A criterion of the existence of an embedding of a monothetic monoid into a topological group

Abstract Using properties of unitary Cauchy filters on monothetic monoids, we prove a criterion of the existence of an embedding of such a monoid into a topological group. The proof of the sufficiency is constructive: under the corresponding assumptions, we are building a dense embedding of a given monothetic monoid into a monothetic group.


Introduction
The problem of embedding a topological semigroup into a topological group was investigated by a number of authors. A list of papers published before 1990 is contained in [9]. In particular, a necessary and su cient condition of the existence of such an embedding is proved in [5]. However, it is technical and very cumbersome. A detailed study of the case that a right reversible cancellative topological semigroup embeds as an open subsemigroup in a topological group, is contained in [10]. For commutative semigroups, another condition of the existence of such an embedding has been earlier found in [11]. Amongst of papers published later, we mention the paper [8] where the case of locally compact semigroups is considered. There are also many papers on embeddings of topological semigroups into Lie groups (see, for example, [9]).
In this paper, we prove a necessary and su cient condition under which a monothetic monoid can be embedded into a topological group. Unlike the cited above papers, our construction does not use the group of quotients of the given monoid. The required group is a unitary extension of this monoid, i.e. such its extension where all its unitary Cauchy lters converge. The concept of such a lter was de ned and studied in [1] and [2]. It is a generalization of the concept of a fundamental sequence of reals. The construction of such an extension has been proposed in [3].
In the rst section, we study properties of unitary Cauchy lters on a monothetic monoid whose topology satis es some additional conditions. In the second one, we use the proved properties of these lters in order to show that the mentioned above extension is the required topological group. This group is monothetic, and this embedding is dense.
Unfortunately, we could not write a completely self-contained paper: the number of places where it was necessarily to refer to previous papers [1], [2], and [3] of this series was too large.

C-lters on monothetic monoids
In this preparatory section, we study the topology of a monothetic monoid satisfying some additional requirements and prove properties of C-lters on it. We only consider Hausdor topological groups and semigroups and use multiplicative notations for them.
A) Let G be a Hausdor topological monoid. Its topology is said to be non-viscous (see [1]), if, for any neighborhood U of the neutral element of G and for any a ∈ G, there exists a neighborhood V of a such that, for any elements a , a ∈ V, if the equation a x = a or the equation xa = a has a solution, then this solution belongs to U. Proposition 1.1. If G is Hausdor and compact and, for any a ∈ G, x = is the only solution of each of the equations ax = a and xa = a, then the topology of G is non-viscous.
Proof. Consider equations of the form ax = a. Suppose that there exist a ∈ G and a neighborhood U of such that, for any neighborhood V of a, there exist a , a ∈ V such that the equation a x = a has a solution which lies outside U. Then there exist nets {a α }, {a α } convergent to a and a net {xα} lying outside U such that a α xα = a α for all α. If x is a cluster point of {xα}, then x ≠ and ax = a.
It is evident that the topology of a topological group and the relative topology of any submonoid of a monoid with a non-viscous topology are non-viscous, too.
The product of any family of topological monoids has a non-viscous Tychono product topology if and only if the topology of each factor is non-viscous. Indeed, denote the given family by {Gα} α∈A and its product by G. For any α ∈ A, there exists a standard algebraic and topological embedding of Gα into G. Therefore, if the topology of G is non-viscous, then it is also true for Gα. Conversely, let the topologies of all Gα be non-viscous and U an arbitrary neighborhood of the neutral element ∈ G. For any α ∈ A, there exists a neighborhood Wα of the neutral element α ∈ Gα such that Wα ≠ Gα only for a nite number of α and W = Wα ⊂ U. Let now a = {aα} be an arbitrary point of G and, for any α ∈ A, Vα be a neighborhood of aα in Gα such that x ∈ Wα for any solution x of the equation a x = a or the equation xa = a with coe cients a , a from Vα. If Wα = Gα, then we take Vα = Gα. Then V = Vα is a neighborhood of a in G, and if a , a are coe cients from V and x is a solution of one of the equations a x = a or xa = a , then x ∈ W ⊂ U.
(R, +) with the topology of the Sorgenfrey line (= the right half-open topology whose a base at each point a consists of intervals [a; a+ϵ[ with ϵ > ) is an example of a cancellative topological monoid whose topology is not non-viscous.
For commutative cancellative monoids, one can give another interpretation of this property. Let G be such a monoid and B a subset of G × G consisting of pairs (a , a ) for which the equation above has a solution. It is evident that this solution is unique. If (a , a ) and (a , a ) are such pairs and x and x are the corresponding them solutions, then a a x x = a a . It means that B is a submonoid of G × G containing its diagonal. Consider now the map ϕ : B → G taking any element (a , a ) ∈ B into the unique solution x of the corresponding equation a x = a . It also follows from the formula above that ϕ is a homomorphism which takes the diagonal of G × G into the point . The property of the topology of G to be non-viscous means that ϕ is continuous at any point of the diagonal.
The following statement shows that the concept of a non-viscous topology can be useful. Proposition 1.2. Any topological monoid with a non-viscous topology which is algebraically a group, is a topological group.
Proof. Denote again the considered monoid by G and show that the mapping g → g − sending any g ∈ G to its inverse is continuous on G. Let g be an arbitrary element of G and W an arbitrary neighborhood of g − . Find a neighborhood U of such that g − U ⊂ W and a neighborhood V of g such that the requirement of the de nition of a non-viscous topology is satis ed for a = g and for this U. For any g ∈ V, g g − is a solution of the equation xg = g and lies in U. Hence, g − = g − (g g − ) ∈ W.
Let now Γ be a cancellative monothetic monoid with a generator p, i.e. p is its element such that the set {p n : n ∈ N} is dense in Γ (see [4]). Such a monoid is commutative. In the following, we always assume that its topology τ satis es the following conditions: (i) the underlying space of Γ is T ; (ii) this topology is non-viscous. Consider now another property of this topology. For an arbitrary commutative topological monoid G, let B be a base of its topology at its identity. It was proved in [7] that the family of all sets of the form gU with g ∈ G, U ∈ B forms a base of a topology on G, and the multiplication in this monoid is continuous in this topology. We will call this topology the GKO-topology corresponding to the initial topology of G. It follows from the continuity of the multiplication in G that its GKO-topology is ner than the initial one.
Returning to the monothetic monoid Γ, denote by Θ its submonoid {p n : n ∈ N = N ∪ { }}, by τ| the restriction of the topology τ on this submonoid and by j its embedding into Γ. When talking about elements from Θ, we will often omit the symbol j and write, for example, p ∈ Γ. However, if F is a lter on Θ, then F and the lter on Γ with a base {j(F) : F ∈ F} (we denote it by j(F)) are di erent lters. Proposition 1.3. The topology τ| coincides with the corresponding to it GKO-topology on Θ.
Proof. Check that the topology τ| is ner than its GKO-topology, i.e. for any neighborhood U of in Θ and for any n ∈ N , there exists a neighborhood V of p n in Θ such that V ⊂ p n U. Indeed, let V be a neighborhood of p n which only contains elements of the form p k with k ≥ n, and V its neighborhood such that x ∈ U for any solution of the equation Then p k−n ∈ U for any p k ∈ V and V ⊂ p n U.
B) Here, we consider C-lters on Θ. We refer the reader to papers [1], [2] and [3] containing all necessary de nitions and proofs of properties of such lters for arbitrary Hausdor topological monoids. The contents of papers [1] and [2] are also summarized in the rst section of paper [3]. We retain the main notations used in these papers. Quote, nevertheless, the basic de nitions.
A net S = {sα} α∈A in a Hausdor topological monoid G is called a C-net if, for each neighborhood U of , there exists α ∈ A and, for each α ≥ α , there exists α ∈ A such that s α ∈ Usα U for all α ≥ α .
Here, the hat on top denotes the topological closure (in G and, in the following, in Γ). Omitting this accent, we obtain a de nition of a strict C-net. It is a generalization of the concept of a fundamental sequence of reals for topological monoids.
Filters corresponding to (strict) C-nets are called (strict) C-lters. Their direct de nition is: A lter F on G is called a C-lter if the set M U (F) = {g ∈ G : UgU ∈ F} belongs to F for any neighborhood U of . To de ne a strict C-lter, it is necessary to consider sets M s U = {g ∈ G : UgU ∈ F}. If G is commutative, it is su cient to consider gU instead of UgU.
For given C-lters F , F , we write F ≥ F if M U (F ) ∈ F for any neighborhood U of . It is proved in [1] that it is a quasi-order relation. For the corresponding C-nets S , S , S ≥ S means that, for any U, the set of all sα from S for which elements of S eventually lie in Usα U, contains all su ciently remote elements of this net. F and F are said to be equivalent (we write F ≈ F ) if F ≥ F and F ≥ F . The notations F ≥ s F (S ≥ s S ) and F ≈ s F (S ≈ s S ) have a similar meaning for strict C-lters and strict C-nets.
By Proposition 2.3 from [1], for any C-lter F, the intersection of all C-lters which are equivalent to it, is a C-lter, too. It is called the least one in this class (or ⊂-least) and is denoted by F lst . For any point x ∈ G,ẋ denotes the lter consisting of all subsets containing x. It is a strict C-lter.
Some of C-lters on Θ converge in Θ. For these lters and for some others, their images under the map j converge in Γ, and such ones can exist that their images under this map diverge. Proof. These statements follow immediately from Proposition 2.1 from [2] since Θ is dense in Γ. Proof. It is straightforward that the following properties are equivalent: a) a C-lter F converges to x; b) the corresponding ⊂-least C-lter F lst converges to x (see Corollary 1.10 from [1]); c) any closed set from F lst contains x; d) any set from F lst contains x (see the previous Proposition); e) F lst ⊂ẋ; f) F lst ≈ẋ (see We will say that a C-lter F on Θ is trivial if there exists x ∈ Θ such that F ⊂ẋ. By Propositions 2.2 and 1.9 from [1], in this case, the lters F andẋ are equivalent, and F converges to x. A C-lter F on Θ is trivial if and only if there exists x ∈ Θ such that members of the corresponding net S(F) are frequently equal to x. Such nets are said to be trivial, too. Remark 1.10. For convergent strict C-lters on Θ, the notions ≈ and ≈ s do not coincide. Indeed, by Propositions 1.12 and 1.14 below, for any x ∈ Θ, there exists a non-trivial strict C-net Sx converging to x. The lters F(Sx) andẋ are equivalent as C-lters but not as strict ones since F ≥ sẋ does not take place. In particular, it implies that, by Propositions 2. 2

and 2.3 from [1], the intersection of these lters is a C-lter converging to x but not a strict one.
A strict C-lter F on Θ converges to x ∈ Θ if and only ifẋ ≥ s F. Indeed, it is straightforward that the following properties are equivalent: a) F converges to x; b) any neighborhood of x belongs to F; c) the set xU belongs to F for any neighborhood U of in Θ; d) x ∈ M s U (F) for any such U; e) M s U (F) ∈ẋ for any such U; f)ẋ ≥ s F. Lemma 1.11. Let S = {p mα } α∈A with mα ∈ N for all α be a non-trivial C-net in Θ. Then, for any m ∈ N , there exists α(m) ∈ A such that mα ≥ m for any α ≥ α(m).
Proof. For any k = , . . . , m − , there exists α k such that mα ≠ k for all α ≥ α k . Otherwise, F ⊂ẋ for some x = , . . . , p m− . We can now take as α(m) an arbitrary α with α ≥ α k for all these k. Proposition 1.12. Any non-trivial C-lter on Θ is strict. If F , F are non-trivial C-lters on Θ such that Proof. Let F be such a lter, U an arbitrary neighborhood of in Θ, and x = p n ∈ M U = M U (F). As above, denote Qn = { , p, . . . , p n− }. Since the set xU ∪ Qn is closed by Proposition 1.6 and contains xU, then it contains xU. Therefore, the inclusion xU ∈ F implies xU ∪ Qn ∈ F. For any k = , . . . , n − , it follows from F ⊄ṗ k that there exists a set F k ∈ F such that p k ∉ F k . Set F = (xU ∪ Qn) ∩ F ∩ · · · ∩ F n− . This set belongs to F and lies in xU. It implies xU ∈ F.
Let now V be an arbitrary neighborhood of in Θ and U its neighborhood such that

Corollary 1.13. Each equivalence class of C-lters on Θ contains a strict C-lter.
Proof. The topology τ| is T , therefore, by Corollary 1.10 from [1], either all C-lters from this class converge to the same x ∈ Θ or all diverge in Θ. In the rst case, they are equivalent to the strict C-lterẋ, in the second one, they are non-trivial and that's why strict ones. Proposition 1.14. (i) For any element a ∈ Γ, there exists a non-trivial C-lter F on Θ such that a = lim j(F).
(ii) If F , F are non-trivial C-lters on Θ (they are strict ones) such that lim j(F ) = lim j(F ) (the condition "non-trivial" is automatically satis ed if this limit does not belong to Θ), then F ≈ s F .
Proof. (i) First, let a / ∈ Θ or a = and {nα} α∈A be a net in N such that the net D = {p nα } α∈A converges to a. Prove that D is a C-net. Denote by U an arbitrary neighborhood of in Γ and nd a neighborhood V of a possessing, for this U, the property from the above de nition of a non-viscous topology. There exists α ∈ A such that p nα ∈ V for all α ≥ α . For each α ≥ α , there exists α ≥ α such that n α > nα if α ≥ α . Then p n α −nα ∈ U and p n α ∈ p nα U for such α and α , i.e. D is a strict C-net. It is non-trivial since it cannot cluster to any p n with n ∈ N. To nd a required lter for a = p n with n ∈ N, it su ces to move the net D corresponding to a = by means of the translation x → p n x.
(ii) Denote a = lim j(F ) and consider nets S(F ), S(F ) corresponding to the given lters. For the same U and V, if p m is an element of the rst net lying in V, then, by Lemma 1.11, elements p m of the second one eventually satisfy the conditions m ≥ m and p m ∈ V. Therefore, p m ∈ p m U and S(F ) ≥ s S(F ).
Also hereinafter, it will be sometimes more convenient for us to consider nets instead of lters. In particular, for any element t from Γ \ Θ, we will use the net is the set of all couples (W , p n ) each of which is composed of a neighborhood W of t and of an element of the form p n lying in W.
It is straightforward that t = lim j(S(t)) and j(S(t)) does not have other cluster points. For any δ = (W , p n ), denote m(δ) = min {k ∈ N : p k ∈ W}. It only depends on W, and we write sometimes m(W). This m(δ) is an increasing and unbounded function on be a lter on Θ corresponding to the net S(t). It is generated by all sets of the form Let now F be a C-lter on Θ such that j(F) converges to the same t. Then any neighborhood W of t belongs to j(F). Therefore, the set W ∩ Θ belongs to F for any W and F ⊃ F(S(t)). All the sets xU are open, therefore, it su ces to prove that the interior of the set M s U in Θ belongs to F for any U. For given U, let V be a neighborhood of in Θ such that p m−n ∈ U follows from p m , p n ∈ V and m ≥ n ≥ . Take an arbitrary x = p k ∈ M s V . Then xV ∈ F. Denote by S = S(F) = {xα} α∈A the strict C-net in Θ corresponding to F and by y = p n , n ≥ k, an arbitrary element from xV. There exists α ∈ A such that α ≥ α implies xα = p mα ∈ xV and mα ≥ n. For such α, we have p mα−k , p n−k ∈ V and p mα−n ∈ U, i.e. xα ∈ yU. Hence, In particular, it means that M s V lies inside M s U and that's why the interior of M s U in Θ belongs to F. In the following, we will also use the next lemma. Lemma 1.17. For any C-lter F on Θ and for any neighborhood U of (in the topology τ|), there exists an open (in this topology) V ∈ F such that x ∈ U for any a , a ∈ V and for any solution x of the equation a x = a lying in Θ.
Proof. If F converges in Θ to a point a, then we can take as V a neighborhood of a possessing this property. It exists since the topology τ| is non-viscous.
Let now F diverge in Θ. Since any C-lter contains the corresponding ⊂-least one, then we may assume that F is itself a non-trivial ⊂-least C-lter. For given U, denote by U a neighborhood of in Θ such that y ∈ U for any x , x ∈ U and any solution y of the equation x y = x . Let S(F) = {sα} α∈A be the strict C-net in Θ corresponding to F. Then there exists α ∈ A such that, for any α ≥ α , there exists α such that s α ∈ sα U for any α ≥ α . Fix any such α and take arbitrary a = s α , a = s α with α , α ≥ α . Denote by l, m, n naturals such that sα = p l , a = p m and a = p n . By Lemma 1.11, we may assume that m, n ≥ l. If m > n, then the equation a x = a does not have solutions lying in Θ. If n ≥ m, then x = p n−m is the only solution. In this case, we have p m ∈ p l U , p n ∈ p l U and p n−m is also a solution of the equation a x = a with a = p m−l and a = p n−l from U . It implies that x ∈ U. Now, the set F = {sα} α≥α belongs to F, and an arbitrary open member of F lying in F is suitable as V. It exists by the previous proposition.
Then it is easy to see that F is open and F ⊃ F O. C) In this section, we prove the main lemma of this paper. Lemma 1.19. For any divergent strict C-net T in Γ, there exists a strict C-net S in Θ such that j(S) ≥ T. Proof. Denote T = {tσ} σ∈Σ . We may assume that tσ ∉ j(Θ) for all σ since, otherwise, there would exist a ner than T net T which would be the image under the map j of a net S in Θ. By Proposition 2.2 from [1], this net T would be an equivalent to T C-net in Γ, and S would be a C-net by Proposition 1.7. If this net would be trivial and its members would be frequently equal to some x ∈ Θ, then it would be also true for T and T. However, this is impossible since T is divergent and can not cluster by Corollary 1.10 from [1]. Hence, S would be the required strict C-net by Proposition 1.12 above.
It follows now from this assumption that, for any σ, there exists a strict C-net Sσ = S(tσ) = {s σδ } δ∈∆σ in Θ constructed as it was described before Lemma 1.13 and such that lim j(Sσ) = tσ. The directed set ∆σ is the set of all couples (W , p n ) each of which consists of a neighborhood W of tσ and of an element of the form p m lying in W. Observe that any Sσ is non-trivial, and, therefore, the statement of Lemma 1.11 is true for it.
Using these nets Sσ, we will construct the required strict C-net S. For that, consider the set F of all maps f : Σ → σ∈Σ ∆σ such that f (σ) ∈ ∆σ for any σ. Denote by B the set of all couples of the form (σ, f ) with σ ∈ Σ, f ∈ F. For couples β = (σ , f ), β = (σ , f ), set β ≤ β if σ ≤ σ and f (σ) ≤ f (σ) for all σ ∈ Σ. Then it is straightforward that (B, ≤) is a directed set. For any β = (σ, f ) ∈ B, set s β = s σ f (σ) . We will prove that {s β } β∈B is the required net S. Remark 1.20. We used a well-known construction here. For example, see exercise 1.6.B from [6]. First, show that S is a strict C-net. For any neighborhood U of in Θ, we need to nd β = (σ , f ) ∈ B such that, for any β ≥ β , there exists β such that s β ∈ s β U for β ≥ β . Denote by U a neighborhood of in Θ such that U ⊂ U. Since Θ is a T -space, its topology has a base consisting of canonically open sets, i.e. of open sets which coincide with the interiors of their closures in this space. Therefore, we may assume that U and U are canonically open. Set now U = Int U . Here, the symbol Int and the hat on top denote the interior and the closure of a considered set in Γ. U is a neighborhood of in Γ containing U = U ∩ Θ as a dense subset. For any σ ∈ Σ, let Vσ be a neighborhood of tσ such that x ∈ U for any solution of the equation a x = a with coe cients from Vσ. Set f (σ) = (Vσ , p kσ ) where kσ = min {k ∈ N : p k ∈ Vσ}. It is a function from F. Choose σ so that, for any σ ≥ σ , there exists σ such that t σ = u σσ tσ with u σσ ∈ U for any σ ≥ σ . It is possible since T is a strict C-net in Γ.
Let now β = (σ, f ) ≥ β = (σ , f ). It follows from f ≥ f that s β = s σ f (σ) ∈ Vσ. All members of Sσ have the form p m with m ∈ N , and these exponents m form an increasing and unbounded function on ∆σ. The set Vσ contains all far enough members of Sσ, and the equation s β x = s σδ has a solution lying in U ∩ Θ = U for such members of this net. It means that the set s β U contains all these members. It implies, in particular, that the closure of the set s β U in Γ contains tσ.
Find β corresponding to β. Take σ as its rst co-ordinate. In order to nd its second co-ordinate f , consider the set s β U . For an arbitrary σ ≥ σ , its closure in Γ contains t σ . If an element u from s β U lies su ciently close to tσ and an element v from U lies su ciently close to u σσ , then their product uv belongs to V σ . Therefore, the set s β U ∩ V σ are non-empty. Similarly to the previous arguments, it is easy to prove that there exists δ (σ ) ∈ ∆ σ such that s σ δ ∈ s β U ⊂ s β U for all δ ≥ δ (σ ). Set now f (σ ) = δ (σ ) for σ ≥ σ and f = f for other elements from Σ.
For an arbitrary β = (σ , f ) ≥ β , it follows from σ ≥ σ and f ≥ f that f (σ ) ≥ δ (σ ) and, therefore, It remains to prove that j(S) ≥ T. Let U and U be neighborhoods of in Γ such that U ⊂ U. We need to nd β ∈ B such that, for any β ≥ β , there exists σ ∈ Σ such that the inclusion t σ ∈ s β U holds for all σ ≥ σ .

Properties of a weakly unitary completion of the monoid Θ. The main theorems
A) In this section, we prove our rst theorem.
Before starting, recall the necessary de nitions and some results of papers [1], [2], and [3]. A Hausdor topological monoid is said to be weakly unitarily complete if all its strict C-lters converge. Let X be a Hausdor monoid, Y a weakly unitarily complete monoid and f : X → Y an algebraic and topological embedding. The couple (f , Y) is called a weakly unitary completion of X if Y properly contains no weakly unitarily complete submonoid containing f (X).
For an arbitrary commutative Hausdor monoid on a T -underlying space, a construction of its weakly unitary completion was described in paragraph 3 of paper [3]. For the monoid Θ, it looks as follows. Denote byΘ the set of all equivalence classes of C-lters on Θ. It was proved above that each of them contains a strict C-lter. In place of classes of C-lters, one can consider the corresponding to them ⊂-least C-lters.
For an arbitrary C-lter F on Θ, we denote by [F] the corresponding point fromΘ. There exists a canonical embedding i of Θ intoΘ: for any x ∈ Θ, i(x) is the class of the lterẋ.
Since Θ is commutative, the product of C-lters on Θ is such a lter, too (see Corollary 2.6 from [2]). For arbitrary such lters F , F , denote by [F ]•[F ] the class of the lter F F . This is a commutative multiplication inΘ (see [2], Proposition 2.9). Its neutral element Θ is equal to i( ), and i is an identity preserving map. Show that it is an algebraic embedding of monoids. Indeed, it is evident that the inclusion(x x ) ⊃ (ẋ ) lst (ẋ ) lst is true for any x , x ∈ Θ. By Proposition 2.2 from [1], these C-lters are equivalent, and their corresponding ⊂-least lters coincide. Therefore, i( The underlying set of Θ endowed with the family Σ of all C-lters on this monoid forms a Cauchy space whose convergence structure de nes a Hausdor topology on Θ (see [1], Theorem 3.2). Since τ| is T , then this so-called unitary topology is ner than τ|.
There exists a familyΣ of lters on the setΘ such that this set endowed with this family forms the Wyler completion of (Θ, Σ). Topological properties of this space are studied in [2]. In particular, the convergence structure of the space (Θ,Σ) de nes a Hausdor topology on the setΘ which is said to be natural. For any lter F ∈ Σ, the lter i(F) belongs toΣ and converges to the point [F] in the natural topology, and each point ofΘ is the limit of such a lter. In the couple of topologies (the unitary one, the natural one), the map i is a homeomorphism of Θ onto an open subspace ofΘ (see Theorem 1.9 from [2]). In the following proof, we use also some of the ideas of the proofs of Theorems 3.2 and 3.7 from [3] but not these theorems themselves since we are going to get a stronger result. In particular, we construct another topology onΘ which we denote, as in [3], by τc, and use a family {Ox}x ∈Θ of lters of open subsets of Θ for that.
Let F be a strict C-lter on Θ belonging to an equivalence class of C-ltersx. If it converges to a point x ∈ Θ, then we denote by O F the set of all neighborhoods of this point x in the space Θ. If F diverges in Θ (it is non-trivial in this case), then O F is the set of open subsets of Θ belonging to the corresponding ⊂-least C-lter F lst . It is also divergent, non-trivial and strict by Proposition 1.12. In any case, O F lies in F and only depends onx. Therefore, we also denote it by Ox.
Show that Ox = Ox impliesx =x . Let F , F be strict C-lters from classesx , respectively,x Consider two cases. First, let F converge to a point x ∈ Θ. Then Ox = Ox means that F converges to the same point, and it involvesx =x by Corollary 1.9. If both these lters diverge in Θ, then, by Proposition 1.16, non-trivial ⊂-least C-lters (F ) lst and (F ) lst have the same bases, andx =x again.
De ne now the topology τc. Similarly to section 3.A) from [3], for any non-empty open V ⊂ Θ, set It is easy to check that V * ∩ V * = (V ∩ V ) * for arbitrary open V , V ⊂ Θ, and it means that the family of all sets of the form V * forms a base of a topology onΘ which we denote by τc. We could de ne this topology di erently. By Corollary 1.9, any convergent to x ∈ Θ C-lter is equivalent toẋ. Therefore,Θ is the union of its subset i(Θ) and its subset Y consisting of classes of divergent C-lters on Θ or, it is equivalent, of ⊂-least divergent C-lters on this monoid. All these lters are strict and have bases consisting of closed sets (Propositions 1.12 and 1.8 above). The latter property means that we are in the situation which was described in section 2 C) of paper [3]. Therefore, we can de ne a topology on the set i(Θ) ∪ Y as follows. For any open V ⊂ Θ, we denoted by V * the set i(V) ∪ {F ∈ Y : V ∈ F}. For ⊂-least divergent C-lters F on Θ and open subsets V ⊂ Θ, the requirements V ∈ O F and V ∈ F coincide. Moreover, the inclusions x ∈ V and V ∈ O i(x) are equivalent for any x and for any open V from Θ. Hence, these di erent de nitions of sets V * lead to the same result, and, for X = Θ, the space (Θ, τc) coincides with the space υX which was de ned in section 2.C) of paper [3]. In the following, we will also use the denotation υΘ for this space.
The latter description of the topology τc is more suitable for the characterization of closed sets. As in [3], for each closed Z ⊂ Θ, denote Z * =Θ \ (Θ \ Z) * . This closed in the topology τc subset consists of i(Z) and of all point [F] ∈ Y such that Z ∩ F ≠ ∅ for any F ∈ F lst . Each closed in this topology subset ofΘ is an intersection of subsets of the form Z * . The equality i(H) = (H) * holds for any subset H ⊂ Θ. The accent tilde on the left side denotes the closure in the topology τc, and, as above, the line on top on the right side denotes the closure in the topology τ| on Θ.
Show now that the topology τc is coarser than the natural topology or coincides with it. It involves, in particular, that i is a dense embedding of Θ intoΘ endowed with the topology τc, any lter of the form i(F) where F is a C-lter on Θ, converges in this topology, and any point fromΘ is the limit of such a lter. Indeed, letx ∈Θ be an arbitrary point and V * its neighborhood in the topology τc from the base above. First, assume that there exists x ∈ Θ such thatx = i(x). Thenx Proof. We prove this result in a sequence of several steps.

The map i is an algebraic and a topological embedding.
It is a topological embedding of the monoid Θ into the monoid (Θ, •, τc) since V * ∩ i(Θ) = i(V) for any open V ⊂ Θ, i.e. the restriction of the topology τc onto i(Θ) coincides with its initial topology τ|. It is also noted above that this embedding is an algebraic one.
The topology τc is T . First, check that it is T . Letx ,x be di erent points ofΘ and F , F the corresponding ⊂-least C-lters. If the rst of them converges in Θ to a point x and the second one converges to another point or diverges, then there exists a neighborhood V of x which does not belong to F . Therefore, x belongs, andx does not belong to V * .
If F diverges in Θ and that's why does not cluster there (see Corollary 1.10 from [1]), while F converges to a point x ∈ Θ, then there exists an open set V ∈ F which does not contain x. Otherwise, by Proposition 1.16, all members of F would contain x. Therefore,x belongs, andx does not belong to V * again.
Finally, if both the lters F and F diverge in Θ, then there exists an open V ∈ F which does not belong to F . Otherwise, again by Proposition 1.16, we would have F ⊂ F , and it would imply to F ≈ F (see Let nowx be an arbitrary point ofΘ and V * its arbitrary neighborhood in the space (Θ, τc) belonging to the base above and corresponding to an open V ⊂ Θ. Find a neighborhood W * of this point such that First, let the ⊂-least C lter F on Θ corresponding tox converge to x ∈ Θ. Since the topology τ| on Θ is T , there exist a neighborhood W of x and a neighborhood U of in Θ such that UW ⊂ V. Then W * is a neighborhood ofx in the space (Θ, τc). Show that its closure in (Θ, τc) lies in V * and start with the following remark. For anyx ∈Θ, the family Ox is a lter in the set of open subsets of Θ. Its open subsets W and Θ \ W have an empty intersection and can not belong the same lter Ox, i.e. W * ∩ (Θ \ W) * = ∅. It implies that W * lies in the closed in the topology τc subsetΘ \ (Θ \ W) * = (W) * which consists of i(W) ⊂ i(V) ⊂ V * and of all pointsx from Y corresponding to ⊂-least strict C-lters possessing the property that intersections of all their members with W are non-empty. If F is one of these lters, then M s U (F) ∩ W ≠ ∅, and that's why there exists y ∈ W such that yU ∈ F. It implies that UW ∈ F, V ∈ F and the pointx corresponding to F lies in V * .
Let now F diverge. It is a strict non-trivial ⊂-least C-lter, and V ∈ F . By Propositions 1.8, 1.16 and 1.18, there exist an open W ∈ F and a neighborhood U of in Θ such that UW ⊂ V. Then the previous arguments show that this W is required.
The multiplication • is continuous in the topology τc. For anyx ,x ∈Θ and for an arbitrary W ∈ Ox •x , we have to nd V ∈ Ox and V ∈ Ox such that V ∈ Ox , V ∈ Ox imply W ∈ Ox •x for anyx ,x ∈Θ.
Let F , F be strict C-lters from classesx , respectively,x . Then the classx •x contains the strict C-lter F = F F . First, we show that, for any W ∈ O F , there exists W ∈ O F and a neighborhood U of in Θ such that W UU ⊂ W. Indeed, if F converges to some point and W is a neighborhood of this point, then the existence of these W and U follows from the fact that the underlying space of Θ is T . If F diverges and W belongs to the corresponding ⊂-least non-trivial C-lter F lst , then it follows from Propositions 1.8, 1.16 and 1.18.
Find now the sets V and V . There exist sets V ∈ F , V ∈ F such that V V ⊂ W . If the lter F diverges in Θ, then, for any neighborhood U of , the set V U is open in Θ and belongs to (F ) lst . Indeed, (F ) lst diverges, too, and that's why both these lters are non-trivial. Now, it follows from Proposition 1.12 that there exists a point a ∈ V such that aU belongs to (F ) lst , and V U ∈ (F ) lst . Therefore, V U ∈ O F for any U . Choose U so that U ⊂ U, and set V = V U . If the lter F converges to a point x ∈ Θ, then x ∈ V and V = V U is a neighborhood of x and belongs to O F . The choice of V is similar. In both these cases, If G converges to x, then x ∈ V V and xU is a neighborhood of x. Since xU ⊂ V V U ⊂ W and W is open, then W is a neighborhood of x, too, and, therefore, W ∈ Ox •x . If G diverges in Θ, then, as above, V V ∈ G implies V V U ∈ G lst and W ∈ Ox •x again.

Thus, (Θ, •, τc) is a topological monoid on a T -underlying space. Since i is a dense algebraic and topological embedding, then it is monothetic with a generator i(p).
The topology τc is non-viscous. Take an arbitrary neighborhood of the neutral element Θ . We may assume that it has the form U * where U is a neighborhood of the neutral element Θ . Denote by U a neighborhood of Θ such that the set U U is contained in U. Letǎ be an arbitrary point ofΘ and F the corresponding ⊂least C-lter on Θ. Find an open subset V ⊂ Θ such that the set V * is a neighborhood ofǎ satisfying the requirement of the de nition of a non-viscous topology from the beginning of section 1 A). If F diverges in Θ, then V is an open member of F from lemma 1.17 corresponding to U in place of U. If F converges to a point x ∈ Θ, then V is a neighborhood of this x corresponding to U by the de nition of a non-viscous topology. In both these cases, V ∈ Ox.
Consider now a solution x of the equation a x = a in (Θ, •) with a , a ∈ V * . Denote by F and G arbitrary strict C-lters from the classes a and x, respectively, and set F = F G. It is a strict C-lter from the class a . It follows from a , a ∈ V * that V belongs to the corresponding lters O F ⊂ F and O F ⊂ F . Since it belongs to the product of the lters F and G, there exist members F and G of these lters, respectively, such that F and F G are subsets of V. It follows now from the choice of U and V that G ⊂ U and so U ∈ G. As above, if G diverges in Θ, then U belongs to the corresponding ⊂-least C-lter G lst , U ∈ O G and x ∈ U * . If G converges to g ∈ Θ, then g belongs to the closure of U , and therefore it lies in U. Hence, U ∈ O G and x ∈ U * again.
The monoid (Θ, •) is cancellative. Letx ,x ,x ∈Θ andx •x =x •x . Denote by S = {s α } α∈A , S = {s β } β∈B and S = {s γ } γ∈C strict C-nets in Θ such that i(S ) converges tox , i(S ) converges tox , and i(S ) converges tox . All members of these nets have the form p m with m ∈ N . Since τc is non-viscous, for any neighborhood U of Θ in Θ, there is a neighborhood V * of the pointx •x ∈Θ such that the pair U * , V * satis es the requirement of the de nition of a non-viscous topology. There exist α ∈ A, β ∈ B, γ ∈ C such that, for α ≥ α , β ≥ β , γ ≥ γ , the elements i(s α s β ) and i(s α s γ ) belong to V * . Both these elements have the form i(p m ), m ∈ N , therefore, for any such α, β and γ, one of these elements can be obtained from the other by multiplication with a factor of the same form. This factor lies in U * ∩ i(Θ) = i(U). Since i is an algebraic embedding, it means that there exists u αβγ ∈ U such that s α s β = s α s γ u αβγ or s α s β u αβγ = s α s γ . Θ is cancellative, and we may cancel the rst factors. Since U is arbitrary, the topology τc is Hausdor , and the multiplication • is continuous, the obtained formula implies that the nets i(S ) and i(S ) have equal limitsx andx .
Thus, (Θ, •, τc) possesses the same properties which were assumed for Γ. Therefore, all statements which were proved in section 1) for Γ, remain valid for this monoid. In particular, it is true (with the replacement of j by i) for Proposition 1.7 and Lemma 1.19.
The monoid (Θ , •, τc) is weakly unitarily complete. Indeed, let T = {tσ} σ∈Σ be its strict C-net. By Lemma 1.19, there exists a strict C-net S in Θ such that i(S) ≥ T. It was noted above that the C-net i(S) converges for any C-net S in Θ. Therefore, the net T converges by Corollary 1.10 from [1].
(Θ, •, τc) is a complete topological group. First, prove that all elements from i(Θ) are invertible, and the function i(p) n → i(p) −n is continuous on i(Θ). Let a strict C-net D = {p nα } α∈A in Θ converge to Θ . Then the net D = {p nα− } α∈A is a strict C-net, too, the net i(D ) converges in (Θ, τc), and i(p) • q = Θ holds for its limit q, i.e. q = i(p) − . It means that i(p) −n exists for any n ∈ N. To prove the continuity of the function i(p) n → i(p) −n on i(Θ), take an arbitrary n ∈ N and denote by W an arbitrary neighborhood of i(p) −n and by U a neighborhood of Θ such that i(p) −n • U ⊂ W. Find a neighborhood V of a = i(p) n corresponding to U for this point a in the de nition of a non-viscous topology. It exists since the topology τc is non-viscous. If i(p) m ∈ V for some m ∈ N , then i(p) n−m ∈ U and i(p) −m ∈ W.
Show now that all elements of (Θ, •) are invertible. Let Da = {p nα } α∈A be a C-net in Θ such that the net i(Da) converges to a given a ∈Θ. Then it follows from the proved continuity of the function i(p) n → i(p) −n that i(Da) − = {i(p) −nα } α∈A is a strict C-net in (Θ, •, τc), and it is evident that a • h = for its limit h. Since (Θ, •) is cancellative, there exists an only h with this property.
Thus, (Θ, •, τc) is algebraically a group. Therefore, it is a topological group by Proposition 1.2. Note also that the continuity of the function x → x − at any point x ∈Θ can be proved similarly to the above argument for the case x ∈ i(Θ).

B)
In this section, we prove the main theorems of this paper.

Theorem 2.2. A monothetic monoid can be (algebraically and topologically) embedded into a topological group if and only if it is cancellative and its topology is T and non-viscous.
If these conditions are satis ed, then the constructed above weakly unitary completion (Θ, •, τc) of the submonoid Θ = {p n : n ∈ N } generated by the generator p of the initial monoid is such a topological group, the natural embedding of this monoid into this group is dense, and this group is monothetic.
Proof. The necessity is evident. To prove the su ciency, show, keeping the previous notation, that the map i can be continued up to an algebraic and topological dense embedding ϕ : Γ → (Θ, •, τc).
Let g be an arbitrary element from Γ. By Corollary 1.9 and Proposition 1.14, there exists a unique ⊂-least Clter F on Θ such that j(F) converges to g. Denote by ϕ(g) the point [F] ofΘ corresponding to F. It is evident that this map ϕ : Γ →Θ coincides with i on Θ. It is injective since g ≠ g implies F ≉ F , but it is not necessarily surjective since some C-lters of the form j(F) can diverge in Γ. It is an algebraic embedding of monoids since the multiplication • corresponds to the multiplication of C-lters and the limit of the product of convergent lters in Γ is equal to the product of their limits.
It now su ces to show that ϕ is a continuous open map of Γ onto ϕ(Γ). Since Γ is a T -space, there exists its base B consisting of canonically open sets, i.e. such that U = Int U for any U ∈ B, where, as above, the hat on top and the symbol Int denote the closure and the interior of the corresponding subset of Γ. The intersections of the form U = U ∩ Θ with U ∈ U form a base of Θ, and the corresponding sets of the form U * form a base in (Θ, •, τc). We will only consider these bases.
Prove that ϕ(U) = ϕ(Γ) ∩ U * and ϕ − (U * ) = U for any U ∈ U. Let g ∈ U ∈ U. For g ∈ U , the corresponding lter F isġ lst , and it implies that U ∈ O F and ϕ(g) ∈ U * . Conversely, if g ∈ Θ and ϕ(g) ∈ U * , then U ∈ O F , and it means that g ∈ U ⊂ U since O F consists of neighborhoods of g in Θ.
Let now g ∈ U \ U . Then U ∈ F since F is a lter on Θ and j(F) converges to g. Since F diverges and U is open in Θ, then U ∈ O F and ϕ(g) = [F] ∈ U * . Conversely, if g ∉ Θ and ϕ(g) ∈ U * , then U ∈ O F ⊂ F. As it was showed in Lemma 1.15, the lter F coincides with the lter F(S(g)) (see text before the proof of this lemma), and there exists a neighborhood W of g such that W = W ∩ Θ is contained in U . W is dense in W and U is dense in U, and it implies that W = W and U = U. Now, the following inclusions are true: W ⊂ Int W ⊂ Int U = U, i.e. g ∈ U. It completes the proof. Corollary 2.3. Let G be the product of any family of monothetic monoids endowed with the Tychono product topology. If there exists an algebraic and topological embedding of G into a topological group, then G and all its factors are cancellative and have T non-viscous topologies. If G is cancellative and has a T non-viscous topology, then there exists its dense algebraic and topological embedding into a topological group.
Proof. If there exists an algebraic and topological embedding of G into a topological group, then G is cancellative and has a T non-viscous topology. Therefore, each factor of G has these properties, too. If G has these properties, then each its factor has them, too, and there exists its dense algebraic and topological embedding into a topological group. The product of these embeddings is a required embedding of G.
Show now that the correspondence Γ → (Θ, •, τc) possesses the following universal property. Theorem 2.4. Under the same notation and assumptions about Γ as above, let G be an arbitrary complete commutative topological group and f : Θ → G (f : Γ → G) an algebraic and topological embedding of the monoid Θ (respectively, of the monoid Γ). Then there exists a continuous homomorphismf of the topological group (Θ, •, τc) into G such that f =f • i (respectively, f =f • ϕ where ϕ is the constructed above embedding of Γ into (Θ, •, τc)).
Proof. First, consider the case when f is a map of Θ, and de ne the mapf . Letx be a point ofΘ, [F] the corresponding to it equivalence class of C-lters on Θ, and F ∈ [F]. Then f (F) is a C-lter on G, i.e. a Cauchy lter of its standard uniformity. Hence, this lter converges, and we denote its limit byf (x). This limit does not depend on the choice of F in [F] since equivalent C-lters have equal limits (see Corollary 1.10 from [1]) and equivalent images under homomorphisms (Proposition 2.1 from [2]). Ifx ∈ i(Θ), i.e. F is equivalent to the lterẋ for some x ∈ Θ, then lim f (F) = f (x) and f =f • i. Show thatf is a homomorphism. Consider arbitraryx ,x ∈Θ and denote by F = {F α } α∈A , F = {F β } β∈B the corresponding C-lters on Θ. Then the corresponding tox •x C-lter F F is generated by all sets of the form F α F β with α ∈ A, β ∈ B. Since f is a homomorphism, the lter f (F F ) is generated by sets f (F α F β ) = f (F α ) * f (F β ), α ∈ A, β ∈ B. Here, * denotes the multiplication in G. Therefore, f (F F ) = f (F ) * f (F ) andf (x •x ) =f (x ) *f (x ).
Let now f be an algebraic and topological embedding of Γ. Then f • j is an embedding of Θ, and there exists a continuous homomorphismf : (Θ, •, τc) → G such that f • j =f • ϕ • j. The continuous maps f anď f • ϕ coincide on Γ since they coincide on its dense subset j(Θ).