ∈ Pj, rj + tj + xj ∈ Ij, ∀j ∈ J =⇒ rj ⊕cj xj ≈ Ij, rj ∈ Rj, xj ∈ Ij =⇒ rj ∈ Ij ( Ij Ck.h (Rj, ⊕cj , �j) ) =⇒ (rj)j∈J ∈ ∏ j∈J Ij. � Proposition 4.4. Let {Rj}j∈J be a family of semirings. Suppose that Pj ⊆ Rj be nonempty such that RjPj ⊆ Pj and PjRj ⊆ Pj and xj + Pj = Pj + xj, for all xj ∈ Rj and for all j ∈ J. Then I is a k-hyperideal of ( ∏ j∈J Rj, ⊕r, �) ( resp. ( ∏ j∈J Rj, ⊕l, �)) if and only if I = ∏ j∈J Ij such that for all j ∈ J, Ij /k.h (Rj, ⊕rj , �j), (resp. 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