Characterizing Topologically Dense Injective Acts and Their Monoid Connections

Abstract

In this paper, we explore the concept of topologically dense injectivity of monoid acts. It is shown that topologically dense injective acts constitute a class strictly larger than the class of ordinary injective ones. We determine a number of acts satisfying topologically dense injectivity. Specifically, any strongly divisible as well as strongly torsion free -act over a monoid is topologically dense injective if and only if is a left reversible monoid. Furthermore, we establish a counterpart of the Skornjakov criterion and also identify a class of acts satisfying the Baer criterion for topologically dense injectivity. Lastly, some homological classifications for monoids by means of this type of injectivity of monoid acts are also provided.

1. Introduction and Preliminaries

Mathematical models of important notions in theoretical computer science and physics, such as automata and dynamical systems, can be represented by acts over semigroups or monoids. In the literature, various categorical properties of acts have been studied, including injectivity. The study of injective acts began with Berthiaume [1], who established that every act possesses an injective envelope. Since then, many authors have continued the work on such classes of acts, similar to the injectivity of modules over rings. Several generalizations of injective acts with respect to subclasses of monomorphisms other than weak injectivity can be found in many papers. For example, quasi-injective acts were considered in [2, 3]. Giuli [4] studied injectivity with respect to sequentially dense monomorphisms of acts over the monoid , and these notions were later generalized to acts over any arbitrary semigroup in [5]. Zhang et al. [6, 7] classified monoids by C-injectivity and CC-injectivity, which are injectivities relative to all inclusions whose domains, and both domains and codomains, respectively, are cyclic. Shahbaz [8] studied -injectivity in the category of acts, where is an arbitrary subclass of monomorphisms. Recently, Sedaghatjoo and Naghipoor [9] investigated classes of acts that are injective with respect to all embeddings with indecomposable domains or codomains. Vital injectivity for modules first appeared in [10, 11], and McMorris [12] investigated vital injectivity of acts over a monoid with zero. This is injectivity with respect to all embeddings of vital right ideals into , where right ideals of have the property that, for each non-zero , there exists a cancellable element for which .

In this paper, we extend the notion of vital right ideal to a new concept called topologically dense right ideal and more generally, topologically dense subact, and explore injectivity of acts with respect to all embeddings of topologically dense right ideals and topologically dense subacts (relative to the set of all subacts of an act which forms a topology on that act).

We prove that the category of acts over a left reversible monoid or a monoid containing a left zero element has enough topologically dense injectives. We show that the class of topologically dense injective acts is strictly larger than that of usual injective ones and identify a condition under which they coincide. Using some algebraic concepts, we find a number of acts satisfying topological dense injectivity. Particularly, any strongly divisible as well as strongly torsion free -act is topologically dense injective if and only if is a left reversible monoid.

Skornjakov [13] presented a criterion for injectivity of acts with a fixed element, which states that it is enough to consider injectivity with respect to all inclusions into cyclic acts. We provide a counterpart of the Skornjakov criterion for topological dense injectivity of all acts (with or without a fixed element).

The Baer criterion states that weak injectivity is equivalent to injectivity. In [14, 15], some classes of acts satisfying this criterion were found. Motivated by these studies, we find a class of acts for which the Baer criterion holds; that is, topologically dense injectivity and weak topologically dense injectivity are the same.

We also investigate the behavior of (weak) topologically dense injectivity of acts with respect to products, coproducts, and direct sums. Finally, we explore some kinds of weak topologically dense injectivity and topologically self-dense injectivity and present some homological classifications for monoids.

First we give some preliminaries needed in the sequel. Let be a monoid. By a (right) S-act or act overS, we mean a set together with a map , such that for all , and . A subset of is called a subact of if for all and . An element for which for all is said to be a fixed element of . Clearly, is an -act with the operation as the action. Let and be two -acts. A mapping is called a homomorphism if for all . The category of all -acts as well as all homomorphisms between them is denoted by Act-. In this category, monomorphisms are exactly one-to-one homomorphisms. A subset of a monoid is called a right ideal of if for any and . A congruence on an -act is an equivalence relation on for which implies that for and . An -act is called decomposable if there exist proper subacts and of such that and . Otherwise, is called indecomposable. An element is called left zero, if for all . The notion of a right zero element is defined similarly. Also is called zero if it is left zero as well as right zero. Note that the zero element, if exists, is unique. Throughout, stands for a monoid unless otherwise stated. For undefined terms and notations about -acts, we refer to [16].

2. Topologically Dense Injectivity in Act-

In this section, the notion of topologically dense injective act is introduced. We characterize some classes of acts satisfying such kind of injectivity. Moreover, Skornjakov and Baer criteria are studied for topologically dense injectivity of acts as well.

For proceeding, first note that the set of all subacts of an -act including and forms a topology on , which has been studied in [17]. Regarding this topology, the closure of an open set (subact) , denoted as , is the set of all elements for which the intersection of and every open set containing is non-empty, that is, . So is topologically dense, or briefly dense, in if , i.e., if for every , there exists such that . In this case, is said to be a dense extension of . Also is closed in if . So considering as an -act, a dense right ideal is a right ideal of which is a dense subact of , that is, for every , there exists such that . For any -acts and , a homomorphism is said to be a dense homomorphism if Im is a dense subact of . A dense homomorphism which is a monomorphism is called a dense monomorphism. In this case, we say that is densely embedded into .

It is easily checked that is a dense subact of an -act if and only if for each non-empty subact of . In particular, the intersection of a right ideal and a dense right ideal of a monoid is non-empty. Furthermore, if is a commutative monoid or contains zero, then every right ideal of is dense.

Recall from [12] that a right ideal of a monoid (with zero) is “vital” if for every (non-zero) , there exists a cancellable element such that , and an -act is “vital injective” if it is injective relative to all vital right ideals into . This and a view of dense subacts motivate us to generalize these notions in the category Act-, as follows.

Definition 1. Let be an -act. Then is said to be topologically dense injective, or simply densely injective, if it is injective with respect to all dense monomorphisms, that is, for any dense monomorphism and a homomorphism there exists a homomorphism such that . Also is called weakly densely injective if it is injective relative to all dense right ideals into .

Clearly, an -act is densely injective if and only if any homomorphism from a dense subact of an -act can be extended to . So we may consider dense injectivity with respect to dense embeddings (inclusions) instead of dense monomorphisms.

Remark 2. Let be a dense subact of an -act . It is clear that any fixed element of (if exists) is also a fixed element of . So if an -act is dense in an injective extension, then contains a fixed element since each injective act has a fixed element.

Clearly, any injective act is densely injective. The following example shows that these two notions are actually different. It also demonstrates that, in contrast to the case of injectivity of acts, a densely injective act does not necessarily contain a fixed element. Furthermore, not all -acts are (weakly) densely injective.

Example 1. (i)Let be a group. Then any -act contains no proper dense subact since if is a dense subact of , for every , there exists such that and so , which means that . This implies that each -act is densely injective. Indeed, if is a dense subact of and is a homomorphism, then and hence extends . So if is an -act with no fixed element, in particular let be a non-trivial group as an act over itself, then it is densely injective but not injective.(ii)Let . Then is not a densely injective -act. To see this, consider the -act with usual multiplication as the action. Clearly, is a dense subact of . Now it is easy to see that the identity mapping is not extended to .(iii)If as an -act is weakly densely injective, then contains a left identity. Indeed, considering the dense embedding , there is a retraction which implies is a left identity of .(iv)Consider the monoid . Using (ii), is not a weakly densely injective -act.

It is well known that the category Act- has enough injectives (with respect to all monomorphisms). In fact, for any -act , the cofree -act with the action for all and , is injective and is embedded into (see [16], Theorem 3.1.5 and Corollary 3.1.6).

In what follows, our aim is to investigate whether Act- has enough densely injectives where is a commutative monoid. In fact, we construct a densely injective dense extension for any -act. To this aim, let us give some preliminaries.

Let be a commutative monoid and be an -act. Set

We show that is a subact of the cofree act . Let and . Then there exists such that for all . Using the commutativity, for all we getwhich means that . Now we have the following.

Theorem 3. For a commutative monoid , the -act is densely injective.

Proof. Let be an -act, be a dense subact of , and be a homomorphism. Fix an element . For any define a mapping byfor any in the following diagram:

We show that . Since is dense in , for some . This implies that and so there exists for which for any . Take . Then for any , noting and being a homomorphism, we haveNow it is easily seen that is a homomorphism which extends , as required.

Corollary 4. Let be a commutative monoid and an -act. Then the -act is a densely injective dense extension of .

Proof. Let be an -act. Using Theorem 3, the -act is densely injective. It suffices to prove that is densely embedded into . Define by , for any . Note that for any . Clearly, is a monomorphism. It remains to show that is a dense subact of . Let . Then there exists such that for any . This implies that and so which completes the proof.
Here we recall the notion of pushout in a category. Let and be two morphisms of a category . The pair with is called a pushout of the pair if(i)(ii)For any pair with and , there exists a unique morphism such that and , i.e., the following diagram is commutative:

Proposition 5. In Act -, pushouts transfer dense monomorphisms, that is, for a pushout diagram

if is a dense monomorphism, then so is .

Proof. Recall from [16] that , is the congruence relation on generated by all pairs , , is the natural epimorphism, and are coproduct injections. We show that is a dense monomorphism. By [17], is a monomorphism. So it suffices to show that is dense. Let . Then for some , or for some . In the former case, we have . In the latter case, using that is dense, there exist and with and hence .
By a dense retract of an -act , we mean a dense subact of together with a homomorphism from to which maps identically. Also is called densely absolute retract if is a dense retract of each of its dense extensions. Clearly, a dense retract of any densely injective -act is densely injective.
In light of Proposition 5 and [18], Lemma 3.5(i), the following result is obtained.

Theorem 6. Let be an -act. Then the following assertions are equivalent:(i) is densely injective.(ii) is densely absolute retract.

Recall that an extension of an -act is essential if any homomorphism is a monomorphism whenever so is . Every minimal injective extension of an -act is said to be an injective envelope of which is isomorphic to any injective essential extension of . Moreover, for every -act there exists an injective envelope which is unique up to isomorphism and we denote it by . The reader is refereed to [1] for more details on these basic concepts. By a densely injective envelope of an -act , we mean a densely injective essential dense extension. The category Act- is said to have enough densely injective envelopes if each -act admits a densely injective envelope.

A monoid is called left reversible if any two right ideals of have a non-empty intersection. In particular, every commutative monoid is left reversible.

As we know, any injective -act has at least one fixed element. One the other hand, for an -act , does not have two fixed elements by [19], Proposition 1. So if has no fixed element, then has only one fixed element. In the following, this unique element is denoted as 0.

Theorem 7. Let be an -act. Then the following assertions hold:(i)If has a fixed element, then is a densely injective envelope of .(ii)Let be a left reversible monoid. If has no fixed element, then is a subact of which is a densely injective envelope of .

Proof. (i)It is clear that is densely injective. So it suffices to show that is dense in . Using [19], Corollary 2, has no fixed element and then is dense in by [19], Proposition 1.(ii)Let . Using [19], Proposition 1, is a non-empty right ideal of . If the right ideal is non-empty, then by left reversibility, which contradicts the assumption. So , which means that is a subact of . Moreover, it follows from [16], Lemma III.1.16, and [19], Proposition 1, that is an essential dense extension of . Using Theorem 6, it suffices to show that is densely absolute retract. To this end, let be a dense extension of . Consider the following diagram:
in which is the inclusion map. Since is injective, there exists a homomorphism that commutes the diagram. For any , if , then which contradicts the fact that is a dense extension of . Thus and so is densely absolute retract. Hence, is a densely injective envelope of .

Corollary 8. If has a left zero element or is a left reversible monoid, then the category Act- has enough densely injective envelopes.

Remark 9. The class of all dense extensions of -acts are clearly composition closed, that is, if is a dense subact of and is a dense subact of , then is dense in . Then, in view of [18], Theorem 3.8 (v), any densely injective envelope of an -act is a minimal densely injective dense extension.

In view of Remark 9 and Theorem 7 (i), we get the following.

Corollary 10. Let be an -act with a fixed element. Then is injective if and only if it is densely injective.

A well-known criterion for injectivity of acts with a fixed element is the Skornjakov criterion stating that it suffices to verify the injectivity relative to all inclusions into cyclic acts, in which the fixed element plays an important role (see [13]). As for dense injectivity, we present an analogous criterion which needs no fixed element.

Theorem 11 (Skornjakov criterion for dense injectivity). An -act is densely injective if and only if it is injective relative to all dense embeddings into cyclic -acts.

Proof. We prove the non-trivial assertion. Let be an -act satisfying the assumption. Consider an -act , a dense subact of , and a homomorphism . We have to show that there exists a homomorphism which extends . Let   be a dense subact of , , and . is non-empty since . Consider a partial order relation on as follows: and .
For any chain in , the pair where for is an upper bound. By Zorn’s lemma there exists a maximal element in . We shall show that . Then, of course, extends .
Suppose that . Then there exists . Since is dense in , there exists such that and so . Set and . We claim that is a dense subact of . Take any . Using the fact that is dense in , we get for some and hence . It follows from hypothesis that there exists a homomorphism such that . Set . Define byfor every . Since , is well-defined and clearly a homomorphism. Also . Moreover, since is dense in and , it is dense in and which contradicts the maximality of .

Recall from [9] that an -act is said to be indecomposable codomain injective or InC-injective for short, if it is injective with respect to all embeddings into indecomposable acts. By [9], Corollary 2.8, an -act is InC-injective if and only if it is injective relative to all embeddings into cyclic acts. Then, using Theorem 11, we get the following.

Corollary 12. Any InC-injective -act is densely injective.

The following result follows from Corollary 12 and [9], Proposition 2.13.

Proposition 13. If any densely injective -act is injective, then is not a left reversible monoid or contains a left zero.

Lemma 14. The following assertions are equivalent for a monoid :(i) is left reversible.(ii)Any right ideal of is indecomposable.(iii)Any right ideal of is dense.(iv)All subacts of indecomposable -acts are indecomposable.(v)Any two right ideals of whose union is dense have a non-empty intersection.(vi)Any dense right ideal of is indecomposable.

Proof. (i) (ii) (iii) (i), (iv) (ii) and (i) (v) (vi) are obvious.
(i) (iv) Follows from [9], Proposition 2.2.
(vi) (i) Suppose that there exist right ideals and of such that . Set which is non-empty. Consider a partial order relation on as follows:Let be a chain in . Clearly, is an upper bound. By Zorn’s lemma there exists a maximal element in . We claim that is a dense right ideal of ; otherwise, there exists such that . It is clear that , so , which is a contradiction. Thus is dense in and , which contradicts the assumption.

Theorem 15. Let be a left reversible monoid. Then(i)Any dense injective -act is InC-injective.(ii)Any dense injective -act is injective if and only if has a left zero element.

Proof. (i)Consider the following diagram: in which is a dense injective -act and is a non-empty subact of a cyclic -act . Consider the non-empty right ideal of . By Lemma 14, is dense in and so is dense in . Indeed, for any , since is dense in , there exists such that and thus . Now since is dense injective, there exists a homomorphism such that .(ii)If has a left zero element, then by Corollary 10, any dense injective -act is injective. For the converse, since is left reversible, has a left zero element by Proposition 13.Let be an -act. The -act with a fixed element 0 adjoined to is denoted by .

Proposition 16. Let be a left reversible monoid and be a densely injective -act. Then.

Proof. If has a fixed element, then by Corollary 10, . Now let have no fixed element. By Theorem 7 (ii), is a dense injective envelope of which implies .
In what follows, a class of densely injective acts is obtained. To this end, let us list some preliminaries.
The notions of torsion free and divisible -acts are known and defined by using the right and left cancellable elements of , respectively (see [16]). In [2], torsion freeness and divisibility are considered in a much stronger sense (without imposing the cancellability properties on elements of ) which we call here strong torsion freeness (see also [20]) and strong divisibility defined as follows.
Let be an -act. Then is called strongly torsion free if for any and for any , the equality implies . Also we say that is strongly divisible if for each , that is, for any , there exists such that .

Lemma 17. Let be a left reversible monoid and be an -act. Then is strongly torsion free if and only if so is .

Proof. Suppose that is torsion free. If is not torsion free, then there exist such that but . Define a relation on byWe show that is a congruence on . The reflexivity and symmetry are clear. For transitivity, let for . Then there exist such that and . Since is left reversible, there exist such that and so , which means that . Let and ; then there exists such that . Left reversibility of gives that there exist with so that . Thus , as desired. Now, since and , . Using [16], Lemma 3.1.15, where . This implies the existence of with and such that , which contradicts being strong torsion free of . The converse is clear.

Lemma 18. Let be a strongly divisible as well as strongly torsion free -act. Then is closed in each of its strongly torsion free extension.

Proof. Let be a strongly torsion free extension of and . Then there exists such that which implies for some . Since is strongly torsion free, and hence .

Theorem 19. Let be a left reversible monoid. Then any strongly divisible as well as strongly torsion free -act is densely injective.

Proof. By Lemma 17, is a strongly torsion free -act and by Corollary 8, has a densely injective envelope . Since is an essential extension of , there is a monomorphism which implies that is a strongly torsion free -act. Now we are done using Lemma 18.

A densely injective act over a left reversible monoid is not necessarily strongly divisible nor strongly torsion free. For this, consider the monoid which is a densely injective -act (see Example 4 (i)) but not strongly divisible nor strongly torsion free. In the next section, we discuss the converse of Theorem 19 (see Proposition 39).

In view of Corollary 10 and Theorem 19, a class of injective acts is characterized in the following.

Corollary 20. Any strongly divisible and strongly torsion free act with a fixed element over a left reversible monoid is injective.

Theorem 21. For an -act , any strongly torsion free dense extension of is essential.

Proof. Suppose that is a homomorphism such that is a monomorphism. Let for . Since is a dense subact of , there exist such that and so and are non-empty. Since is a dense subact of , and are dense right ideals of and so . So there exists which means that . Then and so . Now since is strongly torsion free, and hence is a monomorphism.

The next result presents a criterion for an injective extension of an -act to be an injective envelope.

Corollary 22. If is an injective strongly torsion free dense extension of an -act , then is an injective envelope of .

Example 2. (i)Consider and as -acts with usual multiplication as the actions. Then is a dense extension of . Moreover, is strongly torsion free and strongly divisible. Then, using Corollary 20, is injective. Now it follows from Corollary 22 that is an injective envelope of .(ii)Consider and as -acts with usual addition as the actions. Then is a dense extension of . Moreover, is strongly torsion free and strongly divisible. Then Theorem 19 implies that is densely injective but not injective because it has no fixed element.(iii)By Theorem 21, is an essential extension of . Now using part (ii) and Proposition 16, since is left reversible, we conclude that is an injective envelope of -acts and .

The condition that weak injectivity coincides to injectivity is known as the Baer criterion for injectivity. However, although this condition holds for injectivity of modules over a ring with unit, it fails for injectivity of acts over an arbitrary monoid (see [16]). The next result gives a class of acts satisfying this criterion. As we shall see in the last section, Baer criterion also fails for dense injectivity of acts (see Example 3).

Theorem 23. Let be a strongly torsion free -act. Then is densely injective if and only if it is weakly densely injective.

Proof. It is clear that each densely injective -act is weakly densely injective. For the converse, let be weakly densely injective. Assume that is a dense subact of a cyclic -act and is a homomorphism. We show that there exists a homomorphism which extends . Let . Then there exists such that and hence which means that is dense in . Consider a homomorphism given by . So there is a homomorphism which extends . Define by for any . Let ; then there exists such that . Soand hence , which means that is well-defined. Now it is not difficult to check that is a homomorphism which extends .

In view of Lemma 14 and Theorem 23, we have the following.

Corollary 24. Let be a left reversible monoid and be a strongly torsion free -act. Then is densely injective if and only if it is weakly injective.

Theorem 25. Let be a left reversible monoid. Then an -act is injective if and only if it is densely injective as well as injective relative to all closed subacts.

Proof. Let be a subact of an -act and be a homomorphism. Set . It is clear that is a dense subact of and is a closed subact of . Now since is dense injective, there exists a homomorphism such that . Moreover, since is closed injective, there exists a homomorphism such that . Hence, , which means that is injective. The converse holds trivially.

3. Products, Coproducts, and Direct Sums of (Weakly) Densely Injective -Acts

This section is devoted to study the behavior of (weak) dense injectivity of acts with respect to products, coproducts, and direct sums. The product of a family of -acts is their Cartesian product with the componentwise action, and the coproduct is their disjoint union with natural action. As usual, we use the symbols and for product and coproduct, respectively. For a family of -acts with a unique fixed element 0, the direct sum is defined to be the subact of the product consisting of all such that for all except a finite number.

The following result shows that (weak) dense injectivity well-behaves under products as usual.

Proposition 26. Let be a family of -acts. Then the product is (weakly) densely injective if each is (weakly) densely injective. The converse also holds if each has a fixed element.

Proof. See [8], Theorem 3.24.

It is known that the usual injectivity is not transferred from a coproduct of acts to all of its components in general. For instance, taking a non-trivial group , the -act where is injective, whereas is not an injective -act. In contrast to the case of injectivity, the next result shows that the dense injectivity is inherited from coproducts to their components.

Proposition 27. Let be a family of -acts. If the coproduct is (weakly) densely injective, then so is each .

Proof. Assume that is densely injective. Let . We show that is densely injective. Let be a dense subact of and consider the diagram

where is a homomorphism and is the canonical injection. Since is densely injective, there exists a homomorphism such that . We claim that . Let there exist and , such that . Since is dense in , for some and so . On the other hand, . Then which is a contradiction. Now considering , we get . The proof for weak dense injectivity is the same.

Recall that all coproducts of injective -acts are injective if and only if is left reversible (see [16], Propositions 3.1.13 and 3.5.4). In the following, a counterpart of these results for (weak) dense injectivity of acts is presented.

Theorem 28. The following statements are equivalent for any monoid :(i)All coproducts of (weakly) densely injective -acts are (weakly) densely injective.(ii) is (weakly) densely injective.(iii)For some -act , is (weakly) densely injective.(iv) is left reversible.

Proof. We just need to prove the assertion for dense injectivity.
The implication (i) (ii) (iii) is trivial.
(iii) (iv) Using a same method to the proof of Proposition 2.12 (iii) (iv) in [9] for the densely injective case, the result is obtained.
(iv) (i) Let be a densely injective -act for each . We apply Theorem 11 to prove that is densely injective. Suppose that is a dense subact of a cyclic -act and is a homomorphism. Moreover, consider the epimorphism , the right ideal of , and in the following diagram:

Note that is dense in . Indeed, for any and since is dense in , there exists such that and so . We claim that there exists for which . Otherwise, and are non-empty for some , which clearly gives that is a decomposable subact of . Using [16], Lemma 1.5.36, this implies that and hence are decomposable which contradicts the left reversibility of . This gives the existence of such that . Since is densely injective by the assumption, can be extended to a homomorphism . Hence, taking we have , as required.

Now, we are ready to prove the converse of Theorem 19.

Proposition 29. For a monoid , if any strongly divisible as well as strongly torsion free -act is (weakly) densely injective, then is left reversible.

Proof. Consider the -act which is clearly strongly divisible and strongly torsion free. It follows from the hypothesis that is (weakly) densely injective. Now, using Theorem 28, the assertion holds.

Note that each -act with trivial action, i.e., for any , is densely injective. As for the injectivity of acts with trivial actions, we have the following.

Proposition 30. Let be a non-singleton -act with trivial action. Then the following statements are equivalent:(i) is injective.(ii) is principally weakly injective.(iii) is left reversible.

Proof. (i) (ii) Trivial.
(ii) (iii) Let be a principally weakly injective -act. If is not left reversible, there exist principal right ideals and of such that . Fix two distinct elements . Consider the homomorphism defined by It is easily checked that can not be extended to which contradicts the assumption.
(iii) (i) Let be a left reversible monoid. Since is isomorphic to the coproduct of singleton -acts, it is injective by [16], Proposition 3.1.13.

The following result follows clearly from [8], Theorem 3.30.

Theorem 31. Let be a family of -acts with a unique fixed element 0 such that the direct sum is (weakly) densely injective. Then each is (weakly) densely injective.

A monoid is called (densely) Noetherian if every (dense) right ideal of is finitely generated. It is easy to check that a monoid is (densely) Noetherian if and only if it satisfies the ascending chain condition on its (dense) right ideals, that is, for every ascending chainof (dense) right ideals of , there exists such that .

Lemma 32. A monoid is Noetherian if and only if it is densely Noetherian.

Proof. It is clear that any Noetherian monoid is densely Noetherian. Conversely, suppose that is not Noetherian, so there exists a right ideal of which is not finitely generated. Set is a non-finitely generated right ideal of . Consider a partial order relation on as follows: .
Let be a chain in . Clearly, is an upper bound. By Zorn’s lemma there exists a maximal element in . We claim that is a dense right ideal of ; otherwise, there exists such that . Thus whence and then is finitely generated, which is a contradiction. Thus is a non-finitely generated dense right ideal of , which means that is not densely Noetherian.

Theorem 33. For a monoid with zero, the following conditions are equivalent:(i)Each direct sum of densely injective -acts is densely injective.(ii)Each direct sum of weakly densely injective -acts is weakly densely injective.(iii) is Noetherian.

Proof. (i) (iii) Follows from Corollary 10 and [3], Theorem 1.
(ii) (iii) Follows from [8], Theorem 3.34 and Lemma 32.

Theorem 34. Let have a left zero element. Then each direct sum of densely injective -acts is densely injective if and only if each direct sum of densely injective -acts is a retract of their direct product.

Proof. Since has a left zero element, is a dense subact of . So we are done using Theorem 6.
By Proposition 26, each direct product of densely injective -acts is densely injective and clearly every retract of a densely injective -act is densely injective.

4. Classifying Monoids by (Principal, fg-) Weak Dense and Self-Dense Injectivities

In this section, we study some usual types of weak dense injectivity and self-dense injectivity of acts. By means of these notions, some homological classification results for monoids are also obtained.

Definition 35. An -act is called principally (fg-) weakly densely injective if it is injective relative to all principal (finitely generated) dense right ideals into .

Remark 36. (i)Let be an -act. Similarly to the case of weak injectivity of acts, one has is (fg-) weakly densely injective if and only if for any (finitely generated) dense right ideal of and any homomorphism there exists such that for any .(ii)As we mentioned in Lemma 14, if is left reversible, then every right ideal of is dense. Then, in this case, (principal, fg-) weak dense injectivity and (principal, fg-) weak injectivity coincide. Therefore, in view of [16], Examples 3.4.6, 3.5.6, the notions of principal weak dense injectivity, fg-weak dense injectivity and weak dense injectivity are actually different.

Let be an -act. Then any is called a dense element if for every , there exist such that (i.e., ). Clearly, is dense if and only if is a dense subact of . So every element of is dense element if and only if every subact of is dense. An element is called regular if there exists such that . If all dense elements of are regular, then is called a densely regular monoid. The monoids and are examples of densely regular monoid.

Using [9], Proposition 2.1, the next result is immediate.

Proposition 37. Let be a left reversible monoid. Then an -act is indecomposable if and only if all elements of are dense.

Proposition 38. Let be an -act. Then the following statements are equivalent:(i) is principally weakly densely injective.(ii)For any dense principal right ideal of and any homomorphism there exists such that for any .(iii)For a dense element and any with , for some .

Proof. The proof is similar to that of [16], Proposition 3.3.2.

Corollary 39. The following assertions hold for any monoid :(i)Let be a right congruence on . The factor act is principally weakly densely injective if and only if for any dense element and any for which , implies , there exists such that .(ii)A right ideal , of is principally weakly densely injective if and only if for any dense element and any for which , implies , there exists such that . In particular, if is a principally weakly densely injective dense right ideal of , then is a regular element.

An -act is said to be densely divisible if for any left cancellable dense element , that is, for any there exists such that .

Corollary 40. Every principally weakly densely injective act is densely divisible.

The converse of Corollary 40 is not generally true (see [16], Example 3.3.11).

The next three theorems are proved similarly to the well-known homological classification results for monoids by different kinds of weak injectivity of acts which can be found, for example, in [16].

Theorem 41. The following conditions are equivalent:(i)All -acts are principally weakly densely injective.(ii)All dense right ideals of are principally weakly densely injective.(iii)All finitely generated dense right ideals of are principally weakly densely injective.(iv)All principal dense right ideals of are principally weakly densely injective.(v) is densely regular.

Theorem 42. The following conditions are equivalent:(i)All -acts are fg-weakly densely injective.(ii)All dense right ideals of are fg-weakly densely injective.(iii)All finitely generated dense right ideals of are fg-weakly densely injective.(iv) is a densely regular monoid whose dense finitely generated right ideals are principal.

Theorem 43. The following conditions are equivalent:(i)All -acts are weakly densely injective.(ii)All dense right ideals of are weakly densely injective.(iii)All dense right ideals of have an idempotent generator.(iv) is a densely regular principal dense right ideal monoid.

The following example shows that the Baer criterion fails for dense injectivity of acts as injectivity (see [7], Example 12).

Example 3. Let be a monoid with the multiplication table:

It is easily checked that is a densely regular principal dense right ideal monoid. Then all -acts are weakly densely injective by Theorem 43. Consider the monocyclic right congruence on . Clearly, and is a dense subact of the -act . We show that the -act is not densely injective. Consider the following diagram:

Suppose that there exists a homomorphism extending . If , then and which is a contradiction. Similarly, the case yields also a contradiction.

A monoid is said to be self-densely injective if is densely injective as an -act.

In the following, we study self-dense injectivity property for monoids.

Let be a dense right ideal of and . For , put . Then is a (non-empty) dense right ideal of . Define a relation on by

Then is a right congruence on .

Using Theorem 11, a same argument to [16], Theorem 4.5.3, gives the following result.

Theorem 44. A monoid is self-densely injective if and only if for any dense right ideal of and any homomorphism there exists such that for all and , implies .

Analogous to [16], Theorems 4.5.10, 4.5.11, 4.5.12, we have the next three results.

Theorem 45. The following conditions are equivalent:(i)All principal dense right ideals of are densely injective.(ii) is a densely regular self-densely injective monoid.

Theorem 46. The following conditions are equivalent:(i)All finitely generated dense right ideals of are densely injective.(ii) is a densely regular self-densely injective monoid whose dense finitely generated right ideals are principal.

Theorem 47. The following conditions are equivalent:(i)All dense right ideals of are densely injective.(ii) is a densely regular self-densely injective principal dense right ideal monoid.

Recall from [16] that an idempotent is called right special if for any right congruence on there exists such that and , implies .

Using Theorem 11, the next result for dense injectivity is similar to the injectivity case. The proof is just an adaptation of the proof of [16], Theorem 4.5.13.

Theorem 48. All -acts are densely injective if and only if is a densely regular principal dense right ideal monoid all idempotents of which are special.

Finally, the following example shows that self-dense injectivity does not imply self-injectivity.

Example 4. (i)Consider the monoid . Then it follows from Theorem 48 that is a self-densely injective monoid. Since has no zero, it is not self-injective.(ii)Each non-trivial group is a self-densely injective but not self-injective act over itself (see Example 1 (i)).

5. Conclusion

The examination of injectivity concerning various classes of monomorphisms in a category holds significant importance across multiple mathematical domains. Numerous authors have explored this concept within diverse categories, each pertaining to distinct classes of monomorphisms. This paper delves into the exploration of topological dense injectivity within monoid acts. We establish that the category of acts over a left reversible monoid or a monoid that includes a left zero element possesses enough topologically dense injectives. It is revealed that topologically dense injective acts form a class strictly larger than the class of ordinary injective acts and we identify a condition under which they coincide. We pinpoint several acts that meet the criteria for topological dense injectivity. Specifically, a strongly divisible as well as strongly torsion free -act over a monoid is topologically dense injective if and only if is a left reversible monoid. Moreover, we give a counterpart of the Skornjakov criterion and identify a class of acts that adhere to the Baer criterion for topological dense injectivity. Finally, we present various homological classifications for monoids based on this form of injectivity in monoid acts.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.