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© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 32 (2021). Number 1, pp. 103–126
DOI:10.12958/adm1741

The structure of g-digroup actions

and representation theory∗

J. G. Rodríguez-Nieto, O. P. Salazar-Díaz,

and R. Velásquez

Communicated by A. V. Zhuchok

Abstract. The aim of this paper is to propose two possible

ways of defining a g-digroup action and a first approximation to

representation theory of g-digroups.

1. Introduction

Lie’s third theorem asserts the existence of a bijection between local
Lie groups and its finite dimensional Lie algebra in the following way: the
tangent space at the identity of any Lie group is a Lie algebra, and also,
that to any finite dimensional Lie algebra over the real or the complex
numbers, corresponds the tangent space of a connected Lie group unique
up to finite coverings. On the other hand, we have the so called Leibniz
algebra introduced by A. M. Bloh in [1] and later rediscovered by Loday in
1993 in [9]. A Leibniz algebra is a non associative K-algebra M endowed
with a bracket product [ · , · ] that satisfies the Leibniz identity,

[x, [y, z]] = [[x, y] , z] + [y, [x, z]] ,

∗The first and the second author were supported by the Hermes project: Algunos
aspectos algebraicos y geométricos de los digrupos generalizados, code 45519. The
third author was supported by the CIEN research project: Sobre el segundo grupo de
cohomología en superalgebras de Jordan, code 2019-26870.

2020 MSC: 20B10, 20B30, 20C99, 20K99, 20M10, 20M18, 20M30, 20N99.
Key words and phrases: digroups, groups, semigroups, Abelian digroups, sym-

metric digroups, actions and representations.

https://doi.org/10.12958/adm1741


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104 The structure of g-digroup actions

for every x, y, z ∈ M . When the bracket product is skew-symmetric
the Leibniz identity becomes the Jacobi identity, therefore M adopts a
Lie algebra structure. Conversely any Lie algebra is obviously a Leibniz
algebra.

Following the idea of a possible extension of Lie’s third theorem for
Leibniz algebras, J. L. Loday proposed the so called Coquecigrue problem,
which consists in finding an appropriate structure that generalizes the
concept of Lie group and whose algebra is the corresponding Leibniz
algebra. A crucial aspect to consider for a possible solution to the previous
problem is to determine the correct generalization of the concept of group.
The hypothetical structure for the Leibniz case has been called Leibniz
group, Loday group, or Coquecigrue.

Further results on the Coquecigrue problem and the third Lie Theorem
are given by Monterde and Ongay [11]. The first approximation to the
solution of the Coquecigrue problem was proposed independently by M.
Kinyon [7], R. Felipe [4], and K. Liu [8], which is a generalization of
the group structure with two products and it has been called a digroup.
Digroups form an important variety of algebras arising from dimonoids
introduced by Loday [9]. We refer the reader to the literature [18], [20], [19]
and [21] for some results on dimonoids.

Besides, in [17] O. Salazar-Díaz, R. Velásquez and L. A. Wills-Toro
studied a slightly different structure, allowing non-bilateral inverses, called
generalized digroup, that from now on we call g-digroup.

Solutions to the Coquecigrue problem were given by S. Coves [3] (local),
J. Mostovoy (categorical) [12], and M. Bordemann and F. Wagemann
(Augmented Leibniz algebras) [2].

Due to the nature of the extension of groups to g-digroups, one might
think that many definitions and results on group theory can be directly
extended to g-digroups, see [14] and [15] in which we extend the concepts
of free groups and order and we introduce the concept of tensor product,
however, unexpected results could decline the balance, for example, in [15]
we proved that Lagrange’s theorem is not always true for g-digroups, so
we have proposed some variants of Sylows’s theorems. Following this line
of extending results, in the current paper we introduce, the concept of
classical action of g-digroups from two points of view, one is by considering
a connection with the representation theory of g-digroups stated in [14],
the other one is by extending the definition of action for digroups given
in [5]. The latter way is not natural and the orbits are not well defined,
therefore a theorem like Burnside’s formula is not achieved yet.



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Rodríguez-Nieto, Salazar-Díaz, Velásquez 105

This paper is organized as follows. In section 2, we recall the concepts
of g-digroups, g-subdigroups, free g-digroups and the symmetric g-digroup
following papers [17], [14] and [15]. In section 3 we introduce the concept
of classical action of g-digroups which is a natural extension of the one
given in group theory, we prove Burnside’s formula for g-digroup actions
and a version of Cayley’s theorem, weaker than the one given in [15],
orbit spaces are also introduced. We end the section with an extension
of the notion of digroup action, introduced by H. Guzman and F. Ongay
in [5], to the generalized case, and we use this to get a first approach to
representation theory for g-digroups. Such definition of action does not
give a construction of orbits like in the group theory case.

2. g-digroups

In this section we give a short review of some definitions and results
about g-digroups, for a deeper study see [14] and [17].

Definition 1. A set D is called a g-digroup (generalized digroup) if it
has two binary operations ⊢ and ⊣ over D, which are associative (each
separately), and satisfy the conditions:

1) x ⊢ (y ⊣ z) = (x ⊢ y) ⊣ z
2) x ⊣ (y ⊣ z) = x ⊣ (y ⊢ z),

(x ⊢ y) ⊢ z = (x ⊣ y) ⊢ z
3) There exists (at least) an element e in D, such that for all x in D,

x ⊣ e = x = e ⊢ x.
The elements that satisfy this condition are called bar-units and the
set of bar-units in D, denoted by E (or ED), is called the halo of D.

4) For the fixed bar-unit e, we have that for each x in D there exist
x−1
re and x−1

le
in D (the right-inverse of x and the left-inverse of x,

respectively) such that x ⊢ x−1
re = e and x−1

le
⊣ x = e.

Let ξ ∈ E be a bar-unit. We define the sets of left and right inverses,
denoted by Gξ

l and Gξ
r, respectively, as follows

Gξ
l = {x−1

lξ
| x ∈ D} and Gξ

r = {x−1
rξ

| x ∈ D}. (1)

It is not hard to prove that (Gξ
l ,⊣) and (Gξ

r,⊢) are isomorphic groups
with identity ξ, see [17, Theorem 2]. A g-digroup is called trivial if it
consists of only bar units.

The following proposition summarizes some basic and important prop-
erties of the binary operations ⊣ and ⊢. Its proof can be found in [17].



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106 The structure of g-digroup actions

Proposition 1 ([17]). Let D be a g-digroup and a fixed bar unit ξ. Then,
for all x, y in D,

(1) Given x ∈ D and ξ ∈ E, we have that

(x−1
lξ

)−1
lξ

= (x−1
rξ

)−1
lξ

= ξ ⊣ x

(x−1
rξ

)−1
rξ

= (x−1
lξ

)−1
rξ

= x ⊢ ξ.

(2) For all x, y ∈ D and for every ξ, η ∈ E, we have that y ⊣ x−1
rξ

= y ⊣

x−1
lη

and x−1
lξ

⊢ y = x−1
rη ⊢ y.

(3) The inverse of the products are (x ⊢ y)−1
lξ

= (x ⊣ y)−1
lξ

= y−1
lξ

⊣ x−1
le

and (x ⊢ y)−1
rξ

= (x ⊣ y)−1
rξ

= y−1
rξ

⊢ x−1
re .

Since the proof of the following theorem comes from the results given
in [17], we omit it.

Theorem 1. Let (D,⊢,⊣) be a g-digroup. For any ξ, ζ ∈ E

a) Gξ
l = ξ ⊣ Gζ

l and Gξ
r = Gζ

r ⊢ ξ,

b) Gξ
l
∼= Gζ

l
∼= Gξ

r
∼= Gζ

r,

c) ξ ⊣ D = Gξ
l and D ⊢ ξ = Gξ

r, which implies Gξ
l ⊣ D = Gξ

l and

D ⊢ Gξ
r = Gξ

r.
d) E is a Gξ

l -set respect to the action defined by (a, ζ) 7→ a •l ζ := a ⊢

ζ ⊣ a−1, for all a ∈ Gξ
l and ζ ∈ E.

As it is shown in [17], D can be characterized as

D =

•⋃

ξ∈E

Gξ
l =

•⋃

ξ∈E

Gξ
r

Let’s recall that if D and D′ are g-digroups, a map φ : D → D′ is a
g-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 g-digroup isomorphism.

The upcoming statement not only describes a way to construct g-
digroups but also motivates a second characterization of g-digroups, that
is an extension of the results of M. Kinyon (see [7]) and F. Ongay (see [13]).
Its proof can be found in [17].



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Rodríguez-Nieto, Salazar-Díaz, Velásquez 107

Theorem 2. Let D be a g-digroup, let E be the set of bar units and let
Gξ

l be the set of left inverses respect to some ξ ∈ E. Then Gξ
l × E is a

g-digroup, with operations

(a, α) ⊢ (b, β) := (a ⊣ b, a •l β) and (a, α) ⊣ (b, β) := (a ⊣ b, α), (2)

isomorphic to D respect to the isomorphism ϕl : D −→ Gξ
l × E, defined

by ϕl(x) = (ξ ⊣ x, x ⊣ x−1
lξ

), with inverse ϕ−1
l : Gξ

l × E −→ D, given by

(a, α) 7→ α ⊣ a.

Theorem 2 and formulas (2) establish a bijective correspondence be-
tween g-digroups and G-sets. This correspondence is easily understood
because of the fact that if E is a G-set, then G×E becomes the g-digroup
(G × E,⊢,⊣), with a little modification of the binary operations given
in (2) as follows. For any (w, ξ) and (u, η) in G× E,

(w, ξ) ⊢ (u, η) = (wu,w • η) and (w, ξ) ⊣ (u, η) = (wu, ξ). (3)

Conversely, any g-digroup D can be uniquely splitted, up to isomorphisms,
as a cartesian product Gξ

l × ED in such a way that its halo is the set
{ξ} × ED. Respect to this decomposition of g-digroups, next theorem
describes how the respective g-digroup homomorphisms are affected under
such kind of factorization, its proof is done in [17].

Theorem 3. Let Ψ : D −→ D′ be a g-digroup homomorphism. Then,

there exists an unique homomorphism Ψ′ : Gξ
l ×E −→ Gξ′

l ×E′ such that
the diagram

D
Ψ

−−−−→ D′

ϕl

y
yϕ′

l

Gξ
l × E −−−−→

Ψ′

Gξ′

l × E′

commutes, where Ψ′ ≡ (ϕ, µ), with

a) the map ϕ : Gξ
l −→ Gξ′

l , where ϕ(a) = ξ′ ⊣ Ψ(a), is a group
homomorphism.

b) the map µ : E −→ E′, defined as µ(α) = Ψ(α) is an equivariant
map, i.e.

µ(x • α) = Ψ(x) • µ(α) and µ(a • α) = ϕ(a) • µ(α),

for all α ∈ E, all x ∈ D and all a ∈ Gξ
l .



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108 The structure of g-digroup actions

In this way, g-digroup homomorphisms Ψ : G×E → G′ ×E′ between
g-digroups can be described as Ψ = (ϕ, λ), where ϕ : G→ G′ is a group
homomorphism and λ : E → E′ is an equivariant function.

Theorem 3 extends the correspondence between g-digroups and G-sets
to a bijective correspondence between categories.

Definition 2. A subset H of a g-digroup (D;⊢,⊣) is said to be a g-
subdigroup of D, denoted by H 6 D, if H with the restricted operations
⊢|H and ⊣|H to H, is itself a g-digroup.

It is not hard to prove that the set of bar units of a g-subdigroup H of
a g-digroup D is EH = E∩H , where E is the set of bar units of D, and the

groups of left and right inverses for any ξ ∈ EH , are Γξ
l =

(
Gξ

l ∩H
)
6 Gξ

l

and Γξ
r =

(
Gξ

r ∩H
)
6 Gξ

r, respectively. Moreover, H 6 D iff there exist

Γ 6 Gξ
l and an invariant Γ-set ∆ ⊂ E (Γ •∆ = ∆), such that H ∼= Γ×∆

(see [17, Lemma 6] for more details).
The following definition was given in [14].

Definition 3. Let X be a subset of a g-digroup (D,⊢,⊣). For X− we
mean the set of all inverses, right and left, with respect to all bar units in
D of all elements in X. In other words, if E denotes the halo of D,

X− = X−
l ∪X−

r ,

where X−
l :=

⋃
e∈E{x

−1
le

| x ∈ X} and X−
r :=

⋃
e∈E{x

−1
re | x ∈ X}.

The g-subdigroup of D generated by X, denoted by
〈
X
〉
, is the set of

all elements of D of the form

(
g1 ⊢ · · · ⊢ gp

)
⊢ y ⊣

(
h1 ⊣ · · · ⊣ hk

)
, (4)

where gt, hn and y are in X± = X ∪ X−, for every t = 1, 2, . . . , p and
n = 1, 2, . . . , k.

As in [10], we denote the word (4) in
〈
X
〉

by g1 · · · gp y̆ h1 · · ·hk. In
this way

g1 ⊢ · · · ⊢ gk = g1 · · · gk−1 ğk and g1 ⊣ · · · ⊣ gk = ğ1 · · · gk−1gk.

It is not hard to prove that

g1 · · · gp y̆ h1 · · ·hk ⊢ u1 · · · gt x̆ v1 · · · vs

= g1 · · · gp y h1 · · ·hku1 · · · gt x̆ v1 · · · vs



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Rodríguez-Nieto, Salazar-Díaz, Velásquez 109

and
g1 · · · gpy̆h1 · · ·hk ⊣ u1 · · · gtx̆v1 · · · vs

= g1 · · · gpy̆h1 · · ·hku1 · · · gtxv1 · · · vs.

Let X be a non empty set and F (X) the free group generated by X,
this is, the set of all words in X±. See [6] for details on this definition.
Let FD(X) = F (X)×X ×F (X) and, as above, consider ux̌a := (u, x, a),
where u, a ∈ F (X) and x ∈ X, thereby the set FD(X) can be denoted by
F (X)X̌F (X).

The binary maps ⊣,⊢ given in [14] are now rewritten as follows:

ux̌a ⊣ vy̌b = ux̌avyb

ux̌a ⊢ vy̌b = uxavy̌b

for all u, v, a, b ∈ F (X) and x, y ∈ X. It is not hard to prove that the set
FD(X) together with the binary operations ⊣,⊢ is a g-digroup. The set
of bar units of FD(X) is given by

E(X) = {vy̌b | vyb = e}

and its inverses are of the form

(ux̌a)−1
l(vy̌b)

= vy̌ba−1x−1u−1 = vy̌b(uxa)−1, (5)

(ux̌a)−1
r(vy̌b)

= a−1x−1u−1vy̌b = (uxa)−1vy̌b, (6)

where e is the unit of the free group F (X) and w−1 is the inverse of w
in F (X).

Theorem 4. ([14]) Let X be a countable set. Then FD(X) is a free
g-digroup, with i : X →֒ FD(X) given by i(x) = x̌.

Proof. Let f : X → D be a set function from X into a g-digroup D.
For a fixed bar unit ξ in D, consider the function fξ : X → Gξ

l , where
fξ(x) = ξ ⊣ f(x). The well definition of this function comes directly from

Theorem 1. Thus, there exists a group homomorphism ϕfξ : F (X) → Gξ
l ,

such that ϕfξ(x) = ξ ⊣ f(x) and ϕfξ(e) = ξ. It is clear that, for every

reduced word xδ1i1 · · ·x
δn
in

in F (X)

ϕfξ(x
δ1
i1
· · ·xδnin ) = ξ ⊣ f(xi1)

δ1 ⊣ · · · ⊣ f(xin)
δn , (7)

where

f(xij )
δj =





f(xij ) ⊣ · · · ⊣ f(xij ) (δj-times) if δj > 0

f(xij )
−1
lξ

⊣ · · · ⊣ f(xij )
−1
lξ

(−δj-times) if δj < 0
.



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110 The structure of g-digroup actions

Now, we can define the function ϕξ : FD(X) → D as follows

ϕξ(ux̌v) = ϕfξ(u) ⊢ f(x) ⊣ ϕfξ(v).

Thereby, if u = xδ1i1 · · ·x
δn
in

and v = yβ1
t1

· · · yβm

tm
in F (X), we have that

ϕξ(ux̌v) = f(x1i)
δ1 ⊢ · · · ⊢ f(xin)

δn ⊢ f(x) ⊣ f(yt1)
β1 · · · f(ytm)

βm . (8)

Therefore, from Proposition 1, ϕξ does not depend on the choice of the
bar unit ξ, so we denote ϕξ by ϕ. The proof of the fact that ϕ is a g-
digroup homomorphism is done in [14, Theorem 7]. The proof ends with
the following equality

ϕ(x̌) = ξ ⊢ f(x) ⊣ ξ = f(x).

It can be easily proven that if D is another free g-digroup on the set
X, then D is isomorphic to FD(X). Thus, up to isomorphisms, FD(X) is
unique.

Let us consider the subset X̌ ⊂ FD(X) of all x̌ := ex̌e, x ∈ X. From
the equations (5) and (6), we have that

x̌−1
l
x−1x̌

= x−1x̌x−1 = x̌−1
r
x̌x−1

.

In this way we define, for all x ∈ X and n ∈ Z,

x̌n⊢ =





x̌ ⊢ · · · ⊢ x̌ = xn−1x̌ if n > 0

x̌−1
l
x−1x̌

⊢ · · · ⊢ x̌−1
l
x−1x̌

= xnx̌x−1 if n < 0
. (9)

Similarly, we define x̌n⊣. Thus, the following two equalities arise.

x̌n⊣ ⊢ w = x̌n⊢ ⊢ w and w ⊣ x̌n⊣ = w ⊣ x̌n⊢.

Therefore, for the proof of the following proposition we use x̌n instead x̌n⊢.

In Proposition 2 of [22], one class of generalized digroups is constructed
which, under certain conditions, gives free generalized digroups. Indeed, if
in Proposition 2 of [22] suppose that n = 3 and G = F (X2), where F (X2)
is the free group on X2, we obtain the construction F (X2)×X2 × F (X2)
of the free generalized digroup.

Proposition 2. The free g-digroup FD(X) is generated by the set X̌.



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Rodríguez-Nieto, Salazar-Díaz, Velásquez 111

Proof. Let ux̌v ∈ FD(X). Then, there exist g1, · · · , gp, h1, · · · , hk ∈ X
and ni ∈ Z, i = 1, · · · , p and mj ∈ Z, j = 1, · · · , k, such that

ux̌v = gn1
1 · · · g

np
p x̌hm1

1 · · ·hmk

k .

It is not hard to prove that

ux̌v = ǧ1
n1 ⊢ · · · ⊢ ǧp

np ⊢ x̌ ⊣ ȟ1
m1

⊣ · · · ⊣ ȟk
mk . (10)

As a consequence, FD(X) is generated by X̌.

Let X be a non empty set and let Sym(X) denote the symmetric
group on the set X. Let Aut(FD(X)) be the set of all bijective g-digroup
homomorphisms from FD(X) onto itself.

Theorem 5. The group Sym(X) is isomorphic to a subgroup of
Aut(FD(X)).

Proof. Let f be an element of Sym(X). Then, f̌ : X̌ → FD(X), defined by
f̌(x̌) = y̌, where y = f(x) extends to an unique g-digroup homomorphism
ϕf : FD(X) → FD(X), such that ϕf (x̌) = f̌(x̌). In order to make the
notation easier, we use f̌(x) instead y̌, for the case when y = f(x).

Let ux̌v ∈ FD(X), from the proof of Proposition 2, there are g1, · · · , gp,
h1, · · · , hk ∈ X and ni ∈ Z, i = 1, · · · , p and mj ∈ Z, j = 1, · · · , k, such
that ux̌v = ǧ1

n1 ⊢ · · · ⊢ ǧp
np ⊢ x̌ ⊣ ȟ1

m1
⊣ · · · ⊣ ȟk

mk . Thus, we have
that

ϕf (ux̌v) = ϕf (ǧ1
n1 ⊢ · · · ⊢ ǧp

np ⊢ x̌ ⊣ ȟ1
m1

⊣ · · · ⊣ ȟk
mk)

= ϕf (ǧ1)
n1 ⊢ · · · ⊢ ϕf (ǧp)

np ⊢ ϕf (x̌) ⊣ ϕf (ȟ1)
m1 ⊣ · · · ⊣ ϕf (ȟk)

mk

= f̌(g1)
n1 ⊢ · · · ⊢ f̌(gp)

np ⊢ f̌(x) ⊣ f̌(h1)
m1 ⊣ · · · ⊣ f̌(hk)

mk

= f(g1)
n1 · · · f(gp)

np f̌(x)f(h1)
m1 · · · f(hk)

mk ,
(11)

where

ϕf (ǧi)
ni =





ϕf (ǧi) ⊢ · · · ⊢ ϕf (ǧi) (ni-times) if ni > 0

(ϕf (ǧi))
−1
lξ

⊢ · · · ⊢ (ϕf (ǧi))
−1
lξ

(−ni-times) if ni < 0
.

A similar equality is true for ϕf (ȟj)
mj .

Consider another function λ ∈ Sym(X), then

ϕλ(ux̌v) = λ(g1)
n1 · · ·λ(gp)

np ˇλ(x)λ(h1)
m1 · · ·λ(hk)

mk .



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112 The structure of g-digroup actions

Thereby, ϕf ◦ ϕλ = ϕf◦λ. Moreover, if 1X is the identity function in
Sym(X), then ϕ1X : FD(X) → FD(X) is the identity function. Thus, for
every f ∈ Sym(X), ϕf ∈ Aut(FD(X)). Hence, the following function

Φ : Sym(X) → Aut(FD(X)), where Φ(f) = ϕf (12)

is well defined and it is a group homomorphism. Let f, λ ∈ Sym(X) such
that Φ(f) = Φ(λ), then for every x̌ ∈ X̌, we have that

Φ(f)(x̌) = Φ(λ)(x̌) ⇐⇒ f̌(x) = λ̌(x) ⇐⇒ f(x) = λ(x).

As a consequence, f ≡ λ.

3. g-digroups action and representation theory

In this section we explore the concept of a g-digroup action from two
points of view, one as a natural extension of group actions and the other
by considering a pair of compatible actions. The first one is motivated
by the definition of a g-digroup representation given in [14, Definition 9]
and the second one from the digroup action introduced by H. Guzmán F.
Ongay in [5].

3.1. Classical actions and Burnside’s formula

We recall the definition of a g-digroup representation.

Definition 4. Let (D,⊢,⊣) and (D′,⊢′,⊣′) be two g-digroups and let
Aut(D) be the set of all bijective self homomorphisms of D. A representa-
tion of D′ on D is a function ϕ : D′ → Aut(D), with ϕ(u′) = ϕu′ , such
that, for every u′, v′ in D′ and every w ∈ D,

ϕu′⊢′v′(w) = ϕu′(ϕv′(w)) = (ϕu′ ◦ ϕv′)(w)

and
ϕu′⊣′v′(w) = ϕu′(ϕv′(w)) = (ϕu′ ◦ ϕv′)(w)

It is well known in group theory that every representation induces
a natural group action. We propose a natural extension of such idea to
g-digroups.

Definition 5 (Classical action). Let (D,⊢,⊣) be a g-digroup and let
M be a set. A classical action is a function ⋄ : D × M → M , with
⋄(x,m) = x ⋄m, such that, for every x, y ∈ D and m ∈M ,



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Rodríguez-Nieto, Salazar-Díaz, Velásquez 113

A1) x ⋄ (y ⋄m) = (x ⊢ y) ⋄m,
A2) (x ⊢ y) ⋄m = (x ⊣ y) ⋄m and
A3) for every bar unit e ∈ D, e ⋄m = m.

In this case, M is called a classical D-set or for simplicity, a D-set.

Note that that if ⊢=⊣, then ⋄ is a group action. An immediate conse-
quence of the previous definition is the following proposition

Proposition 3. If D = G×E is a g-digroup and M is a D-set, then M
is a G-set. Conversely, if M is a G-set, then M can be seen as a D-set.

Proof. For (g, δ) ∈ D, let ∗δ : G ×M → M , be the function g ∗δ m :=
∗δ(g,m) = (g, δ) ⋄m, where ⋄ is a classical action. Since

g ∗δ m = (g, δ) ⋄m
= ((e, ξ) ⊢ (g, δ)) ⋄m
= ((e, ξ) ⊣ (g, δ)) ⋄m
= (g, ξ) ⋄m
= g ∗ξ m,

(13)

we have that ∗ξ = ∗δ, thus we use ∗ instead of ∗ξ.

Let g, h ∈ G, e the identity in G and m ∈M , then

g∗(h∗m) = (g, ξ)⋄((h, ξ)⋄m) = ((g, ξ) ⊣ (h, ξ))⋄m = (gh, ξ)⋄m = gh∗m

and

e ∗m = (e, ξ) ⋄m = m.

Therefore, M is a G-set under the action ∗.

Conversely, assume that M is a G-set under the action ∗, then ⋄ :
D ×M →M , with (g, ξ) ⋄m = g ∗m, is a classical action of D over M .
In fact, let (g, ξ) and (h, δ) in D and m ∈M , then

A1) (g, ξ)⋄ ((h, δ)⋄m) = g ∗ (h∗m) = gh∗m = (gh, g •δ)⋄m = ((g, ξ) ⊢
(h, δ)) ⋄m,

A2) ((g, ξ) ⊢ (h, δ)) ⋄m = gh ∗m = (gh, ξ) ⋄m = ((g, ξ) ⊣ (h, δ)) ⋄m
and

A3) for every (e, ξ) bar unit in D, (e, ξ) ⋄m = e ∗m = m.

Hence, M is a D-set under the classical action ⋄.

It is not hard to verify that ⋄ does not depend on the left inverse of
(g, ξ) chosen. In fact, it does not change if instead of (g, ξ)−1

l(e,ω)
we take

(g, ξ)−1
r(e,ω)

, see [17, Corollary 2].



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114 The structure of g-digroup actions

We know that the symmetric group Sym(D) is a trivial g-digroup in
the sense that ⊣=⊢ and σ ⊢ λ = σ◦λ. Consider the map ϕ : D → Sym(D),
defined by ϕ(x) =: ϕx, where ϕx(y) = x ⋄ y. This map is well defined.
Indeed, let x = (g, ξ) and (h, η), (h′, η′) ∈ D, since

ϕ(g,ξ)((h, η)) = ϕ(g,ξ)((h
′, η′)) ⇔ (ghg−1, g • η) = (gh′g−1, g • η′

⇔ (h, η) = (h′, η′),

then ϕx is a one to one function. Furthermore, for every (h, η) ∈ D,

ϕ(h,ξ)((h, g
−1 • η)) = (h, η),

so ϕ(g,ξ) is a surjective function. Thus, ϕ is well defined.

Proposition 4. The function ϕ : D → Sym(D) is a g-digroup homomor-
phism. Moreover, ϕ is injective if and only if the center Z(G) is trivial
and E has one element.

Proof. The first part is true because of the fact that ⋄ is a classical action
of D into itself. The rest of the proof follows from the equivalence

(g, ξ) ⋄ (h, η) = (g̃, ξ̃) ⋄ (h, η) ⇔ (ghg−1, g • η) = (g̃hg̃−1, g̃ • η),

for every g, g̃, h ∈ G, ξ, ξ̃, η ∈ E. Consequently, g̃−1g ∈ Z(G) and ξ = ξ̃,
for every ξ, ξ̃ ∈ E.

Consider the function ⋄′ : D×D → D, where x ⋄′ y = x ⊢ y is the left
translation. Then, ⋄′ is a classical action. In fact,

A1) x ⋄′ (y ⋄′ z) = x ⊢ (y ⊢ z) = (x ⊢ y) ⋄′ z,
A2) (x ⊣ y) ⋄′ z = (x ⊣ y) ⊢ z = (x ⊢ y) ⋄′ z and
A3) e ⋄′ z = e ⊢ z = z.

Again, let ϕx : D → D, with ϕx(y) = x ⋄′ y. Then, if x = (g, ξ) and
y = (h, η),

ϕ(g,ξ)(h, η) = (gh, g • η).

It is straightforward to prove that ϕx ∈ Sym(D) and so ϕ : D →
Sym(D), where ϕ(x) = ϕx is well defined. Besides, ϕ is a g-digroup
homomorphism, indeed,

ϕx⊢y(z) = (x ⊢ y) ⋄′ z = (x ⊢ y) ⊢ z = x ⊢ (y ⊢ z)

= x ⋄′ (y ⋄′ z) = (ϕx ◦ ϕy)(z)



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Rodríguez-Nieto, Salazar-Díaz, Velásquez 115

and

ϕx⊣y(z) = (x ⊣ y) ⋄′ z = (x ⊣ y) ⊢ z = x ⊢ (y ⊢ z)

= x ⋄′ (y ⋄′ z) = (ϕx ◦ ϕy)(z),

for every x, y, z ∈ D. However, since ϕ(g,ξ) = ϕ(g,η), for every ξ, η ∈ E, ϕ
is not a generalized isomorphism, unless E has one element.

The previous constructions motive the following definition and theo-
rem.

Definition 6. A regular representation of a g-digroup D is a digroup
homomorphism from D into a permutation group.

Theorem 6 (Cayley theorem). There exists a faithful regular representa-
tion of a g-digroup D if and only if D is a group.

It is not to hard to see, doing a similar proof as the one of the previous
theorem, that there is not a generalized isomorphism from a g-digroup D
onto a group G unless at least D is itself a group.

Theorem 7. If M is a D-set under the action ⋄, then

γ : D → Aut(M), with γ(x)(m) = γx(m) = x ⋄m,m ∈M,

extends to an unique representation Ψ of D on D′, where D′ = FD(M) is
the free g-digroup on M . Moreover, if ψ : D → Aut(D′) is a representation,
then D′ can be seen as a D-set in the classical sense of group theory.

Proof. Suppose that M is a D-set under an action ⋄ : D × M → M .
Then, for every x ∈ D, γx :M →M , where γx(m) = x ⋄m, is a bijective
function. From Theorem 5, Φ(γx) ∈ Aut(FD(M)). Thus, we can define
the composite function

Ψ := Φ ◦ γ : D → Aut(FD(M)), with Ψ(x) = ϕγx

Let wm̌u ∈ FD(M), with w = m
ei1
i1
m

ei2
i2

· · ·m
eit
it

and u =

m
ej1
j1
m

ej2
j2

· · ·m
ejs
js

, free words in FD(M). Then, from (11),

Ψ(x)(wm̌u) = ϕγx(wm̌u)

= γx(mi1)
ei1 · · · γx(mit)

eit ˇγx(m)γx(mj1)
ej1 · · · γx(mjs)

ejs .
(14)



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116 The structure of g-digroup actions

In addition,

γx(mij )
eij = (x ⋄mij )

eij =: x ⋄m
eij
ij

and γx(mji)
eji

= (x ⋄mji)
eji =: x ⋄m

eji
ji
.

(15)

On the other hand, for every x ∈ D and every w ∈ F (X), we define

γx(w) := γx(mi1)
ei1γx(mi2)

ei2 · · · γx(mit)
eit . (16)

Equations (15) and (16) summarize the natural extension of ⋄ to
a classical action of D over the free group F (M). Thus, we have the
reformulation of Equation (14) as follows

Ψ(x)(wm̌u) = γx(w) ˇγx(m)γx(u).

From (14)-(16), we have that

Ψ(x)(wm̌u ⊢ ŵňû) = Ψ(x)((wmuŵňû))

= (γx(w)γx(m)γx(u)γx(ŵ) ˇγx(n)γx(û))

= (γx(w) ˇγx(m)γx(u)) ⊢ (γx(ŵ) ˇγx(n)γx(û))

= Ψ(x)(wm̌u) ⊢ Ψ(x)(ŵňû).

and

Ψ(x)(wm̌u) ⊣ ŵňû) = Ψ(x)(wm̌uŵnû)

= γx(w) ˇϕx(m)γx(u)γx(ŵ)γx(n)γx(û)

= (γx(w) ˇϕx(m)γx(u)) ⊣ (γx(ŵ) ˇϕx(n)γx(û))

= Ψ(x)(wm̌u) ⊣ Ψ(x)(ŵňû).

Moreover, since the words γx(w) and γx(ŵ), forw = m
ei1
i1
m

ei2
i2

· · ·m
eit
it

∈

F (M) and ŵ = m
el1
l1
m

el2
l2

· · ·m
elr
lr

∈ F (M), are also free words in F (M),
then the assumption γx(w) = γx(ŵ), implies t = s and, up to rearrange-
ment of the respective letters, γx(miv) = γx(mlv), so miv = mlv , for every
v = 1, · · · , t. Thereby, Ψ(x) ∈ Aut(FD(M)).

Let x, y ∈ D, from (15) and (16)

γx⊢y(w) = ((x ⊢ y) ⋄m
ei1
i1

)((x ⊢ y) ⋄m
ei2
i2

) · · · ((x ⊢ y) ⋄m
eit
it

)

= ((x ⋄ (y ⋄m
ei1
i1

))((x ⋄ (y ⋄m
ei2
i2

)) · · · ((x ⋄ (y ⋄m
eit
it

))

= γx((y ⋄m
ei1
i1

)(y ⋄m
ei2
i2

) · · · (y ⋄m
eit
it

))

= γx(γy(m
ei1
i1

· · ·m
eit
it

))

= γx(γy(w).



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Rodríguez-Nieto, Salazar-Díaz, Velásquez 117

Therefore,

Ψ(x ⊢ y)(wm̌u) = (γx⊢y(w) ˇγx⊢y(m)γx⊢y(u))
= (γx(γy(w)), γx(ϕy(m)), γx(γy(u)))

= Ψ(x)(γy(w) ˇγy(m)γy(u))
= Ψ(x)(Ψ(y)(wm̌u)),

thus, Ψ : D → Aut(FD(M)), is a representation. Let Ψ̂ another represen-
tation of D on FD(M), such that Ψ̂(x)(wǔv) = (γx(w) ˇϕx(u)γx(v)), for
every w, u, v ∈M . Then Ψ̂ ≡ Ψ.

The reciprocal comes directly from the fact that x ⋄m := Ψ(x)(m) is
a classical action of D over FD(M).

Now, we introduce the notions of orbits and stabilizers, natural concepts
associated to actions.

Definition 7. Let M be a D-set under the action ⋄. The orbit of m ∈M
is the set

OD
m = {x ⋄m | x ∈ D}.

The set of all elements in D that leave m fixed is called the stabilizer of
m and denoted by StabD(m).

We have the following theorem.

Theorem 8. With the above notation. Let D = G×E be a g-digroup and
M be a D-set. Then,

(a) the family F = {OD
m | m ∈M} is a partition of M ,

(b) StabD(m) is a g-subdigroup of D,
(c) OD

m = OG
m, where OG

m is the orbit of m under the action ∗ and
(d) StabD(m) = StabG(m)× E, where StabG(m) is the stabilizer sub-

group of m under the action ∗.

Proof. (a) Let k ∈ OD
m ∩OD

n , then there exist x, y ∈ D, such that x ⋄m =
y ⋄ n. Thus, m = (x−1

le
⊣ y) ⋄m ∈ OD

n . In a similar way we prove that

n ∈ OD
m. Therefore, OD

m = OD
n . The equality M = ∪n∈MOD

m comes from
the definition of orbits.

(b) Let x, y ∈ StabD(m), then (x ⊢ y) ⋄m = x ⋄ (y ⋄m) = m and
(x ⊣ y) ⋄ m = x ⋄ (y ⋄ m) = m, so StabD(m) is closed under ⊢ and ⊣,
moreover, StabD(m) contains the halo of D. If x−1

le
and x−1

re are the left
and right inverses of x ∈ StabD(m) respect to the bar unit e, then

x−1
le

⋄m = x−1
le

⋄ (x ⋄m) = (x−1
le

⊣ x) ⋄m = e ⋄m = m



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118 The structure of g-digroup actions

and
x−1
re ⋄m = x−1

re ⋄ (x ⋄m)
= (x−1

re ⊢ x) ⋄m

= ((x−1
le

⊢ e) ⊢ x) ⋄m

= ((x−1
le

⊣ e) ⊢ x) ⋄m

= (x−1
le

⊢ x) ⋄m

= (x−1
le

⊣ x) ⋄m = m.

As a consequence StabD(m) is a g-subdigroup of D.
(c)

OD
m = {(g, ξ) ⋄m | (g, ξ) ∈ F}

=Eq. (13) {(g, ξ) ⋄m | g ∈ G}
= {g ∗m | g ∈ G} = OG

m.

(d)

StabD(m) = {(g, ξ) ∈ D | (g, ξ) ⋄m = m}

=Eq. (13) {g ∈ G | (g, ξ) ⋄m = m} × E
= {g ∈ G | g ∗m = m} × E
= StabG(m)× E.

Let D = G × E be a finite g-digroup and M be a D-set. We define
the index of StabG(m) in D as the rational number [D : StabD(m)] =

|D|
|StabD(m)| . Thus, from the previous theorem, [D : StabD(m)] = [G :

StabG(m)], therefore, we have the following equation

[D : StabD(m)] =
∣∣OD

m

∣∣ . (17)

The proof of the following lemma is a direct consequence of Equation
(13), then we omit it.

Lemma 1. Let M(g,ξ) be the set of all m ∈ M fixed by (g, ξ). Then
M(g,ξ) =M(g,η), for every ξ, η ∈ E and g ∈ G. Thus, M(g,ξ) =Mg, where
Mg is the set of all m ∈M fixed by g under the action ∗.

We end this section with the following version of Burnside’s formula.
As we note, its proof is the same as the one for groups.

Theorem 9 (Burnside’s formula for g-digroup actions). Let D = G× E
be a finite g-digroup and let M be a D-set. If r is the number of orbits in
M under D, then

r |D| =
∑

(g,ξ)∈D

∣∣M(g,ξ)

∣∣ .



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Rodríguez-Nieto, Salazar-Díaz, Velásquez 119

Proof. Let N be the number of pairs ((g, ξ),m) such that (g, ξ) ⋄m = m,
then

N =
∑

(g,ξ)∈D

∣∣M(g,ξ)

∣∣ . (18)

On the other hand, for m ∈M , there are | StabD(m)| pairs ((g, ξ),m)
such that (g, ξ) ⋄m = m, then

N =
∑

m∈M |StabD(m)| .

From Equation (17),

∑
m∈M |StabD(m)| =

∑
m∈M |D| /

∣∣OD
m

∣∣
= |D|

∑
m∈M 1/

∣∣OD
m

∣∣ .
If O ⊂M is an orbit, then

∑
m∈O 1/

∣∣OD
m

∣∣ = 1, thus

N = |D| r. (19)

Hence, from Equations (18) and (19), we have that

r |D| =
∑

(g,ξ)∈D

∣∣M(g,ξ)

∣∣ .

A more direct proof of Burnside’s formula can be gotten by applying,
directly, Burnside’s formula for groups.

3.2. g-digroup actions

In this section we explore, among other things, an extension of digroup
actions proposed by Guzmán and Ongay in [5] to g-digroup actions.

Definition 8 (d-equivariant). Let D = G× E be a g-digroup and M be
a G-set with action ∗. A function ϕ : E ×M →M is called d-equivariant
if for every g ∈ G, ξ and δ in E and m ∈M ,

(a) ϕ(g · ξ, g ∗m) = g ∗ ϕ(ξ,m) (equivariant), and
(b) ϕ(ξ, ϕ(δ,m)) = ϕ(ξ,m) (idempotent).

The following definition is equivalent to the one given in [5, Defini-
tion 3].

Definition 9 (g-digroup action). Let D be a g-digroup. A set M is said
to be a D-set if there exist two functions

⊳, ⊲ : D ×M →M, (20)

such that, for all x, y ∈ D and m ∈M ,



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120 The structure of g-digroup actions

1) x ⊲ (y ⊲ m) = (x ⊢ y) ⊲ m,
2) x ⊳ (y ⊳ m) = (x ⊣ y) ⊳ m,
3) there exists a bar unit e ∈ D, such that e ⊲ m = m,
4) x ⊲ (y ⊳ m) = (x ⊢ y) ⊳ m and
5) x ⊳ (y ⊲ m) = (x ⊣ y) ⊳ m.

Any couple of functions ⊲ and ⊳ that satisfies 1)-5) is called left-action or,
for simplicity, action of D over M .

Since, for every x, y ∈ D and every m ∈M ,

(x ⊣ y) ⊲ m
3)
= (x ⊣ y) ⊲ (e ⊲ m)
1)
= ((x ⊣ y) ⊢ e) ⊲ m
= ((x ⊢ y) ⊢ e) ⊲ m
1)
= (x ⊢ y) ⊲ (e ⊲ m)
3)
= (x ⊢ y) ⊲ m,

(21)

condition 3) is equivalent to
3′) For every bar unit ê ∈ D and every m ∈M , ê ⊲ m = m.

This is because, if we assume 3) and ê is a bar-unit in D, then

ê ⊲ m = (ê ⊣ e) ⊲ m
(21)
= (ê ⊢ e) ⊲ m = e ⊲ m = m.

The left side in conditions 1), 2), 4) and 5) corresponds to the four
choices of couples (⊲, ⊲), (⊳, ⊳), (⊲, ⊳) and (⊳, ⊲). The right side of these
conditions, at least for the cases 1) and 2) is very natural, but for the
cases 4) and 5) we might think that we can consider other possibilities.
Some of these possibilities imply that ⊳ = ⊲, i. e., the g-digroup action
becomes a classical action.

An example of the previous analysis is summarized in the following
proposition.

Proposition 5. With the above notation.
(a) If in the action definition, we change condition 4) by

x ⊲ (y ⊳ m) = (x ⊢ y) ⊲ m.

Then, ⊳ = ⊲.
(b) Moreover, if in the action definition, we change condition 5) by

x ⊳ (y ⊲ m) = (x ⊣ y) ⊲ m.

Then, ⊳ = ⊲.



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Rodríguez-Nieto, Salazar-Díaz, Velásquez 121

Proof. (a) Let x, y ∈ D and m ∈M such that x ⊲ (y ⊳ m) = (x ⊢ y) ⊲ m,
then

x ⊲ m = (e ⊢ x) ⊲ m
= e ⊲ (x ⊳ m)
= x ⊳ m.

So, we have that ⊳ = ⊲.

(b) Let x, y ∈ D and m ∈M such that x⊳ (y ⊲m) = (x ⊣ y)⊲m. Thus,

x ⊲ m = (x ⊣ e) ⊲ m
= x ⊳ (e ⊲ m)
= x ⊳ m.

Hence, we have that ⊳ = ⊲.

The following theorem shows a connection between g-digroup actions
and group actions.

Theorem 10. Let D = G× E be a g-digroup. Then, every G-set M is a
D-set under the actions

⊳ : D ×M →M and ⊲ : D ×M →M, (22)

defined as follows: for every (g, ξ) ∈ D and m ∈M ,

⊳((g, ξ),m) = ⊲((g, ξ),m) = g ∗m, (23)

where ∗ is the action of G defined over M .

Conversely, if M is a D-set, with actions ⊳ and ⊲, then D is a G-set
under the action

∗ξ : G×M →M, where ∗ξ (g,m) = (g, ξ) ⊲ m, (24)

where ξ ∈ E.

Proof. For the first part we have to prove that the functions defined by
equation (23) satisfy the action conditions. Let (g, ξ) and (h, δ) ∈ D and
let m ∈M . Since ⊳ = ⊲ we only have to verify 1), 3) and 5).

1) We have

(g, ξ) ⊲ ((h, δ) ⊲ m) = (g, ξ) ⊲ (h ∗m)
= g ∗ (h ∗m) = (gh) ∗m = (gh, g(δ)) ⊲ m
= ((g, ξ) ⊢ (h, δ)) ⊲ m.



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122 The structure of g-digroup actions

3) The bar units of D are of the form (e, ξ), where e is the identity of
G. So, we have that (e, ξ) ⊲ m = e ∗m = m.

5) We have

(g, ξ) ⊲ ((h, δ) ⊲ m) = (g, ξ) ⊲ (h ∗m)
= g ∗ (h ∗m) = (gh) ∗m = (gh, ξ) ⊲ m
= ((g, ξ) ⊣ (h, δ)) ⊲ m.

Conversely, since ⊲ is a function, so is ∗ξ for any ξ ∈ E. Let g, h ∈ G and
e be the identity in G, then

g ∗ξ (h ∗ξ m) = (g, ξ) ⊲ ((h, ξ) ⊲ m)
= ((g, ξ) ⊢ (h, ξ)) ⊲ m
= ((g, ξ) ⊢ (h, ξ)) ⊲ ((e, ξ) ⊲ m)
= (((g, ξ) ⊢ (h, ξ)) ⊢ (e, ξ)) ⊲ m
= (((g, ξ) ⊣ (h, ξ)) ⊢ (e, ξ)) ⊲ m
= ((g, ξ) ⊣ (h, ξ)) ⊲ ((e, ξ) ⊲ m)
= (gh, ξ) ⊲ m
= (gh) ∗ξ m.

Besides, e ∗ξ m = (e, ξ) ⊲ m = m. Thus, ∗ξ is an action of G over M .

We have the following characterization which is the same to the one
given in [5], up to the definition of d-equivariance.

Theorem 11 (A characterization of a g-digroup action). Let D = G×E be
a g-digroup and M be a G-set under the action ∗. Then, M can be endowed
with a D-set structure in which ∗ = ∗ξ is the action defined by Equation
(24) if and only if there exists a d-equivariant function ε : E ×M →M .

Proof. Suppose that M is a D-set under the actions (⊳, ⊲). We define the
function

ε : E ×M →M,

where ε(ξ,m) = (e, ξ) ⊳ m. Indeed,

ε(h · ξ, h ∗m) = (e, h · ξ) ⊳ h ∗m
= (e, h · ξ) ⊳ ((h, ξ) ⊲ m))
5)
= ((e, h · ξ) ⊣ (h, ξ)) ⊳ m
= (h, h · ξ) ⊳ m
= ((h, ξ) ⊢ (e, ξ)) ⊳ m
4)
= (h, ξ) ⊲ ((e, ξ) ⊳ m)
= (h, ξ) ⊲ ε(ξ,m)
= h ∗ ε(ξ,m).



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Rodríguez-Nieto, Salazar-Díaz, Velásquez 123

Besides, ε(ξ, ε(η,m)) = (e, ξ) ⊳ ((e, η) ⊳ m) = (e, ξ) ⊳ m = ε(ξ,m). Then,
we have proven that ε is a d- equivariant function.

Conversely, suppose that ε : E ×M →M is d-equivariant. We define
(g, ξ) ⊲ m = g ∗m and (g, ξ) ⊳ m = ε(ξ, g ∗m). Thus, we have to verify if
they satisfy the g-digroup action conditions.

1) For every (g, ξ), (h, δ) ∈ D and m ∈M ,

(g, ξ) ⊲ ((h, δ) ⊲ m) = g ∗ (h ∗m)
= (gh) ∗m
= (gh, g · δ) ⊲ m
= ((g, ξ) ⊢ (h, δ)) ⊲ m.

2) Let (g, ξ), (h, δ) ∈ D and m ∈M , then

(g, ξ) ⊳ ((h, δ) ⊳ m) = (g, ξ) ⊳ (ε(δ, h ∗m))
= ε(ξ, g ∗ ε(δ, h ∗m))
= ε(ξ, ε(g · δ, g ∗ (h ∗m)))
= ε(ξ, ε(g · δ, (gh) ∗m)))
= ε(ξ, (gh) ∗m)
= (gh, ξ) ⊳ m
= ((g, ξ) ⊣ (h, δ)) ⊳ m.

3) The bar units of D are of the form (e, ξ), where e is the identity of
G. So, we have that, (e, ξ) ⊲ m = e ∗m = m.

4) Let (g, ξ), (h, δ) ∈ D and m ∈M , then

(g, ξ) ⊲ ((h, δ) ⊳ m) = (g, ξ) ⊲ (ε(δ, h ∗m))
= g ∗ ε(δ, h ∗m)
= ε(g · δ, g ∗ (h ∗m))
= (gh, g · δ) ⊳ m
= ((g, ξ) ⊢ (h, δ)) ⊳ m.

5) For every (g, ξ), (h, δ) ∈ D and m ∈M , we have that

(g, ξ) ⊳ ((h, δ) ⊲ m) = (g, ξ) ⊳ h ∗m
= ε(ξ, g ∗ (h ∗m))
= ε(ξ, (gh) ∗m))
= (gh, ξ) ⊳ m
= ((g, ξ) ⊣ (h, δ)) ⊳ m.

It is well known in group theory that there is a strong connection
between the symmetric group and the G-sets. Here we have something



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124 The structure of g-digroup actions

similar given in Proposition 6. Before the statement of such corollary, we
recall the definition of the symmetric g-digroup, for more details see [16].

Let A be non-empty set and let G 6 Sym(A). If X is a G-set, with
action (g, ξ) 7→ g • ξ, we define the map

T : Sym(A)×X → End(A×X)
(α, ξ) 7→ T(α,ξ),

whit T(α,ξ)(b, η) = (α(b), ξ), ∀(b, η) ∈ A×X.
Over the set TG := {T(g,ξ) | g ∈ G and ξ ∈ X} we define the binary

maps ⊣ and ⊢, by

T(g,ξ) ⊣ T(h,η) := T(gh,ξ) and T(g,ξ) ⊢ T(h,η) := T(gh,g•η).

Theorem 12 (The symmetric generalized digroup). The set (TG,⊢,⊣)
is a (symmetric) g-digroup with halo TI = {T(Id,ξ) | ξ ∈ X}. For any bar
unit T(Id,e) ∈ TI, the left and right inverses of T(g,ξ) ∈ TG are

T−1
(g,ξ)le

= T(g−1,e) and T−1
(g,ξ)re

= T(g−1,g−1•e).

As it was mentioned, the following proposition asserts the connection
between g-action and the symmetric g-digroups.

Proposition 6. Let A,X and a TG as previously given, then A×X is a
TG-set.

Proof. It is not hard to prove that TG is isomorphic to the g-digroup
G × E, where (g, ξ) ⊢ (h, η) = (gh, φg(η)) and (g, ξ) ⊣ (h, η) = (gh, ξ).
Consequently, we have to prove that A×X is a G× E-set, but it is an
immediate consequence of the fact that A× E is naturally a G-set and

ε : E × (A× E) → A× E,

where ε(ξ, (a, η)) = (a, ξ) is a d-equivariant function.
(a)

ε(φg(ξ), g ∗ (a, η)) = ε(φg(ξ), (g(a), φg(η)))
= (g(a), φg(ξ))
= g ∗ (a, ξ) = g ∗ ε(ξ, (a, η)).

(b)
ε(ξ, ε(η, (a, δ))) = ε(ξ, (a, η)) = (a, ξ) = ε(ξ, (a, δ)).

In the following definition we introduce the concepts of orbits and
isotropic g-subdigroup like in [5, Definition 4].



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Rodríguez-Nieto, Salazar-Díaz, Velásquez 125

Definition 10. Let D be a g-digroup and let M be a D-set. We define
the ⊲-orbit and the ⊳-orbit of an element m ∈ M as the sets O⊲

m =
{x ⊲ m | x ∈ D} and O⊳

m = {x ⊳ m | x ∈ D}, respectively. We also define
Dl

m = {x ∈ D | x ⊲ m = m} and Dr
m = {x ∈M | x ⊳ m = m}.

The sets O⊲
m and O⊳

m are not orbits in the sense of group actions.
Thereby, we could consider the orbit of m as the set OD

m = O⊲
m ∪O⊳

m, but
it is unknown if they partition M .

The proof of the following proposition is equal to the one given for
[5, Proposition 7], thus we omit it.

Proposition 7. The set Dl
n is a g-subdigroup of D, called the left isotropic

g-subdigroup of D. Moreover, Dm = Dl
m∩Dr

m is also a g-subdigroup, called
the bilateral isotropic g-subdigroup of D.

Acknowledgements

The authors thank the referee for the valuable comments. The third
author acknowledges to Vicerrectoría de Investigaciones, Universidad de
Antioquia.

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Contact information

José Gregorio

Rodríguez-Nieto,

Olga Patricia

Salazar-Díaz

Escuela de Matemáticas
Universidad Nacional de Colombia
Medellín, Colombia
E-Mail(s): jgrodrig@unal.edu.co,

opsalazard@unal.edu.co

Raúl Velásquez Instituto de Matemáticas
Universidad de Antioquia
Medellín, Colombia
E-Mail(s): raul.velasquez@udea.edu.co

Received by the editors: 17.12.2020
and in final form 14.08.2021.

mailto:jgrodrig@unal.edu.co
mailto:opsalazard@unal.edu.co
mailto:raul.velasquez@udea.edu.co

	Rodríguez-Nieto, Salazar-Díaz, Velásquez