Augmented, free and tensor generalized digroups

Abstract The concept of generalized digroup was proposed by Salazar-Díaz, Velásquez and Wills-Toro in their paper “Generalized digroups” as a non trivial extension of groups. In this way, many concepts and results given in the category of groups can be extended in a natural form to the category of generalized digroups. The aim of this paper is to present the construction of the free generalized digroup and study its properties. Although this construction is vastly different from the one given for the case of groups, we will use this concept, the classical construction for groups and the semidirect product to construct the tensor generalized digroup as well as the semidirect product of generalized digroups. Additionally, we give a new structural result for generalized digroups using compatible actions of groups and an equivariant map from a group set to the group corresponding to notions of associative dialgebras and augmented racks.


Introduction
The digroup structure is introduced by M. Kinyon [2], R. Felipe [3] and K. Liu [4] as a non trivial extension of the concept of group, with the purpose of giving an answer to the so called Coquecigrue problem which is supposed to provide a generalization of the third Lie theorem for Leibniz algebras, see [5].
A slightly di erent structure studied in [1] is called generalized digroup. It doesn't request bilateral inverses for its elements. This concept is corresponding to what is called Digroups in [6].
For digroups with bar units that generate bilateral inverses (see [7]) several authors propose di erent generalizations of the notion of digroup. For instance, in [8], J. D. H. Smith shows that any digroup with bilateral inverses is equivalent to what he calls a ( + )-diquasigroup (Theorem 10.8). His proof uses digroups generated by two groups that act in a commutative way over a set. This idea is similar to a work developed in [9] which leads to express associative dialgebras in terms of bimodules over associative algebras and equivariant maps.
In addition, in [2], M. Kinyon proves that any digroup generates a rack and it is natural to think that it can be extended to generalized digroups. Since any rack can be generated by a group acting over a set, with the action commuting with the conjugation and an equivariant map, that gives another motivation to explore what we call here augmented generalized digroups, a construction that provides another characterization of generalized digroups.
Due to the fact that augmented racks give set theorical solutions to the quantum Yang-Baxter equation (see [10]), and that augmented generalized digroups can be de ned, it is possible to study relations between the Yang-Baxter equations and generalized digroups that could procure solutions. These ideas are being explored by the authors in a work in progress.
Finding free structures is a central problem in abstract algebra. For the case of dimonoids we can nd constructions in several works, for example see [6,8,[11][12][13]. The free generalized digroup is exhibited in the present work and it is done following Loday's ideas for free dimonoids (see [5]).
The semidirect product of groups (see [15]) induces our de nition of tensor generalized digroup and its representations. The cyclic generalized digroup and generalized semidirect product result naturally from the discussion involved.
The paper is organized as follows. In Section 2, we review the basic theory and the notions of subdigroup and normality in the sense of [1] and we introduce the de nitions of anti-homomorphisms and involutions over generalized digroups.
We nish Section 2 with the relation of generalized digroups with associative dialgebras and racks, we de ne augmented generalized digroups, and we also show that each generalized digroup can be expressed in such a way.
Section 3 is dedicated to study the construction of the free generalized digroup and to show some properties of this new structure.
In the last section, we introduce the notions of generalized tensor digroups and generating sets and we nish with the concept of the semidirect product of generalized digroups.

Some results about Generalized Digroups
In this section we brie y recall some de nitions and results about generalized digroups, for a deeper study see [1]. We also review some properties and we introduce the notion of anti-homomorphism and involution for this structure. Finally we introduce the notion of augmented generalized digroups.

. General results
We summon up the de nition, some basic properties and a way to look up generalized digroups.

De nition 1.
A set D is a generalized digroup if it has two binary associative operations and over D, such that they satisfy the following conditions: It is clear that a group (G, ·) can be seen as a generalized digroup by considering = · = .
The elements that satisfy condition 3. are called bar-units and the set of them, denoted by E, is called the halo of D. For any bar-unit ξ ∈ E, we denote the sets of left and right inverses with respect to ξ by G ξ l and G ξ r , respectively. A generalized digroup that consists only of bar units or that is a group is called a trivial generalized digroup, here we exhibit a non trivial generalized digroup.

For x, y in D, the inverses of the products are
where * represents any of the products.
It is easy to see that the halo also corresponds to any of the following sets Since the proof of the following theorem comes from the results given in [1], we omit it. Theorem 1. Let (D, , ) be a generalized digroup. For any ξ ∈ E, (G ξ l , ) and (G ξ r , ) are isomorphic groups with unit ξ . Moreover, it is true that for any ξ , ζ ∈ E 1. G ξ l = ξ G ζ l and G ξ r = G ζ r ξ , As it is shown in [1] a characterization of D is given below In order to describe another way of looking at a generalized digroup let's recall that if D and D are generalized digroups, a map ϕ : D → D is a generalized digroup homomorphism if for any x, y ∈ D ϕ(x y) = ϕ(x) ϕ(y) and ϕ(x y) = ϕ(x) ϕ(y).
In addition, if ϕ is a bijection, then ϕ is a generalized isomorphism and D is isomorphic to D . The second characterization of generalized digroups is an extension of the results of M. Kinyon (see [2]) and F. Ongay (see [16]).
Theorem 3. Let D, E and G ξ l be as in Theorem 2, then the map φ l : That is, a generalized digroup can be seen as a cartesian product between a G-set E and the group G, with set of bar units {e} × E. With respect to such decomposition of a generalized digroup, next theorem describes a generalized digroup homomorphism.
for all α ∈ E, all x ∈ D and all a ∈ G ξ l .
Now we extend the notion of anti-homomorphism and involution, known in group theory, for generalized digroups.
An anti-homomorphism x → x * from a digroup to itself is called an involution if it is its own inverse, i.e. (x * ) * = x, for all x ∈ D.
A similar result as Theorem 4 can be stated for generalized digroup anti-homomorphisms, we only have to check that the corresponding function Ψ = (ϕ, µ) is in fact an anti-homomorphism.
Some basic properties about involution are given in the following proposition. Now, we consider the involution * : D → D de ned by (g, z) * = (g − , −z). As we can see, the halo of D is {id}× S . And so, for every z ∈ S − , (id, z) * = (id, −z) ≠ (id, z).
We recall the concept of a generalized subdigroup of a generalized digroup D, denoted by S ≤ D, a subset, such that with the products in D restricted to it, is a generalized digroup. It is proven in [1], section 3., that S must satisfy that there are two subsets Γ and ∆ of it such that ∆ is Γ-invariant respect to the action de ned in It is also useful to bring back the de nition of normality, where for a generalized digroup (D; , ) and a subdigroup N, the latter is a normal generalized subdigroup of D, denoted by N D, if x N = N x, for any x ∈ D. Moreover, we have that

. Augmented generalized digroups
This subsection is dedicated to the new characterizacion of generalized digroups mentioned in the introduction.
In the proposition we show that any generalized digroup is equivalent to a generalized digroup generated by compatible actions and an equivariant map. We call the structure that satis es the hypothesis an augmented generalized digroup and we denote it by (G, X, λ, ρ, π).

Proposition 3. Let G be a group, with unit G , and let X be a G-set under the left and rigth compatible actions λ and ρ, i.e. λg commutes with ρ h , for any g, h ∈ G. If there is an equivariant map π : X → G with respect to both actions such that π(X) generates G, then (X, , ) is a generalized digroup with the operations
x y = λ π(x) (y) := π(x) • λ y and x y = ρ π(y) (x) := x •ρ π(y).
Proof. The products and are associative because λ and ρ are actions. From the compatibility of the actions, condition 1 in the de nition of generalized digroup follows. The equivariance of π, with respect to the actions λ and ρ, implies that Similarly, x (y z) = x (y z), for all x, y, z ∈ X; and condition 2 is satis ed. Because π(X) generates G, there are x, y ∈ X such that π(x) · π(y) = G , and therefore E ≠ ∅. Since π(ξ ) = G , then ξ is a bar unit and we have that E is the halo of X.
The proof for x y is analogous.
In a similar way, we get the expressions for the inverses with respect to the bar units ξ , η ∈ E, with ξ = x y and η = x y: We show that any generalized digroup can be seen as in the previous proposition. Using that G e l = e D and G e r = D e, for all e ∈ E (see Theorem 1), and E x = x E, since E is a normal generalized subdigroup, we obtain the following result.
Theorem 5. Let D be a generalized digroup. There exist a group G, two compatible actions over D, λ and ρ, and an equivariant map π : D → G such that D can be seen as an augmented generalized digroup (G, D, λ, ρ, π).
Proof. Given the factorization of a generalized digroup in terms of the set E and the group G ξ l , for an arbitrary ξ ∈ E, we have that G := D/E is a group isomorphic to all groups of inverses and If we de ne the maps λ : we have that λ and ρ are compatible actions from G on D by identities 1. and 2. in De nition 1. Finally, the projection map π : D → G, de ned by π(x) = [x], is an equivariant map with respect to both actions and π(D) generates G. It's easy to see for any x, y ∈ D that x y = λ π(x) (y) and x y = ρ π(y) (x). Now we are going to show some basic properties respect to augmented generalized digroups.
In Theorem 4, for any generalized digroup homomorphism Ψ : for all x ∈ D, and therefore φ is a group homomorphism equivariant with respect to the generalized digroup homomorphism Ψ.
Moreover, this digroup homomorphism satis es for all x, y ∈ D. From these considerations we obtain the following result.
Proof. Let φ : π(X) → Π(Y) be a map de ned by φ π(x) := Π Ψ(x) , for all x ∈ X. From the digroup axioms and generalized digroup homomorphism characterization (see Theorem 4) it follows that (φ, Ψ) is an augmented generalized digroup homomorphism, G ∼ = G ξ l and H ∼ = G ξ l , since the following diagram commutes and for any (g, x, h) ∈ G × X × H the inverses with respect to the bar unit ( G , z, H ) are: Therefore the groups of inverses are isomorphic to the direct product G × H. The actions λ and ρ induce a left action λ and a right action ρ, from G × H on the set G × X × H by The projection map π, given by (g, x, h) → (g, h), is equivariant with respect to the actions λ and ρ. Let (D, , ) be a generalized digroup, since for any ξ ∈ E, we have that for the isomorphic groups G ξ l and G ξ l , the set D is a left G ξ l -set and a rigth G ξ r set, with commutative actions de ned by λg(x) := g x and ρ h (y) := y h, for any g ∈ G ξ l , h ∈ G ξ r and x, y ∈ D, respectively, then (G ξ l × D × G ξ r ) is a generalized digroup isomorphic to D, with isomorphism given by x → (ξ x, x, x ξ ).

The free generalized digroup
In [5] J.-L. Loday constructs the free dimonoid. Later, using a free semigroup FS[X] and the word lenght map A. V. Zhuchok in [11] constructs another free dimonoid which is isomorphic to the one de ned by J. -L. Loday. Both dimonoids don't extend to digroups since A. V. Zhuchok in [12], Theorem 4, p. 833, shows that it's impossible to adjoin a set of bar units (halo) to a Loday's free dimonoid. In this section, we exhibit the free generalized digroup FD(X) for any set X. The basic ideas in the construction are related to the articles [1,7,9], and the book Dialgebras and related operads (see [14]).
In addition we present several properties of FD(X) and relate it with the characterization theorems and augmented generalized digroups.
For the construction of free generalized digroup we use classical results for free structures (see [17]). Let X be a set and F(X) the free group generated by X, that is, the set of all words in X ± , see [15] for details on this de nition. The proof of the following statement is straightforward and therefore we omit it. However, the result is central in this section.

Proposition 4. Let FD(X) := F(X) × X × F(X) with the binary maps and de ned for all x, y ∈ X and u, v, a, b ∈ F(X) by
where the empty word e is the unit of the free group F(X) and w − is the inverse of w in F(X).

Remark 1. The inverses
We want to show now that FD(X) is the free generalized digroup in X, i.e. it's the free element in the generalized digroup category. First, note that the natural immersion X → FD(X) is given by x → (e, x, e).

Theorem 7. (Universal property) Let D be a generalized digroup and let X be a set.
For each function f : X → D, there exists a homomorphism Ψ : FD(X) → D that extends f , that is, such that the following diagram commutes.
Let E be the halo of D and ξ ∈ E xed. For any x ∈ X there is a bar unit η ∈ E such that f (x) ∈ G η l , Since G η l is a group, the inverse of f (x) in G η l is f (x) − lη ∈ G η l , and therefore we use the convention f (x) − := f (x) − lη . We de ne a function φ : F(X) → G ξ l in the following way: for any x ∈ X, φ( ∈ G ξ l , and for any w = x δ i x δ i · · · x δn in ∈ F(X), with x i j ∈ X and δ j = ± , Then φ : F(X) → G ξ l is a group homomorphism. Now, we de ne Ψ : FD(X) → D by Ψ((u, x, a) (a), for all (u, x, a) ∈ FD(X). The map Ψ is a homomorphism. Indeed, Similarly, and

De nition 4. The generalized digroup FD(X) is called the free generalized digroup on X.
The bar units and the inverses in the free generalized digroup have the following properties Remark 2. For FD(X) we have that 1. If y ∈ X and v ∈ F(X) then .
We want to describe the sets in De nition 6 from [1] for the generalized digroup FD(X).

Proposition 5. For any bar unit (v, y, b) ∈ E(X), the group of left inverses G l
(v,y,b) is isomorphic to F(X).

Remark 3. According to the previous result, we can see that G l
We describe now some actions from the group F(X) to the free generalized digroup FD(X). First, we de ne the maps From these de nitions it is simple to see that

Theorem 8. The free generalized digroup FD(X) is a F(X)-set with respect to the actions λ, ρ and γ. Besides, λ and ρ are compatible.
After de ning the action γ, we can see how it works out with the function Ψ, say, Ψ(u, x, a)) Thus, we can conclude Lemma 3. The map Ψ is φ-equivariant with respect to the actions λ, ρ and γ. Now, given a projection map Π : FD(X) → F(X) de ned by (u, x, a) → uxa, we have that Π is surjective and equivariant with respect to the actions λ and ρ. Moreover, for all (u, x, a), (v, y, b) ∈ FD(X), y, b) and therefore Theorem 9. The augmented generalized digroup (F(X), FD(X), λ, ρ, Π) is the free generalized digroup FD(X).
Going back to the units of the free generalized digroup, we classify them in two groups. For that, note that elements of the form (e, z, z − ) and (z − , z, e) are bar units.
We denote by ξwz the bar unit (w, z, (wz) − ) and by ξzc the bar unit ((zc) − , z, c). Thus ξez = (e, z, z − ) and ξze = (z − , z, e). It is simple to see that X, E l B and E l B are equipotent sets.
Using the characterization of the bar units, we can express the inverses in terms of it. For the halo E(X), we consider the following notation E l (X) = {(w, z, (wz) − ) | z ∈ X and w ∈ F(X)} and E r (X) = {((zc) − , z, c) | z ∈ X and c ∈ F(X)}.

Lemma 5.
If in the free generalized digroup FD(X) we denote the action γ by ·, we can see that

e. E(X) is invariant under the action γ.
We review involutions in free generalized digroups. As we have in subsection 2.1, every involution * de ned on a generalized digroup G × E can be projected to a group involution ι : G → G. In the following theorem we describe a way to extend a group involution on F(X) to an involution on FD(X). Its proof comes directly from calculations, then we omit it. Theorem 10. Let ι : F(X) → F(X) be a involution over the free group generated by a set X. The involution ι can be extended to an involution * : FD(X) → FD(X), by (u, x, a) * := (a ι , x ι , u ι ) if and only if x ι ∈ X, for all x ∈ X.

The tensor generalized digroup
In this section we use the construction of the semidirect product of groups given in [15] in order to de ne the tensor generalized digroup and the concept of generating set for a generalized digroup. It is a surprise to prove that the same construction of the semidirect product of groups works for generalized digroups.
It is well known that if G is a group, and ϕ : G → Aut(G ) is a representation of G, with ϕg = ϕ(g), then G × G can be endowed with group structure as follows This product is called, in the literature, the semidirect product of G and G and it is denoted by G ϕ G . As we see above, G ϕ G can also be considered as a generalized digroup by taking This form of constructing generalized digroups from group structures is called the trivial form.
The following theorem gives us a non trivial form to provide the semidirect product G ϕ G with generalized digroup structure. At the end of this section we use this idea to de ne the tensor generalized digroups.

Theorem 11. Let G be a group. Then, the group G ϕ G is a generalized digroup with the following operations
and Since the binary operations and do not depend on the representation ϕ, we use the notation G ⊗ G to refer to G ϕ G as a generalized digroup in a nontrivial form and u ⊗ v instead of (u, v).
Proof. The proof that and are associative binary operations is taken from direct calculations. Now, we verify the condition given in the De nition of [1]. In fact, let u ⊗ u , v ⊗ v and w ⊗ w in G ⊗ G, then It is not hard to verify that the set of bar units of G ⊗ G, i.e., the halo of

De nition 6. The generalized digroup in the theorem is called the tensor generalized digroup of G.
Note that if in the previous theorem we de ne, for every u ∈ G, the bijection φu : G → G, as φu(v) = uv, then we can rewrite the equations (2) and (3) as follows and Let us consider the following interesting fact. If G is obtained by the quotient of the free group F(X) by the relations R i (X) = e, i = , , ..., k, then the tensor generalized digroup G ⊗ G can be described, in an informal way, as the quotient of the tensor generalized digroup F(X) ⊗ F(X) by the relations R i (X) ⊗ R j (X) = e ⊗ e, i, j = , , ..., k, like in the case of group presentations, see [15] for more details.
Consider the following de nition.

De nition 7.
For the free generalized digroup FD(X), we de ne the ber of FD(X) at the distinguished element y of X as the subset F y (X) = F(X) × {y} × F(X) of FD(X).

Proposition 7. The subset F y (X) is a normal generalized subdigroup of FD(X). Moreover, it is isomorphic to F(X) ⊗ F(X).
Proof. The proof that and are associative binary operations on F y (X) comes from direct calculations. Besides, the bar units of F y (X) must be of the form (b − y − , y, b) or (v, y, y − v − ), i.e. (v, y, b) such that vyb is the empty word, and so they belong to F y (X). We end the proof that F y (X) is a generalized subdigroup noting that for every bar unit (v, y, b), The normality of F y (X) is taken directly from the following fact. For every (s, y, d) ∈ F y (X) and all (u, x, a), (w, z, c) in FD(X), we have that: (u, x, a) (s, y, d) (w, z, c) = (uxas, y, dwzc) ∈ F y (X).
In fact, F y (X) satis es a stronger condition than normality. Now, let φ y : F y (X) → F(X) ⊗ F(X) be the function de ned by φ y ((u, y, a)) = uy ⊗ a.
Let X be a non empty set and let X ⊗ be the set of all tensors of the form x ϵ ⊗ , with x ∈ X and ϵ ∈ { , − }.
Here, means that there is nothing in the corresponding position, so it acts as the identity in the free group F(X).
Let u ⊗ v ∈ F(X) ⊗ F(X), with u = x ϵ i · · · x ϵp ip and v = x δ j · · · x δ k j k . From the de nition of and , The previous calculation motivates the following proposition.
Thus the set X of all elements of D of the form g · · · gp y h · · · h k , where g t , hn and y are in X ± = X ∪ X − l,r , for every t = , , ..., p and n = , , ..., k, is a subdigroup of D.
Proof. In order to simplify the notation we use u y v, with u = g · · · gp and v = h · · · h k to represent the elements of X . Since and de ne binary operations on X . Besides, due to the fact that for every bar unit ξ in D, ξ can be represented as for every x, y ∈ X, the set X contains the halo of D.
Let ξ = y y − r ξ x − l ξ x and let u y v ∈ X , with u = g · · · gp and v = h · · · hp. From Proposition 1, Proof. Let ξ be a bar unit of D and f ∈ Aut(D). Since for all u ∈ D there exists v ∈ D such that u = f (v), we have that thus, f (ξ ) is in the halo of D.
Besides, let v = f (u − r ξ ), then f − (v) = u − r ξ . Therefore, u f − (v) = ξ . So, by applying f on the two sides of the previous equation, we have that f (u) v = f (ξ ).
Since f (ξ ) is also a bar unit of D, then v is a right inverse of f (u) with respect to f (ξ ). In other words, v = (f (u)) − r f (ξ ) .