Injective Hulls of Acts over Left Zero Semigroups

Semigroup Forum Vol. 75 (2007) 212–220c© 2007 Springer

DOI: 10.1007/s00233-007-0722-x

RESEARCH ARTICLE

Injective Hulls of Actsover Left Zero Semigroups

M. M. Ebrahimi, M. Mahmoudi, and Gh. Moghaddasi Angizan

Communicated by Steve Pride

Abstract

In this paper, using the notion of “Cauchy sequences”, we study the injec-tivity of acts over left zero semigroups, and give some equivalent conditions toit. We see that Baer criterion is true if an identity is adjoined to the semigroup.Further, we find injective hulls of acts over left zero semigroups.

We hope that the technique used here will be useful to study injectivity ofacts over some other more general semigroups.

Keywords: S -act, Cauchy sequence, injective, essential.

AMS Classification: 08A60, 08B30, 08C05, 20M30.

1. Introduction

One of the very useful notions in many branches of mathematics as well as incomputer sciences is the action of a semigroup or a monoid on a set. In thispaper, we study acts over a left zero semigroup, that is a semigroup whosemultiplication is defined by st = s for all s, t ∈ S . Throughout this paper,unless stated otherwise, S is such a semigroup and S does not have an identityelement (note that if a left zero semigroup has an identity then it is trivial).

Recall that a (right) S -act or S-system is a set A together with a functionα: A × S → A , called the action of S (or the S -action) on A , such that forx ∈ A and s, t ∈ S (denoting α(x, s) by xs), x(st) = (xs)t . If S is a monoidwith identity e , the condition xe = x is added.

An S -act A is called separated if for a �= b in A there exists s ∈ S withas �= bs .

A morphism f : A → B between S -acts A,B is called an equivariantmap if, for each x ∈ X , s ∈ S , f(xs) = f(x)s .

Since idA and the composite of two equivariant maps are equivariant, wehave the category Act-S of all S -acts and equivariant maps between them.

An element a of an S -act A is called a fixed or a zero element if as = afor all s ∈ S . We denote the set of all fixed elements of an S -act A byFixA . We see that the set FixA plays an important role for acts over left zerosemigroups.

The following lemma is easily proved.

Ebrahimi, Mahmoudi and Angizan 213

Lemma 1.1. Let S be a semigroup. Then the following are equivalent:

(i) The binary operation of S is defined by st = s for all s, t ∈ S .

(ii) Considering S as a right S -system, every element of S is a zero.

(iii) For every S -system A , AS ⊆ FixA , where AS = {as: a ∈ A, s ∈ S} .

As a corollary to the above lemma, we see that for a left zero semigroupS , each nonempty S -act A has a fixed element.

The class of all S -acts is clearly a variety. In particular, monomor-phisms in Act-S are exactly one-one equivariant maps. Thus, we can considermonomorphisms of this category as inclusions.

An S -act B containing (an isomorphic copy of) an S -act A as a sub-actis called an extension of A . The S -act A is said to be a retract of B if thereexists an equivariant map f : B → A such that f �A= idA , in which case f issaid to be a retraction.

A is called an absolute retract if it is a retract of each of its extensions.

An S -act A is said to be injective if for every monomorphism h: B → Cand every equivariant map f : B → A there exists a homomorphism g: C → Asuch that gh = f .

A monomorphism h: A → B is called essential if any equivariant mapg: B → C is a monomorphism whenever gh is a monomorphism.

We have the following result from [1] or [3].

Proposition 1.2. The category Act-S has enough injectives, and for any S-act A , the following conditions are equivalent:

i) A is injective.

ii) A is an absolute retract.

iii) A has no proper essential extension.

An extension B of an S -act A is called an injective hull of A if it is anessential extension of A which is also injective.

Finally, recall from [1], [3] that

Proposition 1.3. For an extension B of an S-act A , the following are equiv-alent:

i) B is an injective hull of A .

ii) B is a maximal essential extension of A .

iii) B is a minimal injective extension of A .

So, the injective hull of an S -act A is unique up to isomorphism. Also,we know that injective hulls in Act-S exist (see [1], [3]).

214 Ebrahimi, Mahmoudi and Angizan

2. Cauchy completeness

The notion of a Cauchy sequence is introduced in [6] for a particular class ofacts, called projection algebras, and is used in [8] to study injectivity of theseacts. In this paper we use this notion to study the injectivity of acts over leftzero semigroups.

Generalizing this notion for S -acts, we say that a Cauchy sequence overan S -act A is a sequence (as)s∈S of elements of A with ast = ast , which inthe case where S is left zero we have ast = ast = as , for all s, t ∈ S . Hence,we get the following definition in the case of acts over left zero semigroups.

Definition 2.1. By a Cauchy sequence over an S -act A , where S is left zero,we mean a family (as)s∈S of elements of FixA .

Notice that a Cauchy sequence over A is in fact an equivariant map fromS to A .

Lemma 2.2. For an S -system A over a left zero semigroup S , the set C(A)of all Cauchy sequences over A is a separated S -act with the action of Son it given as γt to be the constant sequence (at)s∈S , for a Cauchy sequenceγ = (as)s∈S and t ∈ S .

Proof. Take t, t′ ∈ S , and a Cauchy sequence (as)s∈S . Then we have

((as)s∈S .t).t′ = (at)s∈S .t

′ = (at)s∈S = (att′)s∈S = (as)s∈S .(tt′)

To see that C(A) is separated, take Cauchy sequences γ = (as)s∈S and γ′ =(a′s)s∈S with γ.t = γ′.t for all t ∈ S . This means that at = a′t for all t ∈ Sand hence γ = γ′ .

Remark 2.3. For a left zero semigroup S , each S -act A , and a ∈ A , thesequence λa = (as)s∈S is Cauchy. In fact, λ(A) = {λa: a ∈ A} is a sub-actof C(A), since λa.t = λat . This is in fact the image of the equivariant mapλ: A → C(A) given by a → λa . Further, λ is one-one if and only if A isseparated, and in this case A ∼= λ(A).

Definition 2.4. By a limit of a sequence (as)s∈S over A in some extensionB of A we mean an element b ∈ B such that bs = as for all s ∈ S .

Notice that limits are not necessarily unique, unless B is separated.

Fact 2.5. For an element b in an extension B of an S-act A, and a sequenceγ = (as)s∈S over A , the following are equivalent:

(i) b is a limit of γ .

(ii) λb = γ .

(iii) γ belongs to λ(A) .

(iv) b is a solution of the sequential system xs = as , s ∈ S .


An interesting fact for Cauchy sequences over acts on a left zero semigroupwhich is not true in general (see [8]):

Lemma 2.6. A sequence (as)s∈S over an S -act A has a limit in some ex-tension B of A if and only if it is a Cauchy sequence.

Proof. Take b as a limit of (as)s∈S . Then for s, t ∈ S , bs = as impliesb(st) = (bs)t = ast which means that as = ast = ast .

Conversely, let (as)s∈S be a Cauchy sequence over A . Then B =A ∪ {(as)s∈S} by the action (as)s∈S .t = at for t ∈ S is an extension of A ,and b = (as)s∈S is a limit of (as)s∈S in B .

Definition 2.7. An S -act A is said to be complete if any Cauchy sequenceover A has a limit in A .

Theorem 2.8. For a left zero semigroup S , the S-act C(A) is complete.

Proof. Let (γs)s∈S be a Cauchy sequence over C(A). One can easily see thateach γs being a fixed element of C(A) is a constant sequence, say γs = (as)t∈Swith as ∈ FixA . Now, the Cauchy sequence γ = (as)s∈S is clearly a limit of(γs)s∈S .

3. Injective S -acts

This section is devoted to the study of injectivity of acts over a left zerosemigroup S . We find some equivalent conditions to injectivity. In particular,we show that the Baer criterion holds in case an identity is adjoined to S .

Theorem 3.1. An S-act A is injective if and only if it is complete.

Proof. (⇒) Let A be injective, and (as)s∈S be a Cauchy sequence over A .Consider the extension B = A ∪ {(as)s∈S} of A by the action (as)s∈S .t = atfor t ∈ S . Since A is injective and hence absolute retract, there exists anequivariant map f : B → A such that f �A= id . Let f((as)s∈S) = a . Thena.t = f((as)s∈S).t = f((as)s∈S .t) = f(at) = at for all t ∈ S and hence a is alimit of (as)s∈S .

(⇐) It is enough to prove that A is an absolute retract. Take B to bean extension of A . Define g: B → A with g �A= id and for b ∈ B \ A takeg(b) = ab where ab is a limit of the Cauchy sequence (as)s∈S with

as =

{bs if bs ∈ Aa0 if bs �∈ A

and a0 ∈ FixA . For the case where for all s ∈ S , bs �∈ A , we take ab = a0 .

To see that g is equivariant on B \ A , take b ∈ B \ A and s ∈ S . Ifbs ∈ A then g(b)s = abs = bs = g(bs). If bs �∈ A then since (bs)t = bs for allt ∈ S , we get g(bs) = abs = a0 = a0s = abs = g(b)s .


Now, as a corollary of the above theorem and Theorem 2.8, we get

Corollary 3.2. For every S-act A , C(A) is injective.

Corollary 3.3. A separated S-act A is injective if and only if it is a retractof C(A) .

Proof. (⇒) By Remark 2.3, the equivariant map λ: A→ C(A) is one-one.Now, since A is injective, there is a retraction f : C(A)→ A .

(⇐) A being a retract of an injective S -act is injective.

Corollary 3.4. An S-act A is injective if and only if the equivariant mapλ: A→ C(A) is onto.

Proof. It is enough to note that λ: A → C(A) is onto if and only if A iscomplete.

Corollary 3.5. For an S-act A, if S and FixA are finite then A is injectiveif and only if Card(A/ ∼) = CardC(A) where ∼ is the equivalence relation onA given by a ∼ b⇔ as = bs , ∀s ∈ S .

Proof. Since the map (A/ ∼) → C(A) given by [a] → (as)s∈S is alwaysone-one, and it is onto if and only if λ: A → C(A) is onto, the result followsfrom the above corollary.

Now recall the following from literature.

Definition 3.6. We say an S -act A is

(i) weakly injective if each equivariant map f : I → A from a right ideal I ofS can be extended to an equivariant map from S to A ;

(ii) I -injective, for a right ideal I of S , if each equivariant map f : I → A isof the form λa for some a ∈ A , that is f(s) = as for s ∈ I ;

(iii) S -injective, if each equivariant map f : S → A is equal to λa for somea ∈ A , that is f(s) = as for s ∈ S .

(iv) absolutely s-pure if it is s-pure in each of its extension B ; in the sensethat each system xs = as , s ∈ S , of equations over A is solvable in Awhenever it is solvable in B .

Theorem 3.7. For an S -act A , the following are equivalent:

(i) A is injective.

(ii) A is an absolute retract.

(iii) A is complete.

(iv) A is absolutely s-pure.

(v) A is I -injective, for each right ideal I of S .

(vi) A is S -injective.


If an identity is adjoined to S , the following condition is also equivalentto the above conditions:

(v) A is weakly injective.

Proof. Parts (i), (ii) and (iii) are equivalent because of Proposition 1.2 andTheorem 3.1.

(iii) ⇒ (iv) If A is complete and xs = as, s ∈ S is a system over Awhich has a solution in an extension B of A then by Lemma 2.6, (as)s∈S is aCauchy sequence over A . Since A is complete, this sequence has a limit a inA . Then, by Lemma 2.5, a is a solution of the given system in A .

(iv)⇒ (v) Take I to be a right ideal of S and f : I → A be an equivariantmap. Put as = f(s) for s ∈ I and as = a0 for s �∈ I , where a0 is a fixedelement of A . Since f is equivariant, (as)s∈S is a Cauchy sequence over A .So, by Lemma 2.6, it has a limit is some extension B of A . By Lemma 2.5,this means that the system xs = as, s ∈ S has a solution in B . So, by (iv),this system has a solution in A , namely a . Now, for s ∈ I , f(s) = as = as .

(v)⇒ (vi) is true since S is itself an ideal of S .

(vi) ⇒ (iii) Let A be S -injective, and (as)s∈S be a Cauchy sequenceover A . Then, f : S → A given by f(s) = as is an equivariant map. So, using(vi), there exists some a ∈ A such that as = f(s) = as and hence a is a limitof the given Cauchy sequence.

If an identity is adjoined to S then weak injectivity coincides with I -injectivity for all right ideals I , and so condition (v) is added to the aboveequivalent conditions.

Remark 3.8. As the above theorem shows, in the case where an identityis adjoined to S , weak injectivity coincides with injectivity; that is the Baercriterion is true for acts over a left zero monoid S . But if S is a left zerosemigroup without identity then weakly injective S -acts are not necessarilyinjective. For example, take a two element left zero semigroup S = {s, t} andan S -act A = {a, b} where both a, b are fixed. Then A is easily weakly injective.But, it is not injective since it is not complete; the Cauchy sequence (a, b) doesnot have a limit in A .

4. Injective hull of S -acts

First we give a criterion to determine essential extensions for acts over a leftzero semigroup.

Definition 4.1. An extension B of A is called s-dense if sb ∈ A for all s ∈ Sand b ∈ B .

Lemma 4.2. An extension B of A is essential if and only if

(i) A is s-dense in B .

(ii) For every b, b′ ∈ B \A , if bs = b′s (∀s ∈ S) , then b = b′ .

(iii) For every b ∈ B \A , a ∈ A , there exists s ∈ S such that bs �= as .


Proof. (⇒) Let b ∈ B and s ∈ S such that bs �∈ A . Take a fixed element a0

in A . Then the set {a0, bs} consisting of two fixed elements is a sub-act of B .Now, considering the Rees congruence θ on B corresponding to this sub-act,we have θ �= ∆ while θ ∩ A2 = ∆. This contradicts essentialness, and so A iss-dense in B .

To prove (ii), take b �= b′ ∈ B \ A with bs = b′s for all s ∈ S .By (i), bS, b′S ⊆ A , and so A ∪ {b, b′} is a sub-act of B . Define the mapf : A ∪ {b, b′} → A ∪ {b} by f(b) = f(b′) = b and f �A= id . Then f is anequivariant map which is not one-one but its restriction to A is one-one. Thiscontradicts the fact that A being essential in B is also essential in A ∪ {b, b′} .

Part (iii) is proved similar to part (ii), replacing b by a .

(⇐) Let g: B → C be an equivariant map which is one-one on A . Ifg(b) = g(b′) then, since A is s-dense in B , for each s ∈ S , we have

bs = g(bs) = g(b)s = g(b′)s = g(b′s) = b′s

So, by (iii), either b, b′ ∈ A and so b = g(b) = g(b′) = b′ or b, b′ ∈ B \ A . Inthe latter case, (ii) gives that b = b′ . Thus, g is one-one, and hence B is anessential extension of A .

Corollary 4.3. A separated S -act B is an essential extension of A iff A iss-dense in B .

Definition 4.4. We call an extension B of an S -act A a Γ-extension of Aif B is of the form A∪ Γ, for some subset Γ of C(A) \ λ(A), where the actionson any element (as)s∈S of Γ are defined by (as)s∈S .t = at , for t ∈ S .

Another corollary of Theorem 4.2 is

Lemma 4.5. Any Γ-extension of A is an essential extension of A .

Proof. Take a Γ-extension B of A . By the definition of B , A is clearlys-dense in B . To prove (ii), let γ = (as)s∈S , γ′ = (a′s)n∈S be in Γ andγs = γ′s,∀s ∈ S . Then, as = a′s,∀s ∈ S , and hence γ = γ′ . Finally, forγ = (as)s∈S ∈ Γ and a ∈ A , let γs = as, ∀s ∈ S . Then, a is a limit point of γin A which is a contradiction.

Now, we are ready to give the injective hull of an S -act A .

Theorem 4.6. For an S-act A, the Γ-extension A∗ = A ∪ Γ , for Γ = C(A) \λ(A) , is the injective hull of A .

Proof. By the above lemma, A∗ is an essential extension of A . So, it isenough to prove that it is also injective. Applying Theorem 3.1, we show thatA∗ is complete. Let γ = (γs)s∈S be a Cauchy sequence over A∗ . If γs ∈ A forall s ∈ S , then either γ has a limit in A and the proof is complete or γ does nothave a limit in A , and so γ belongs Γ and is a limit of γ in A∗ since γt = γt


for all t ∈ S . The other case is that there exists s ∈ S such that γs �∈ A , sayγs = (ast )t∈S . Then γ being Cauchy we have, in particular, γss = γs �∈ A . But,γss = ass ∈ A . This is a contradiction and so this case does not happen.

Finally, for the case of separated S -acts we have

Theorem 4.7. For a separated S-act A, C(A) is the injective hull of A .

Proof. By Corollary 3.2, C(A) is injective. Moreover, the monomorphismλ: A→ C(A) is essential. For, if f : C(A)→ D is an equivariant map such thatf ◦ λ is one-one, then f is also one-one since

f((as)s∈S) = f((bs)s∈S) ⇒ f((as)s∈St) = f((bs)s∈St), ∀t ∈ S⇒ f((at)s∈S) = f((bt)s∈S), ∀t ∈ S⇒ fλ(at) = fλ(bt), ∀t ∈ S⇒ at = bt, ∀t ∈ S⇒ (as)s∈S = (bs)s∈S

References

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[4] Ebrahimi, M. M. and M. Mahmoudi, The category of M -sets, ItalianJournal of Pure and Applied Mathematics 9 (2001), 123–132.

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[6] Giuli, E., On m-separated projection spaces, Appl. Categ. Struc. 2 (1994),91–99.

[7] Kilp, M., U. Knauer and A. V. Mikhalev, “Monoids, Acts and Categories”,de Gruyter, Berlin/New York, 2000.


[8] Mahmoudi, M. and M. M. Ebrahimi, Purity and equational compactness ofprojection algebras, Appl. Categ. Struc. 9(4) (2001), 381–394.

Department of Mathematicsand Center of Excellencein Algebraic and Logical Structuresin Discrete Mathematics

Shahid Beheshti UniversityTehran 19839Iran

[email protected]@sbu.ac.ir

Department of MathematicsShahid Beheshti UniversityTehran 19839IranandSabzevar Teachers Training UniversitySabzevar, [email protected]

Received August 31, 2004and in final form April 23, 2007Online publication August 27, 2007

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Injective Hulls of Acts over Left Zero Semigroups