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Proper Behaviour of Sequential Injectivity of Acts Over SemigroupsM. M. Ebrahimi a; M. Mahmoudi a; L. Shahbaz a

a Department of Mathematics, and Center of Excellence in Algebraic and Logical Structures in DiscreteMathematics, Shahid Beheshti University, G. C. Tehran, Iran

Online Publication Date: 01 July 2009

To cite this Article Ebrahimi, M. M., Mahmoudi, M. and Shahbaz, L.(2009)'Proper Behaviour of Sequential Injectivity of Acts OverSemigroups',Communications in Algebra,37:7,2511 — 2521

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Communications in Algebra®, 37: 2511–2521, 2009Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870802467262

PROPER BEHAVIOUR OF SEQUENTIAL INJECTIVITYOF ACTS OVER SEMIGROUPS

M. M. Ebrahimi, M. Mahmoudi, and L. ShahbazDepartment of Mathematics, and Center of Excellence inAlgebraic and Logical Structures in Discrete Mathematics,Shahid Beheshti University, G. C. Tehran, Iran

Banaschewski defines and gives sufficient conditions on a category ��� and a subclass ��� ofits monomorphisms under which ���-injectivity properly behaves with respect to the notionssuch as ���-absolute retract, ���-essentialness, and the existence of ���-injective hulls.

In this article, taking ��� to be the category of acts over a semigroup S and ���d to bethe class of sequentially dense monomorphisms, introduced and studied by Ebrahimi,Giuli, Mahmoudi, Moghaddasi, and Shahbaz, we study the behaviour of ���d-injectivity(also called sequentially dense injectivity). We show, among other things, that S2 = S

is a sufficient, but not necessary, condition for the behaviour of s-injectivity. Finally, wegive an explicit description of the s-injective hulls of acts when they exist.

Key Words: Sequentially dense; Sequential injective hull; Sequential injectivity.

2000 Mathematics Subject Classification: 08A60; 08B30; 18A20; 18G05; 20M30; 20M50.

1. INTRODUCTION AND PRELIMINARIES

Injectivity is one of the most central notions in algebra, as well as in many otherbranches of mathematics. Banaschewski [1] defines and studies some set of sufficient,but not necessary, conditions on a category � and a subclass � of monomorphismsin � under which �-injectivity properly behaves in the sense that the following threepropositions hold (the definition of the terms will be given in the sequel):

Proposition 1.1 (First Theorem of Proper Behaviour). ForeveryA ∈ �, the followingconditions are equivalent:

(I1) A is �-injective;(I2) A is �-absolute retract;(I3) A has no proper �-essential extension.

Proposition 1.2 (Second Theorem of Proper Behaviour). Every A ∈ � has an�-injective hull which is unique up to isomorphism.

Received June 8, 2007; Revised December 14, 2007. Communicated by V. Gould.Address correspondence to M. Mahmoudi, Shahid Beheshti University, Tehran 19839, Iran;

E-mail: mojganmahmoudi@yahoo.com and m-mahmoudi@cc.sbu.ac.ir

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2512 EBRAHIMI ET AL.

Proposition 1.3 (Third Theorem of Proper Behaviour). For an extension B of A,the following conditions are equivalent:

(H1) B is an �-injective hull of A;(H2) B is a maximal �-essential extension of A;(H3) B is a minimal �-injective extension of A.

Tholen in [14] further studies these relations, and Ebrahimi in [3] studies theproperly behaved injectivity for algebras in a Grothendieck topos.

In this article, we take � to be the category of (right) acts over a semigroupS (see Section 1) and �d to be the class of sequentially dense monomorphismsof (right) S-acts and recall the characterization of �d-essential extensions (seeSection 2). Then, we study �d-injectivity and its behaviour in Section 3. Finally,in Section 4, we characterize �d-injective hulls and explicitly find these extensionsfor acts over some classes of semigroups. Some of these results generalize someof the results in [6, 10, 11]. At the end of Section 4, we give some results on theexistence and nonexistence of �d-injective hulls.

In the rest of this section we briefly recall the definition and the categoricaland algebraic ingredients of the category Act-S of (right) S-acts needed in the sequel.For more information see [5] and [9].

Actions of a semigroup on a set have always been a useful tool to studymathematical structures and have recently captured the interest of some computerscientists, too (see [7] and its references). For this reason and because of its closerelation to the category of sets, one can take the category Act-S of (right) S-acts, fora semigroup S, as the “universe of discourse” to study mathematical notions in it.

Let S be a semigroup and A be a set. If we have a mapping

� � A× S → A

�a� s� �−→ as �= ��a� s��

called the action of S on A, such that �as�t = a�st� for a ∈ A� s� t ∈ S, we call A a(right) S-act or a (right) act over S.

Each semigroup S can be considered as an S-act with the action given by itsoperation. Adjoining an external left identity 1 to a semigroup S, an S-act S1 �= S ∪ �1�is obtained.

Also, recall that an element a of an S-act A is said to be a fixed or a zeroelement if as = a, for all s ∈ S. A fixed element of the semigroup S as an act is calleda right zero of the semigroup, and a left zero z of a semigroup is defined by zs = zfor all s ∈ S, and a zero element, denoted by 0 of S is an element which is both aright and a left zero.

We call an S-act A separated if, for each a �= b in A, there exists 1 �= s ∈ S suchthat as �= bs.

Let A be an S-act and A′ be a subset of A. Then, A′ is called a subact of A,written as A′ ≤ A, if a′s ∈ A′ for all s ∈ S and a′ ∈ A′. A subact of the S-act S is saidto be a right ideal of the semigroup S.

A homomorphism (also called an equivariant map or an S-map) from an S-act Ato an S-act B is a function from A to B such that for each a ∈ A� s ∈ S, f�as� = f�a�s.The category of all right S-acts and S-maps between them is denoted by Act-S.

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BEHAVIOUR OF SEQUENTIAL INJECTIVITY OF ACTS 2513

The class of S-acts is an equational class, and so the category Act-S is completeand cocomplete (has all products, equalizers, pullbacks, coproduct, coequalizers,and pushouts). In fact, limits and colimits in this category are computed as in thecategory Set of sets and equipped with a natural action.

Also, note that a morphism in Act-S is a monomorphism if and only if itis one-one. This follows from the existence of free S-acts. Therefore, we do notdistinguish between monomorphisms of acts and inclusions, and call an S-act Bcontaining (an isomorphic copy of) an S-act A an extension of A. Also epimorphismsin Act-S are exactly the onto S-maps.

2. SEQUENTIALLY DENSE ESSENTIAL MONOMORPHISMS

In this section, recalling (from [4, 8]) the definition of the sequentialclosure operator C on the category of S-acts, we give the class �d of C-densemonomorphisms and recall some of the properties of this class needed in the sequel.Then we recall from [13] the definition, some properties, and the description of�d-essential monomorphisms.

Definition 2.1. The sequential closure C = �CB�B∈Act-S on Act-S is defined as

CB�A� = �b ∈ B � bS ⊆ A�

for any subact A of an S-act B.

It is easily shown that C is a closure operator on Act-S in the sense of [4], whichmeans:CB�A� is a subact of B,A ≤ CB�A�,A1 ≤ A2 ≤ B impliesCB�A1� ≤ CB�A2�� andfor every S-map f � B → D� f�CB�A�� ⊆ CD�f�A�� for each subact A of B.

Notice that the above closure operator also satisfies CB�A ∩ A′� = CB�A� ∩CB�A

′� for A�A′ ≤ B, and for A ≤ B ≤ D, CB�A� = CD�A� ∩ B.For convenience, we sometimes write AB, or simply A, for CB�A�, if no

confusion arises. Notice that, for each subact A of an S-act B, A is the largestsubact of B with the property AS ⊆ A. Now, one has the usual two classes ofmonomorphisms related to the notion of a closure operator as follows.

Definition 2.2. We say that A is sequentially dense or s-dense in B if A = B, and Ais sequentially closed or s-closed in B if A = A.

An S-map f � A → B is said to be s-dense (s-closed) if f�A� is an s-dense(s-closed) subact of B. We denote the class of all s-dense monomorphisms by �d.

Remark 2.3. Notice that:

(a) All isomorphisms are s-dense, and the composition of an isomorphism with ans-dense monomorphism is s-dense;

(b) The composition of s-dense maps is not necessarily s-dense (see the followingTheorem 2.4);

(c) If gf is an s-dense map, then g is clearly an s-dense map, also f is an s-densemap provided that g is a monomorphism;

(d) The composition of an s-dense map with an epimorphism is again s-dense.

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2514 EBRAHIMI ET AL.

Theorem 2.4. For a semigroup S, S2 = S if and only if �d is closed undercomposition.

Proof. Let �d be closed under composition. Then, since the inclusion maps S2 → Sand S → S1 are s-dense, S2 is s-dense in S1. In particular, S = 1S ⊆ S2, and soS2 = S. For the converse, let A ≤ B and B ≤ D be s-dense subacts. Then, D = B = Awhich is equal to A because S2 = S. �

An important notion related to injectivity with respect to monomorphisms orany other class � of morphisms in a category � is essentialness. Recall the followingdefinition and remark from [13].

Definition 2.5. An s-dense monomorphism f � A → B of S-acts is said to be�d-essential (or simply s-essential) if any homomorphism g � B → C is an s-densemonomorphism whenever gf is an s-dense monomorphism.

Remark 2.6. (1) A monomorphism f is �d-essential if and only if it is s-dense aswell as essential (that is, if gf is a monomorphism, then so is g).

(2) Because the composition of two essential monomorphisms is clearlyessential, using Theorem 2.4 we get that the composition of �d-essentialmonomorphisms is an �d-essential monomorphism if S2 = S.

(3) If gf is s-essential and g is a monomorphism, then g is s-essential.

Now, we recall a characterization of s-essential monomorphisms which isneeded in characterizing s-injective hulls. First note the following remark.

Remark 2.7. For an S-act A and a ∈ A, we define the S-map �a � S → A by �a�s� =as for all s ∈ S. Also the set ��a � a ∈ A� is denoted by ��A�. It is clear that ��A� isa subact of the act Hom�S� A�, of all homomorphisms from S to A, with the actiongiven by �fs��t� = f�st�, for s� t ∈ S. Also the assignment � � a �→ �a is an S-mapfrom A to ��A�. Further, � is one-one if and only if A is separated, and in this caseA ��A�.

For a subset of Hom�S� A�\��A�, A�� �= A ∪ is an S-act, where the actionon A�� is given the same as in A for the elements of A and t = �t� for ∈ and t ∈ S, which we call it a -extension of A. Notice that, for the case where A isseparated, A�� Hom�S� A� for = Hom�S� A�\��A�. This is because A ��A�,as stated above.

Proposition 2.8 ([13]; Characterization of s-essential extensions). An S-act B is ans-essential extension of A if and only it is a -extension of A.

We close this section by recalling that the above characterization yields alargest and a maximal s-essential extension of any given S-act.

Definition 2.9. Let A be an S-act. Then:

(i) By a largest s-essential extension of A, we mean an s-essential extension B of Asuch that for each s-essential extension C of A there exists an S-map h � C → Bsuch that h �A is the inclusion map.

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BEHAVIOUR OF SEQUENTIAL INJECTIVITY OF ACTS 2515

(ii) By a maximal s-essential extension of A, we mean an s-essential extension B of Asuch that every monomorphism h � B → C from B to an s-essential extension Cof A for which h �A is the inclusion map, is an isomorphism.

Theorem 2.10 ([13]). Let A be an S-act, and = Hom�S� A�\��A�. Then the S-actA�� is a largest, and a maximal s-essential extension of A.

3. SEQUENTIALLY DENSE INJECTIVITY AND ITS BEHAVIOUR

In this section we study injectivity of right S-acts with respect to the class �d

of sequentially dense monomorphisms and discuss the so called proper behaviourtheorems of injectivity for �d-injectivity.

Definition 3.1. (i) For a class � of morphisms in a category �, we say thatan object A of � is �-injective if for every morphism h � B → C in � and everymorphism f � B → A in �, there exists a morphism g � C → A in � such that gh = f .

(ii) An S-act A is called sequentially dense injective or s-injective if it is�d-injective.

Remark 3.2. (a) If A is an injective S-act, then it is s-injective, but the converse isnot necessarily true. For example, a group S is clearly an s-injective S-act (see alsoTheorem 3.3) but it is not injective, since it does not have a zero element (see [9]Lemma III.1.7).

(b) A retract of an s-injective act is s-injective.

Theorem 3.3. An S-act A is s-injective if and only if Hom�S� A� = ��A�.

Proof. Let A be s-injective and f � S → A be a homomorphism. Then, since S iss-dense in S1, there exists an S-map f̄ extending f . Then, f = �a for a = f̄ �1�.For the converse, let h � B → C be an s-dense monomorphism (inclusion), and letf � B → A be an S-map. Then, since h is s-dense, for every c ∈ C and s ∈ S, cs ∈ Band so f�c is a homomorphism S → A. By hypothesis, there exists ac ∈ A such thatf�c = �ac . Now, define g � C → A as g �B= f and g�c� = ac for c ∈ C\B. Then gis an S-map, since for c ∈ C\B� t ∈ S, g�ct� = f�ct� = f�c�t� = �ac �t� = act = g�c�t.Also, gh = f . �

Further, we recall the following theorem which will be used in the sequel.

Theorem 3.4 ([12]). For a semigroup S, the following are equivalent:

(i) All right S-acts are s-injective;(ii) S is an s-injective S-act;(iii) S has a left identity element.

Now, recalling the three so called proper behaviour theorems of injectivitymentioned in Section 1, we study these theorems for s-injectivity and show thatthe first theorem holds for acts over any semigroup S, but the other two hold

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2516 EBRAHIMI ET AL.

when S2 = S. Thus, the condition S2 = S is sufficient for the proper behaviour ofs-injectivity in the category Act-S. In the final section of the article, we will see thatthese two theorems are not true in general, and the condition S2 = S is not necessaryfor the proper behaviour of s-injectivity of S-acts.

The following lemma which is a counterpart of (E3) in [1] is needed to provethe first proper behaviour theorem.

Lemma 3.5 ([13]). For every �d-morphism f � A → B, there exists a homomorphismg � B → C such that gf is an �d-essential monomorphism.

Definition 3.6. A subact A of an S-act B is said to be a retract of B if there existsan S-map (called a retraction) g � B → A such that g �A= idA. An S-act A is said tobe s-absolute retract if it is a retract of each of its �d-extensions.

Theorem 3.7 (First Theorem of Proper Behaviour). For a semigroup S and everyS-act A, the following are equivalent:

(i) A is s-injective;(ii) A is s-absolute retract;(iii) A has no proper s-essential extension.

Proof. �i� ⇒ �ii� is clear.

�ii� ⇒ �i� Let f � S → A be an S-map. Take the s-dense extension B = A ∪�f� of A with fs = f�s�. By (ii), there exists a homomorphism g � B → A suchthat g �A= idA. Then f = �a for a = g�f�, because f�s� = fs = g�fs� = g�f�s = as.Now, by Theorem 3.3, A is s-injective.

�ii� ⇒ �iii� Let B be an s-essential extension of A. Then, A is s-dense in B andso, by (ii), there exists a retraction g � B → A. Thus g �A, which is equal to idA, is amonomorphism. But, A is essential in B, so g has to be a monomorphism. Now, forb ∈ B, g�g�b�� = g�b� since g�b� ∈ A, and this implies g�b� = b which gives b ∈ A.Thus B = A.

�iii� ⇒ �ii� Let B be an �d-extension of A. Then, by Lemma 3.5, there existsa homomorphism g � B → C such that g �A is s-essential. Applying (iii), g �A is anisomorphism. Now, �g �A�

−1g is a retraction and so A is a retract of B. �

Now, giving a definition, we prove the second theorem of the proper behaviours-injectivity.

Definition 3.8. By a sequentially dense injective hull, or s-injective hull, of an S-actA, we mean an s-essential extension of A which is s-injective.

Remark 3.9. For an S-act A, s-injective hull is unique up to isomorphism (if itexists). To see this, let B and C both be s-injective hulls of A. Then there exists ahomomorphism h � B → C such that h �A= idA, because C is s-injective and A iss-dense in B. From the fact that A is s-essential in B, we get that h is an s-essentialmonomorphism. But, by Theorem 3.7, B has no proper s-essential extension, sinceit is s-injective. So, h is an isomorphism, as required.

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The second theorem of proper behaviour of s-injectivity is about the existenceof s-injective hulls which is proved in the following theorem for S-acts with S2 = S.

Theorem 3.10 (Second Theorem of Proper Behaviour). If S2 = S, then each S-acthas an s-injective hull.

Proof. Consider E�A�, the injective hull of an S-act A (it exists as proved in [2]).We claim that the closure A = CE�A��A� of A in E�A� is the s-injective hull of A.First, A is clearly s-dense in A. Also, A is essential in A, since A is essential in E�A�(apply Lemma III.1.15 in [9], which states that if A ≤ B ≤ C and A is essential inC, then A is essential in B and B is essential in C). Thus A is s-essential in A, byRemark 2.6(1). Secondly A is s-injective. To see this, we apply Theorem 3.3. For ahomomorphism f � S → A, consider the inclusion map A ↪→ E�A�. By injectivityof E�A�, we get a homomorphism f̄ � S1 → E�A� extending f . Now, f = �a wherea = f̄ �1�. Then, since f̄ �1�s = f�s� ∈ A for all s ∈ S, we have a = f̄ �1� ∈ A. ButA = A, because using Theorem 2.4 and since S2 = S, we get that A is s-dense in A

(notice that A is s-dense in A, and A is s-dense in A). �

Finally, we give the third theorem of proper behaviour of s-injectivity. Weneed the following.

Definition 3.11. Let A be an S-act. Then, by a minimal s-injective s-dense extensionof A, we mean an s-dense extension B of A which is s-injective, and every (s-dense)monomorphism k � C → B from an s-injective s-dense extension C of A which mapsA identically is an isomorphism.

Theorem 3.12 (Third Theorem of Proper Behaviour). If S2 = S, then the followingare equivalent for every extension B of an S-act A:

(i) B is the s-injective hull of A;(ii) B is a maximal s-essential extension of A;(iii) B is a minimal s-injective s-dense extension of A.

Proof. �i� ⇒ �ii� Let C be an extension of B and an s-essential extension ofA. Then, applying Remark 2.6(3), C is an s-essential extension of B. But, byTheorem 3.7, B being s-injective has no proper s-essential extension and so C = B.

�ii� ⇒ �i� If B is a maximal s-essential extension of A, then, usingTheorem 2.4, it has no proper s-essential extension. So, by Theorem 3.7, B iss-injective and hence the s-injective hull of A.

�i� ⇒ �iii� If D ≤ B is an s-injective (s-dense) extension of A, since A iss-essential in B, Remark 2.6(3) implies that B is an s-essential extension of D.Then, since D is s-injective, applying Theorem 3.7, we get B = D.

�iii� ⇒ �i� Let Es�A� be the s-injective hull of A which exists byTheorem 3.10. Since B is s-injective, there is an S-map f � Es�A� → B such thatf �A= A ↪→ B. Since A is s-essential in Es�A�, f has to be a monomorphism. So, by(iii), B Es�A�. �

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2518 EBRAHIMI ET AL.

Two other conditions can be added to the equivalent conditions given inTheorem 3.12. To give them, we need the following definition.

Definition 3.13. By a smallest s-injective s-dense extension of A we mean an s-denses-injective extension B of A such that for each s-injective s-dense extension C of A,there exists a monomorphism g � B → C such that g �A is the inclusion map.

Theorem 3.14. Assuming S2 = S, the following conditions are equivalent to theconditions of Theorem 3.12:

(iv) B is a largest s-essential extension of A;(v) B is a smallest s-injective s-dense extension of A.

Proof. Recalling the notations of Theorem 3.12, we have:

�i� ⇒ �iv� is clear.

�i� ⇒ �v� Let B be the s-injective hull of A, and C be an s-injective s-denseextension of A. Since C is s-injective, there exists a homomorphism g � B → C suchthat g �A is the inclusion map. Moreover, g is a monomorphism because B is ans-essential extension of A.

�iv� ⇒ �i� Take Es�A� to be the s-injective hull of A, which exists byTheorem 3.10. Since Es�A� is an s-essential extension of A, and B is a largests-essential extension of A, we obtain an S-map h � Es�A� → B such that h �A is theinclusion map. Now, since A is s-essential in Es�A�, h is a monomorphism and so,since B is an s-essential extension of A, Remark 2.6(3) implies that h is s-essential.But, Es�A� is s-injective, and so, by Theorem 3.7, has no proper s-essential extension.Hence, h is an isomorphism.

�v� ⇒ �i� Suppose Es�A� is the s-injective hull of A which exists byTheorem 3.10. Then, since Es�A� is s-injective and B is a smallest s-injective s-denseextension of A, there exists a monomorphism g � B → Es�A� such that g �A isthe inclusion map. Also, since Es�A� is an s-essential extension, we get that g iss-essential, by Remark 2.6(3). But, B is s-injective and so has no proper s-essentialextension. Thus, g is an isomorphism. �

4. MORE ON s-INJECTIVE HULL

In Section 3 we proved that S2 = S is a sufficient condition for the properbehaviour of s-injectivity, in particular for the existence of s-injective hulls, in thecategory Act-S. Now, one may ask the following two questions: (1) Is there anexplicit description for the s-injective hull of an S-act A? (2) Is S2 = S a necessarycondition for the proper behaviour of s-injectivity? In this section, first we give anexplicit description of the s-injective hull of A, when it exists, and then we will showthat neither S2 = S is a necessary condition for the proper behaviour of s-injectivity,nor an S-act always have an s-injective hull.

Proposition 4.1. Let S be any semigroup and A be an S-act. Then B is the s-injectivehull of A if and only if it is isomorphic to the S-act A�� for = Hom�S�A�\��A�.

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Proof. It follows using Remark 3.9, Theorem 2.10, and the fact that B is thes-injective hull of A if and only if it is a maximal s-essential extension of A byTheorem 3.12. �

As a corollary of Proposition 4.1, Theorem 3.10, and Remark 2.7, we have thefollowing corollary.

Corollary 4.2. Let S be a semigroup with S2 = S and A be an S-act. Then:

(i) A��, where = Hom�S� A�\��A�, is the s-injective hull of A;(ii) If A is separated, then Hom�S� A� is the s-injective hull of A.

By Proposition 4.1, if the sequential injective hull of an act A over a semigroupS exists, then it is of the form A�� for = Hom�S� A�\��A�. And, Theorem 3.10shows that if S2 = S, then A�� is the s-injective hull of A. The question is nowwhether the converse is true or not, and if not for what other kinds of semigroupswe have the same result?

Since, by Proposition 2.8, any A�� is an s-essential extension of A, to answerthe above questions about the existence of the s-injective hulls, one should in factcheck that when A�� is s-injective?

First we give the following lemma.

Lemma 4.3. For an S-act A and = Hom�S�A�\��A�, A�� is s-injective if and onlyif for every S-map f � S → A��, we have f�S� ⊆ A.

Proof. Let A�� be s-injective. Then, by Theorem 3.3, given an S-map f � S→A��,there exists ∈ A�� such that f = �. Then for each s ∈ S, f�s� = s ∈ A.

Conversely, to show that A�� is s-injective, let f � S → A�� be an S-map.Then, by hypothesis, we get f�S� ⊆ A. Now, if f ∈ ��A�, then the proof is complete,otherwise, f ∈ and f = �f , and we again get the result. �

The following two theorems show that S2 = S is not a necessary condition forthe existence of s-injective hulls. Recall that a null semigroup is a semigroup S witha zero element, denoted by 0 such that st = 0 for all s� t ∈ S.

Theorem 4.4. If S is a null semigroup then for each S-act A, A�� for =Hom�S� A�\��A� is the s-injective hull of A.

Proof. Because of Proposition 2.8, it is enough to prove that A�� is s-injective.We apply the above lemma. Take an S-map � � S → A��. Then ��0� is a zeroelement of A��, and hence of A, say ��0� = a0. Now, if there exists s ∈ S such that��s� ∈ , then ��s��t� = ��s�t = ��st� = ��0� = a0, and so ��s� = �a0 �∈ , which isa contradiction. So, ��S� ⊆ A and, by Lemma 4.3, A�� is s-injective. �

Although s-injectivity properly behaves for acts over null semigroups, andalso for semigroups S with S2 = S, the following example shows that there aresemigroups S for which the s-injective hull of an S-act does not exist.

Example 4.5. For acts over infinite cyclic semigroups, s-injective hulls do notgenerally exist. Consider the semigroup S = ���+� as an act over itself. Then, sinceit does not have a left identity element, it is not s-injective (see Theorem 3.4).

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2520 EBRAHIMI ET AL.

Since it is a separated act, ��� Hom�S���. Now, define the homomorphism� � � → ��� by ��1� = id (which belongs to because � does not have aleft identity) and ��k� = k− 1 for k ≥ 2. Then Im��� � � and so ��� is nots-injective.

Our final result shows that for a rather special semigroup S, a right cancellativesemigroup, the null semigroups and semigroups S with S2 = S are the onlysemigroups for which the s-injective hulls of S-acts exist.

Theorem 4.6. If a semigroup S is right cancellative and for each S-act A, thes-injective hull of A exists, then S2 = S.

Proof. First recall that if gf is an s-dense monomorphism and g is a mono-morphism, then f is an s-dense monomorphism (Remark 2.3(c)). If S has a leftidentity (in this case every act is s-injective, by Theorem 3.4), then S2 = S. Now, weassume that S does not have a left identity. To show that S2 = S, we prove thatS2 is s-dense in S1 (see also the proof of Theorem 2.4). We know that S2 iss-dense in S2�� and S2 ≤ S1. Now, by showing that there exists a monomorphismk � S1 → S2�� such that its restriction to S2 is the inclusion of S2 into S2��, andthen applying Remark 2.3(c), we get that S2 is s-dense in S1. To see the existence ofthe mentioned k, we have the following.

Since S2 is s-dense in S and S2�� is s-injective, there exists an S-map j �S → S2�� whose restriction to S2 is the inclusion map. Then, since S is s-densein S1, again by s-injectivity of S2��, there exists an S-map k � S1 → S2�� whoserestriction to S is j. Now we prove that k is a monomorphism. Let k�s1� = k�s2�where s1, s2 ∈ S. Then k�s1s� = k�s2s� for all s ∈ S, which means j�s1s� = j�s2s�, andhence s1s = s2s (because the restriction of j to S2 is the inclusion map). But thisimplies s1 = s2, because S is right cancellative. Also, if k�s� = k�1� for some s ∈ S,then k�s2� = k�s� = k�1�, and so s2 = j�s2� = k�s2� = k�1�. Thus, for all t ∈ S, k�t� =k�1�t = s2t and hence k = �s1 for s1 = s2. Now, for each x ∈ S2� x = k�x� = �s1�x� =s1x. Thus, s1 is a left identity of S2. Now, tt = s1tt for each t ∈ S and, since S is rightcancellative, we have t = s1t for each t ∈ S. Thus, s1 is a left identity of S, whichcontradicts the hypothesis. In all, k is the required monomorphism. �

Remark 4.7. Although we have stated the existence and a description of s-injectivehulls just for the null semigroups and semigroups S with S2 = S, it is easily seen thatit contains a rather large classes of semigroups. In fact, clearly, each of the followingclasses satisfies S2 = S:

(1) Regular semigroups;(2) Semisimple semigroups;(3) Semigroups with a one sided identity;(4) Semigroups S for which the poset Idr�S� of all right ideals of S forms a Boolean

algebra with respect to ∪ and ∩.

There is still an open question concerning s-injective hulls.

Is there a necessary and sufficient condition on S such that all S-acts A haves-injective hulls?

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BEHAVIOUR OF SEQUENTIAL INJECTIVITY OF ACTS 2521

ACKNOWLEDGMENTS

Our sincere thanks goes to the referee for his/her careful reading and helpfulsuggestions.

The last two authors are thankful to Iran National Science Foundation (INSF)and the Center of Excellence in Shahid Behesti University for their financial support.

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