EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 6, No. 2, 2013, 211-221 ISSN 1307-5543 – www.ejpam.com Sequentially Complete S-acts and Baer Type Criteria over Semigroups H. Barzegar Department of Mathematics, Tafresh University, Tafresh, Iran. Abstract. Dedicated to Professor M. Mehdi Ebrahimi on his 65th Birthday Although the Baer Criterion for injectivity is true for modules over a ring with an identity, it is an open problem for acts over a semigroup S (with or without identity). In this work, we study a kind of Baer Criterion for injectivity of acts over a semigroup S. We consider a kind of weak injectivity which we call s -completeness and give some conditions under which s-completeness coincides with injectivity. 2010 Mathematics Subject Classifications: 20M30, 08A60; 08B30 Key Words and Phrases: Sequentially pure, Injectivity, sequentially complete 1. Introduction Throughout this paper S will denote a given semigroup and Recall that, for a semigroup S, a set A is a right S-act (or an S-act) if there is a, so called, action µ : A× S → A such that, denoting µ(a, s) := as, a(st) = (as)t and if S is a monoid with 1, a1= a. A morphism f : A→ B between S-acts A, B is called a homomorphism if, for each a ∈ A, s ∈ S, f (as) = f (a)s. The category of all (right) S-acts and homomorphisms between them is denoted by Act-S. Sequentially complete S-acts are special objects of this category which will be studied here. an S-act A is called pure in an extension B of A if any system of finitely many equations over A has a solution in A whenever this is the case for B. An S-act A is called absolutely pure if it is pure in all of which extensions. A monoid S is said to be completely right pure if all its right S-acts are absolutely pure . Completely right pure monoids have further been studied in literature [see e.g. 4, 7, 8]. A characterization of completely right pure monoids was given in [7], but clearly not satisfactory. Gould in [8] attempts to remedy this by giving a characterization of completely right pure monoids in terms of right ideals and right congruences that is a closer analogue of Proposition 2.1 of [7]. Email address: h56bar@tafreshu.ac.ir http://www.ejpam.com 211 c© 2013 EJPAM All rights reserved. H. Barzegar / Eur. J. Pure Appl. Math, 6 (2013), 211-221 212 Proposition 2.1 of [8] shows that the monoid S is completely right pure if and only if every S-act A is pure with one variable in any extension. Ebrahimi and Mahmoudi [4] has introduced the concept of s-pure monomorphism in the category of Projection Algebras by a system of equations such as xs = as(s ∈ S, as ∈ A). So we are persuaded to study a kind of completely right purity for semigrops for these equations in general in the Category Act-S. The sense of s-complete is equivalent to a kind of injectivity which is called s-injectivity and studied in [12]. Here in, we characterize the semigroups S over which all S-acts are s-complete. Also, every injective S-act is s-complete but the converse is not true in general. The Baer Criterion for injectivity (weak injectivity implies injectivity) of S-acts, which is true for modules over a ring with an identity, is an open problem for acts over a semigroup S (with or without identity). Furthermore in this paper a weaker kind of Baer Criterion (s-completeness implies injectiv- ity) is investigated and some semigroups over all of which every s-complete S-act is injective is introduced. These conclusions are the main part of this article, which appear in section 4. 2. Preliminary In this section we briefly recall the definition and the categorical and algebraic ingredi- ents of the category Act-S of (right) S-acts over a semigroup S and recall sequentially pure monomorphisms in this category. For more information and the notions not mentioned here about this category see, for example, [9]. Recall that an element a ∈ A(t ∈ S) is said to be a fixed element (left zero element) if as = a (ts = t) for all s ∈ S. The S-act A∪ {0} with a fixed element adjoined to A is denoted by A0. Since the class of S-acts is an equational class, the category Act-S is complete (has all products and equalizers) and cocomplete (has all coproducts and coequalizers). In fact, limits and colimits in this category are computed as in the category Set of sets and equipped with a natural action. In particular, for a family {Ai} of S-acts their cartesian product ∏ Ai with the S-action defined by (ai)s = (ais) is the product of a family {Ai} in Act-S. the coproduct of a family {Ai} in Act-S is their disjoint union ∐ Ai = ∪(Ai × {i}) with the action of S defined by (a, i)s = (as, i) for s ∈ S, a ∈ Ai . Recall that for a family {Ai : i ∈ I} of S-acts with a unique fixed element 0, the direct sum ⊕ i∈I Ai is defined to be the subact of the product ∏ i∈I Ai consisting of all (ai)i∈I such that ai = 0 for all i ∈ I except a finite number. We use ⊕ i∈I Ai only for S-acts with unique fixed element. An S-act A is said to be injective if for any monomorphism g : B → C and any homomor- phism f : B→ A there exists a homomorphism h : C → A such that hg = f . An S-act A is said to be weakly injective if it is injective with respect to right ideals of S. In this section we recall the notion of sequentially pure monomorphisms mainly from [1, 2]. For simplicity, we let the letter “s” stand for the prefix “sequentially”. Definition 1. We say that A is s-pure in an extension B of A if every sequential system of equations with constants from A such as ΣA = {xs = as : s ∈ S, as ∈ A} has a solution in A whenever it has a solution in B. The system ΣA is said to be consistent if it has a solution in some extension B of A. H. Barzegar / Eur. J. Pure Appl. Math, 6 (2013), 211-221 213 Note that there is a one to one correspondence between the set of all systems of equations ΣA of the above form on an S-act A and the set of all functions k : S→ A. For any S-act B and b ∈ B, let us denote the homomorphism λb : S → B, defined by λb(s) := bs, by λb. In these notations, we have Lemma 1 ([1]). A map k : S→ A is a homomorphism if and only if there exists an extension B of A and b ∈ B such that k = λb. Remark 1 ([1]). For a subact A of B, the following are equivalent: (i) A is s-pure in B. (ii) For every b ∈ B with bS ⊆ A there is an element a ∈ A with λb = λa. (iii) Every homomorphism k : S→ A is of the form λa for some a ∈ A whenever it is of the form λb for some b ∈ B. Throughout the paper we will opt for one of the three equivalents above for s-purity. The above remark also shows that if one defines à := {b ∈ B : ∃ a ∈ A, λb = λa} and Ā := {b ∈ B : bS ⊆ A}, then A is s-pure in B if and only if Ã= Ā. For more details, see [1]. 3. Sequentially Complete S-acts In this section we study some algebraic and categorical properties of s-complete S-acts and characterize the semigroups S over which all acts are s-complete. The main result of this section is Theorem 1, which shows that s-completeness is equivalent to s-injectivity that is defined and studied in [11]. Also it is equivalent to absolutely s-pure. Some of the following results will be used in the next section. Definition 2. An S-act A is called sequentially complete or (s-complete) if every consistent system ΣA has a solution in A. Theorem 1. For an S-act A, the following are equivalent: (i) A is s-complete. (ii) A is absolutely s-pure ( that is, it is s-pure in each of its extension). (iii) A is s-pure in its injective hull E(A). (iv) A is s-injective (i.e, every homomorphism k : S −→ A is of the form λa for some a ∈ A). (v) Every homomorphism f : S→ A can be extended to a homomorphism f : S1→ A. Proof. (i)⇒ (ii) Let B be an extension of A and for b ∈ B, bS ⊆ A. So ΣA = {xs = bs | s ∈ S} is a consistent system which has a solution a in A. Thus A is s-pure in B. (ii) ⇒ (i) Let the system ΣA has a solution in an s-pure extension B of A. Since A is absolutely s-pure, ΣA has a solution a in A. H. Barzegar / Eur. J. Pure Appl. Math, 6 (2013), 211-221 214 The equivalency of (ii), (iii) and (iv) is obtained from Theorem 2.2 of [10]. The equivalency of (iv) and (v) is obtained from [11], Theorem 2.7. From here to the end of paper we use some parts of Theorem 1 for s-completeness. Remark 2. Having parts (iv) and (v) of the above theorem, one can easily get some relations between s-completeness and injectivity or any types of weak injectivity [see 9]. For example, we have the following: Every injective S-act is s-complete. But the converse is not in generally true, indeed, the monoid S = Z with the usual multiplication, is s-complete as an S-act but it is not divisible. So it is not injective S-act. Also weak injectivity does not imply s-completeness. To show this fact, consider S = (N, min). Since identity homomorphism on N is not of the form λa, then NN is not s-complete. Now let I be a right ideal of S and f : I −→ N be a homomorphism. We want to extend f to S. The case I = S is obvious. Otherwise, I is of the form mS = {ms | s ∈ S}. The homomorphism g : N→ N defined by g(n) = f (mn) is an extension of f . Thus N is a weakly injective as an S-act. Theorem 2. An S-act A is s-complete and weakly injective if and only if for every right ideal I of S, every homomorphism f : I → A is of the form λa for some a ∈ A. Proof. By using Theorem 1 the Sufficiency is clear. We show only the Necessity. Let A be an s-complete and weakly injective S-act. Take a homomorphism f : I → A from a right ideal I of S. Since A is weakly injective, f can be extended to a homomorphism f : S → A. Now, since A is s-complete, f = λa for some a ∈ A, and hence so is f . Lemma 2. If A is s-pure in B and B is s-complete, then A is s-complete. Proof. Let f : S → A be a homomorphism. Since B is s-complete, f is of the form λb for some b ∈ B and since A is s-pure in B, it is of the form λa for some a ∈ A. As a result of Lemma 2, we have the following corollary. But first recall that a subact A of B is a retract of B if there exists a homomorphism, so called retraction, g : B → A such that g|A = idA. Corollary 1. A retract of an s-complete S-act is an s-complete S-act. Proof. [By [1, Lemma 2.4]] If an S-act A is a retract of B then it is s-pure in B. Now we are done by applying Lemma 2. In the next theorem, we mention a characterization of semigroup S over which all acts are s-complete. Definition 3. An S-act A is said to be principally s-complete if A is s-pure in each of its cyclic extension. Theorem 3. For a semigroup S the following are equivalent: H. Barzegar / Eur. J. Pure Appl. Math, 6 (2013), 211-221 215 (i) All right S-acts are s-complete. (ii) All right S-acts are principally s-complete. (iii) S is an s-complete S-act. (iv) S is s-pure in S1. (v) S has a left identity element. (vi) pullbacks preserve s-pure monomorphisms. Proof. The implications (i) =⇒ (ii) =⇒ (iv), (i) =⇒ (iii) and (iii) =⇒ (iv) are obtained by using Theorem 1. By [1, Theorem 3.1], (iv) and (v) are equivalent. (v) =⇒ (i) For every S-act A every homomorphism k : S −→ A is of the form k = λk(1). So by Theorem 1.(iv) the result is true. (v)⇐⇒ (vi) Apply [1, Lemma 3.3]. The following lemma will be used in Corollary 2 and in Section 4. Lemma 3. Let Ai(i ∈ I) be a family of S-acts and S be a finitely generated as an S-act. Then ⊕ Ai is s-pure in ∏ Ai . Proof. Consider S k−→ ⊕ Ai ,→ ∏ Ai such that k = λ{ai}({ai} ∈ ∏ Ai), and S = ⋃n i=1 t iS 1. So there exists a finite subset J ⊂ I such that for every s ∈ S, k(s)i = 0(i 6∈ J). Thus k = λ{bi} for bi = ( ai for i ∈ J 0 for i /∈ J . Now Theorem 1.(iv) completes the proof. Theorem 4. For a semigroup S, the following statements are equivalent: (i) Every direct sum of s-complete S-acts is s-complete. (ii) Every direct sum of s-complete S-acts is s-pure in their direct. Proof. (i) =⇒ (ii) Let {Ai} be a family of s-complete S-acts. Then ⊕ Ai is s-complete and by Theorem 1 it is s-pure in ∏ Ai . (ii) =⇒ (i) This implication is obtained by applying [11, Theorem 3.1], Theorem 1 and Lemma 2. Corollary 2. If the semigroup S is a finitely generated as an S-act, then every direct sum of s-complete S-acts is s-complete. Lemma 4. Let g : S → T be an epimorphism. If right T-act A be s-complete as an S-act, then it is s-complete as a T-act. H. Barzegar / Eur. J. Pure Appl. Math, 6 (2013), 211-221 216 Proof. Let f : T → A be a T-homomorphism. Since A is s-complete as an S-act, then f g = λa for some a ∈ A which implies that for every t ∈ T , f (t) = f (g(st)) = asst = aT g(st) = at t = λa(t). Remark 3. As we saw in Theorem 1, s-completeness is equivalent to s-injectivity which defined in [11]. Some categorical properties such as product, coproduct and direct sum of s-injective S-acts were checked in [11]. 4. Some Baer Type Criteria for Injectivity of S-Acts Although the Baer Criterion for injectivity (weak injectivity implies injectivity) is true for modules over a ring (with an identity), it is an open problem for acts over a semigroup S (with or without identity). In fact, we are not aware of any type of weak injectivity implying injec- tivity of S-acts, in general, other than Skornjakov-Baer Criterion, which says that injectivity with respect to subacts of cyclic acts implies injectivity with respect to all monomorphisms. One line of study in this regard is to investigate the relation between M1-injectivity and injectivity with respect to another subclassM2 of monomorphisms, the results of which may be called the Baer type criteria. Note that if M2 ⊆ M1, then M1-injectivity implies M2- injectivity. The Baer type problem is about the converse of this fact. By using Theorem 1 in this paper in fact we use injectivity only with respect to a monomor- phism S→ S1 which is s-completeness. As we saw in Remark 2, every injective S-acts is s-complete but the converse is not true in general. In this final section, we use s-completeness to give some Baer type results about injectivity of S-acts. We introduce some semigroups over all of which every s-complete S-acts is injective. First recall the following definition from the closure operator given in [6]. Definition 4. For a subact A of B, let Ā := {b ∈ B : bS ⊆ A}. Then, A is said to be s-dense in B if Ā= B. The following definition is a well known definition in the literature as we use here. Definition 5. For a subclass M of monomorphisms we say that the M - morphism f : A→ B is M -essential if every g : B → C is a monomorphism whenever g f is a monomorphism. For simplicity, we say essential whenM is the class of all monomorphism. We have the following result from [3]. Theorem 5. An S-act A is injective if and only if it has no proper essential extension. Henceforth in this section, we have some conditions for which every s-complete S-act is injective. But first recall the following lemma: Lemma 5. [[1]] Any s-dense, s-pure monomorphism has a retraction. H. Barzegar / Eur. J. Pure Appl. Math, 6 (2013), 211-221 217 Theorem 6. If every essential extension is s-dense, then every nonempty right ideal of S has a left zero element. In particular, S has a left zero element and every S-act A has a fixed element. Proof. Let I be a nonempty right ideal of S that does not have a left zero element. By [2], Lemma 4.1, I0 is an essential extension and hence s-dense extension of I . Thus for every s ∈ S, 0= 0s ∈ I which is a contradiction. Similar to the proof of [2, Proposition 3.6(ii) and Lemma 3.9] one gets: Lemma 6. Let A be a subact of B: (i) If C be a subact of B that |C | ≥ 2 and |C ∩A| ≤ 1, then B is not an essential extension of A. (ii) If A and B \ A have fixed element, then B is not an essential extension of A. For a subact A of an S-act B and b ∈ B, we use the notation Ib = {s ∈ S | bs ∈ A}. Also the set of all fixed elements of an S-act B and the set of all left zero elements of a semigroup S are denoted respectively by F ix(B) and Z(S). Corollary 3. Let A have at least one fixed element and B be an s-pure essential (essential) exten- sion of A. Then: (i) F ix(B)⊆ A. (ii) For every b ∈ B, Ib 6= ;. Corollary 4. If S has a left zero element and S is an essential extension of a right ideal I , then Z(S)⊆ I . If S is a left zero semigroup, then I = S. Theorem 7. If every essential extension is s-dense, then every S-act has an s-dense injective (which is injective with respect to s-dense monomorphisms) s-dense extension. Proof. Let A ∈Act-S and ι : A→ E(A) be an injective hull of A(which exists as proved in [3]). So ι is s-dense and E(A) is an s-dense injective s-dense essential extension of A. Lemma 7. Let A have at least one fixed element and B be an s-pure essential extension of A. Then (i) Ā= A. (ii) For each b ∈ B \ A, ; 6= Ib 6= S Proof. (i) By [2, Lemma 4.7.(v)], A is s-pure essential in Ā. Since A is s-dense in Ā, by Lemma 5, A is a retract of Ā which is an isomorphism by essentiality. Thus A= Ā. (ii) By part (i), Ā= A, so Ib 6= S and by Corollary 3, ; 6= Ib. Theorem 8. If A is s-complete S-act and every s-pure essential extension of A is s-dense, then A is injective. H. Barzegar / Eur. J. Pure Appl. Math, 6 (2013), 211-221 218 Proof. Let E(A) be an injective hull of A. Since A is an s-complete and E(A) is an essential extension of A, then E(A) is an s-pure essential extension of A. Thus A is a retract of E(A) by Lemma 5, which implies A is an injective S-act. Theorem 9. If S2 = S, then the following are equivalent: (i) Every s-complete S-act is injective. (ii) Every essential extension is s-dense. (iii) Every s-pure essential extension is s-dense. (iv) Every s-pure essential extension is isomorphism. Proof. (i) ⇒ (ii) Let B be an essential extension of A. By [5, Theorem 3.10], A has an s-dense injective hull such as ι : A→ Ed(A). By Theorem 1, Ed(A) is s-complete and hence it is injective. So there exists g : B→ Ed(A) such that g|A = ι which implies g is a monomorphism. Since g|A = ι is an s-dense monomorphism, it is clear that B is an s-dense extension of A. (iii) ⇒ (iv) Let B be an s-pure essential extension of A. Then A is s-dense in B and by Lemma 5, it is a retract of B. So there is a homomorphism g : B → A such that g|A is a monomorphism, which implies g is an isomorphism. (iv)⇒ (i) It is concluded from Theorem 8. Theorem 10. If for every nontrivial right ideal I of S, Ī 6= I , then every s-complete S-act with at least one fixed element is injective. Proof. Let B be an s-pure essential extension of an s-complete S-act A and b ∈ B \ A. By Lemma 7, ; 6= Ib 6= S. By hypothesis Īb 6= Ib and there exists x ∈ Īb \ Ib. Since A → B is an s-pure essential extension, the inclusion map ι : A→ A∪ {bx} is also s-pure essential and s-dense which is a retraction by Lemma 5. Essentiality of ι implies that it is an isomorphism. So bx ∈ A and x ∈ Ib which is a contradiction. Thus A= B and by Theorem 8, A is an injective S-act. Corollary 5. If S is an infinite monogenic semigroup, then every s-complete S-act with at least one fixed element is injective. Proof. First recall that every infinite cyclic semigroup is isomorphic to (N,+) [see 9]. Let I be a nontrivial right ideal of S. So there exists 1 ≤ n0 ∈ N such that I = {n ∈ N | n0 ≤ n} and S \ I = {1,2, . . . , n0− 1}. Thus Ī = I ∪ {n0− 1}. Now we get the result by using Theorem 10. Corollary 6. If S is a finite monogenic semigroup, then every s-complete S-act with at least one fixed element is injective. Proof. The proof is similar to the proof of Theorem 5. H. Barzegar / Eur. J. Pure Appl. Math, 6 (2013), 211-221 219 Corollary 7. Let S be a semigroup and s0 ∈ S, such that for all s, t ∈ S, st = s0. Then every s-complete S-act is injective. Proof. It is clear that every nonempty right ideal of S is a subset of S containing an element s0 and for every nonempty proper right ideal I of S, I 6= Ī = S. So the proof is complete by Theorem 10. Theorem 11. If every nonempty proper right ideal of S generates by a central idempotent ele- ment, then every s-complete S-act with at least one fixed element is injective. Proof. Let A be an s-complete S-act with at least one fixed element and B be an essential extension of A. By using Theorem 1 and Lemma 7, for every b ∈ B\A, ; 6= Ib 6= S. So Ib = ebS1 such that eb is a central idempotent element. Consider the map g : B→ A defined by g(b) = ( b , if b ∈ A beb , if b 6∈ A Let b ∈ B and s ∈ S. If b ∈ A, g(bs) = bs = g(b)s. If b 6∈ A, g(b)s = bebs. Two cases may happen: Case (1): s ∈ Ib. So s = ebs1 = eb(ebs1) = ebs and g(bs) = bs = b(ebs) = g(b)s. Case (2): s /∈ Ib. Then bs /∈ A and Ibs = ebsS 1. Since (bs)eb = (beb)s ∈ A, eb ∈ Ibs and hence Ib ⊆ Ibs. So ebs = ebs t for some t ∈ S. On the other hand sebs ∈ Ib implies sebs = eb t1(t1 ∈ S) = eb(eb t1) = eb(sebs). Thus g(bs) = (bs)ebs = beb(sebs) = b(ebs)ebs = b(ebs t)ebs = bebs(ebs t) = bebs t = bebs = g(b)s. Therefore g is a homomorphism and since B is an essential extension of A, g is an isomor- phism. So we get the result by Theorem 9. As usual S is a Clifford semigroup [see 9] if each α ∈ S has an inverse element(i.e, there exists α−1 ∈ S such that αα−1α = α−1αα−1) and the set of all idempotents is equal to the set of all central elements. It is easy to check that for every α ∈ S, αα−1 is an idempotent element and αS1 = αα−1S1. So we conclude the following corollary; Corollary 8. Let S be a Clifford semigroup. If each of proper nonempty right ideals of S is principal, then every s-complete S-act with at least one fixed element is injective. The following two corollaries are special cases of clifford semigroup. Corollary 9. Let the semigroup S be a commutative chain with the relation (x ≤ y⇔ x y = x) or (x ≤ y⇔ x y = y). Then every s-complete S-act with at least one fixed element is injective. Corollary 10. Let the semigroup S be a commutative band. Then every s-complete S-act with at least one fixed element is injective. H. Barzegar / Eur. J. Pure Appl. Math, 6 (2013), 211-221 220 An important semigroup satisfying to these corollaries is S = (N, min). The category of all S-acts for this semigroup, so called projection algebras, has been studied by Ebrahimi and Mahmoud [4] and Giuli [6].These types of S-acts are mostly used in computer science. Lemma 8. Let A have a fixed element and B be a proper s-pure essential extension of A. Then for every b ∈ B and every nonempty right ideal J of S, Ib ∩ J 6= ;. Proof. For b ∈ A, Ib = S and the result is obvious. For b /∈ A, let J be a nonempty right ideal of S and Ib∩J = ;. By Corollary 3, Ib 6= ; and F ix(B)⊆ A. It is clear that B′ = {bs|s ∈ J} is a subact of B and B′ ∩ A = ;. By Lemma 6, |B′| = 1 and hence for every s ∈ J , bs = b0 for some b0 ∈ B. Consider s0 ∈ J . Then for every t ∈ S, b0 t = (bs0)t = b(s0 t) = b0. So b0 ∈ F ix(B)⊆ A, which is impossible. Theorem 12. Suppose that every nonempty nontrivial right ideal I of S is maximal. Then every s-complete S-act with at least one fixed element is injective. Proof. Let A be an s-complete S-act and B be an s-pure essential extension of A. Let there exist an element b ∈ B \A. By using Lemma 7, ; 6= Ib 6= S. Consider t ∈ S \ Ib, so Ib ∩ tS1 6= ;. If tS1 6= S, Ib = Ib∩ tS1 = tS1. Thus t ∈ Ib, which is a contradiction. If tS1 = S, then Ib ⊆ tS1 and since Ib is maximal right ideal, Ib = tS. Since A→ B is an s-pure essential extension, the inclusion map ι : A→ A∪ {bt} is also s-pure essential and s-dense which is a retraction by Lemma 5. Essentiality of ι implies that it is an isomorphism. So bt ∈ A and t ∈ Ib which is a contradiction. Thus A= B and by Theorem 8, A is an injective S-act. Corollary 11. If S is a simple semigroup, then every s-complete S-act with at least one fixed element is injective. Corollary 12. Let S be a semigroup with one zero element s0, such that the set of whose ideals be {;, {s0}, S}. Then every s-complete S-act is injective. Theorem 13. Assume that for every proper nonempty right ideal I of S there exists a nonempty right ideal J of S such that I ∩ J = ;. Then every s-complete S-act with at least one fixed element is injective. Proof. we begin by proving S2 = S. Let S2 6= S. Then there exists a right ideal j of S such that S2 ∩ J = ;. Consider x ∈ J . for every s ∈ S, xs ∈ J ∩ S2, which is impossible. Now the proof is straightforward by using Theorem 9 and Lemma 8. Corollary 13. Let S be a Boolean Algebra on ideals(i.e compliment of every right ideal is a right ideal). Then every s-complete S-act with at least one fixed element is injective. Corollary 14. If S is a left zero semigroup, then every s-complete S-act is injective. 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