Normal subdigroups and the isomorphism theorems for digroup
We discuss the notion of normality of a sub-object in the category of digroups. This allows us to define quotient digroups, and then establish the corresponding analogues of the classical Isomorphism Theorems.
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irk-123456789-1557342019-06-18T01:26:27Z Normal subdigroups and the isomorphism theorems for digroup Ongay, F. Velásquez, R.E. Wills-Toro, L.A. We discuss the notion of normality of a sub-object in the category of digroups. This allows us to define quotient digroups, and then establish the corresponding analogues of the classical Isomorphism Theorems. 2016 Article Normal subdigroups and the isomorphism theorems for digroup / F. Ongay, R.E. Velásquez, L.A. Wills-Toro // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 2. — С. 262-283. — Бібліогр.: 7 назв. — англ. 1726-3255 2010 MSC:Primary 20N99. http://dspace.nbuv.gov.ua/handle/123456789/155734 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We discuss the notion of normality of a sub-object in the category of digroups. This allows us to define quotient digroups, and then establish the corresponding analogues of the classical Isomorphism Theorems. |
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Ongay, F. Velásquez, R.E. Wills-Toro, L.A. Normal subdigroups and the isomorphism theorems for digroup Algebra and Discrete Mathematics |
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Normal subdigroups and the isomorphism theorems for digroup |
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Normal subdigroups and the isomorphism theorems for digroup |
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Normal subdigroups and the isomorphism theorems for digroup |
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Normal subdigroups and the isomorphism theorems for digroup |
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Normal subdigroups and the isomorphism theorems for digroup |
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normal subdigroups and the isomorphism theorems for digroup |
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Normal subdigroups and the isomorphism theorems for digroup / F. Ongay, R.E. Velásquez, L.A. Wills-Toro // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 2. — С. 262-283. — Бібліогр.: 7 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT ongayf normalsubdigroupsandtheisomorphismtheoremsfordigroup AT velasquezre normalsubdigroupsandtheisomorphismtheoremsfordigroup AT willstorola normalsubdigroupsandtheisomorphismtheoremsfordigroup |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 22 (2016). Number 2, pp. 262–283
© Journal “Algebra and Discrete Mathematics”
Normal subdigroups and the isomorphism
theorems for digroups
Fausto Ongay∗, Raúl Velásquez∗∗
and Luis Alberto Wills-Toro∗ ∗ ∗
Communicated by V. M. Futorny
Abstract. We discuss the notion of normality of a sub-
object in the category of digroups. This allows us to define quotient
digroups, and then establish the corresponding analogues of the
classical Isomorphism Theorems.
Introduction
Digroups are a generalization of groups that involves two operations.
Their origins can be traced back to the work of J. L. Loday on Leibniz
algebras and dialgebras, and for background on the subject we refer
the reader to the detailed discussion in [5], and the references therein.
Digroups are examples of dimonoids with bar-units considered recently
in some other aspects in the paper of A.V. Zhuchok (see [7]).
As one would expect, there is a natural notion of homomorphism, so
digroups constitute a category, and thus a rather evident question is: what
kind of category do they constitute? Intuitively, one might suspect that
the category of digroups should have more or less the same properties
∗Partially supported by CONACYT, Mexico, research project 106 923
∗∗Partially supported by Universidad de Antioquia, CODI research project “Álgebras
no asociativas”
∗ ∗ ∗Partially supported by Universidad Nacional de Colombia, research project 30232
“Ideales y Derivaciones en Álgebras de Leibniz”
2010 MSC: Primary 20N99.
Key words and phrases: Digroups, Isomorphism Theorems.
F. Ongay, R. Velásquez, L. A. Wills-Toro 263
as that of groups; but since there are some obvious key differences, the
generalization of the classical results for groups is not straightforward.
In this note we shall partially answer this question, by studying the
construction of quotient digroups, and the corresponding analogues of
the usual Isomorphism Theorems for groups (see for instance the classical
text [3]).
A natural scenario for these isomorphism theorems is that of Universal
Algebra (see e.g. [1] or [2]), where one studies algebraic structures with
a finite number of operations, each one having finitely many arguments
(n-ary operations). Digroups —as well as groups— fall of course into
this context, since each has a nullary operation associated with a special
element, and a unary operation of inversion, besides the binary products.
The isomorphism theorems then describe the possibility of defining equiv-
alence classes, or congruences, in such a way that the quotient inherits
an algebraic structure of the same type, starting from a given morphism
or sub-object.
Now, the presence of the nullary operation, either for groups or di-
groups, gives a so-called zero object in the corresponding category, and
this makes them pointed categories. The point is then that while the
isomorphism theorems for general universal algebras are not concerned
with these “pointed properties”, in fact the existence of zero objects re-
stricts the sub-structures in a very concrete manner, and as a rule this
has a direct impact on the way the congruences can be constructed. For
instance, it is well-known for groups and rings that the class of the zero
sub-object, which is the same as the kernel of the morphism, is enough to
determine completely the equivalence classes, and therefore the quotient;
but in general this is not so simple. Thus, our intent here is to give a
detailed answer to the question: what can be said for digroups in regard
to the construction and properties of these congruences? This question
is of interest because of the deep connections between the algebraic and
geometrical properties of digroups.
The paper is organized as follows:
Section 1 is mostly a review of known results about digroups. We
outline first a comparison between the category of digroups and the
category of groups, and this leads us to a natural defintion of normal
subdigroup. The key idea, however, is to recall a useful identification of a
digroup with a G-set admitting a fixed point.
In Section 2 this identification is exploited to obtain a definition of
quotient digroups. The main point is that the algebraic definition of
normality is not as strong in this case as it is in the case of groups.
In Section 3, which is the core of the paper, we then present the
analogues of the three classical Isomorphism Theorems. We give along the
264 Normal subdigroups
way several examples, illustrating the similarities as well as the differences
between the two categories.
Finally, Section 4 contains a few concluding remarks.
We shall use the fairly standard notation H < G to denote that H is
a subgroup, and H ⊳ G to denote that H is a normal subgroup of G.
1. The category of digroups
1.1. Digroups and normal subdigroups
Let us begin with the usual definition of a digroup:
Definition 1. A digroup is a set D provided with two binary operations
⊢,⊣, which are associative and satisfy the compatibility conditions:
x ⊣ (y ⊣ z) = x ⊣ (y ⊢ z) ;
(x ⊢ y) ⊢ z = (x ⊣ y) ⊢ z;
(x ⊢ y) ⊣ z = x ⊢ (y ⊣ z) .
It also possesses a neutral element and inverses in the sense that there
exists a distinguished fixed element e such that for each x, e ⊢ x =
x ⊣ e = x, and for each x there exists a unique element x−1 such that
x ⊢ x−1 = x−1 ⊣ x = e.
Remark 1. The conditions ∀ x, e ⊢ x = x ⊣ e = x define a so-called
bar unit. In contrast to the neutral element in a group, in a non-trivial
digroup (that is, a digroup that is not a group) bar units are not unique,
and the set of these elements is sometimes called the halo of the digroup.
The corresponding definition of morphism is then:
Definition 2. Let D and D′ be digroups, then:
A function Φ : D → D′ is a homomorphism if
Φ (x ⊢ y) = Φ (x) ⊢ Φ (y) , Φ (x ⊣ y) = Φ (x) ⊣ Φ (y) ,
and Φ(e) = e′.
If Φ is a digroup homomorphism, then its kernel is Φ−1(e′).
Finally, the sub-objects in the category are:
Definition 3. Let D be a digroup. Then, S ⊆ D is a subdigroup of D
if e ∈ S and S is closed under the digroup operations (including that of
taking inverses).
F. Ongay, R. Velásquez, L. A. Wills-Toro 265
It is easy to check that if Φ : D → D′ is a digroup homomorphism,
then the image of a bar unit (inverse) is a bar unit (inverse) and the direct
(inverse) image of a subdigroup is a subdigroup.
Remark 2. From the definitions above it is plain that, indeed, digroups
form a pointed category: the trivial group, regarded as a digroup, is
clearly a zero object. Obviously, zero objects can also be identified with
the subdigroup consisting solely of the distinguished bar unit e of a given
digroup D (see, e.g. [4]).
These are all direct adaptations of the corresponding notions in the
classical theory of groups. Thus, based on the analogy with this theory,
a natural choice for the definition of normality would be stability of
the subdigroup under conjugation; but in principle we now have eight
possibilities to define the conjugation x ∗ y ∗ x−1, where ∗ denotes any of
the two digroup operations. They are however not all independent, and
from the digroup axioms we have, for instance, that x ⊣ (y ⊣ x−1) = x ⊣
(y ⊢ x−1), and hence x ⊣ (S ⊣ x−1) = x ⊣ (S ⊢ x−1); thus, we are left
with the following three options: ∀x ∈ D,
x ⊢ S ⊣ x−1 ⊆ S ; x ⊣ S ⊣ x−1 ⊆ S ; or x ⊢ S ⊢ x−1 ⊆ S.
Moreover, no parentheses are needed if we write the conditions in this
way, because x ⊣ (y ⊣ x−1) = (x ⊣ y) ⊣ x−1, etc.; and since in general
x ⊢ y ⊣ x−1, x ⊢ y ⊢ x−1 and x ⊣ y ⊣ x−1 are all different, the three
possibilities listed are indeed not equivalent. We choose the following
option, and in the next sections we shall give arguments explaining why
this is the “correct” definition:
Definition 4. Let D be a digroup. A subdigroup S of D is normal, if ∀
x ∈ D, x ⊢ S ⊣ x−1 ⊆ S.
We notice here that this is in fact equivalent to the following condition:
∀x, x ⊢ S = S ⊣ x. The point of this remark (to be proved in Lemma 2,
after some preliminary work) is that, just as in the case of groups, this
equality gives well-defined equivalence classes on the digroup (that is,
pairwise disjoint subsets of the digroup D), which of course is something
necessary to define a quotient digroup. However, in marked contrast to
the case of groups, in general these equivalence classes do not cover the
whole group.
On the other hand, that this definition is adequate is further supported
by the following result:
Proposition 1. Let Φ : D → D′ be a digroup morphism. Then ker(Φ) is
a normal subdigroup of D.
266 Normal subdigroups
Proof. The proof is similar to the usual one: if we denote S = ker(Φ),
and let x ∈ D be any element, then for y ∈ S
Φ(x ⊢ y ⊣ x−1) = Φ(x) ⊢ Φ(e) ⊣ Φ(x−1) = Φ(x) ⊢ e′ ⊣ (Φ(x))−1 = e′,
because in any digroup e ⊣ x−1 = x−1 ⊣ x ⊣ x−1 = x−1.
We remark however that similar results hold (and the proofs are
even more direct) for the other possible choices of conjugation in the
digroup, so that the property of stability of the kernel of a morphism under
conjugation alone does not suffice to determine uniquely the condition
for normality, and therefore a choice has to be made.
1.2. Digroups and G-sets
To understand the properties of digroups better, it is convenient to
use an alternative presentation, stemming from results due to M. Kinyon
([5]) that we can summarize as follows:
Theorem 1. Given a digroup D, define its set of inverses,
G =
{
y | ∃ x such that y = x−1
}
,
and its set of bar units
E = {y | ∀ x, y ⊢ x = x ⊣ y = x} .
Then G is a group, and the projections
D → G : x 7→ a = e ⊣ x = (x−1)−1 ; D → E : x 7→ α = x ⊣ x−1,
yield a bijection ρ : D → G×E, with inverse ρ−1 : G×E → D given by:
(a, α) 7→ α ⊣ a; a ∈ G,α ∈ E.
The map ρ turns into an isomorphism of digroups if the operations
on G× E are defined as
(a, α) ⊣ (b, β) = (a ⊣ b, α) ; (1)
(a, α) ⊢ (b, β) =
(
a ⊢ b, a ⊢ β ⊣ a−1
)
. (2)
We shall call G the group part (or factor) of D, and E the halo or bar
unit part.
F. Ongay, R. Velásquez, L. A. Wills-Toro 267
Actually, in a digroup D the group G is characterized by the fact
that both operations restricted to G coincide, so that the symbol for the
product might as well (and thus will) be omitted for products of two of
these elements.
Moreover, the expression a ⊢ β ⊣ a−1 determines a left action of G
on E, which we shall write as a · β. Thus, in any digroup D, E is a G-set,
and the neutral element is a fixed point for this action, and the digroup
is completely determined by the action of its group of inverses G on its
bar units E.
Also, this identification between objects in two categories has a nat-
ural counterpart for the morphisms; namely, we can identify a digroup
homomorphism with a pair composed by a group homomorphism and
an equivariant map on the halo. The precise statement is given in the
following lemma:
Lemma 1. Let Φ : D → D′ be a digroup homomorphism; let ρ : D →
G×E and ρ′ : D′ → G′×E′ be the corresponding isomorphisms considered
in Theorem 1. Then, there exists a unique homomorphism Φ′ : G× E →
G′ × E′, that makes the following diagram commute:
D D′
G× E G′ × E′
✲Φ
❄
ρ
❄
ρ′
✲
Φ′
In fact, Φ′ ≡ (φ, µ), where φ : G → G′, a 7→ φ(a) = e′ ⊣ Φ(a), is a group
homomorphism, and µ : E → E′, α 7→ µ(α) = Φ(α) is an equivariant
map, i.e., µ(a · α) = φ(a) · µ(α), for all (a, α) ∈ G× E.
In particular, orbits in E under G are mapped into orbits in E′ under
φ(G) 6 G′.
We can then summarize the above discussion in the following manner
(see [6]):
Theorem 2. A digroup is equivalent to the datum of a G-set E, or
equivalently an action G×E → E, with the only condition for this action
of the existence of a fixed point, ε (in particular, the fixed point for the
action does not need to be unique). Under this equivalence, the neutral
element of the digroup D = G×E is (e, ε), where e is the identity element
of G.
This relation between digroups and G-sets is in fact an equivalence
of categories: each morphism Φ : G× E → G′ × E′ corresponds to a pair
268 Normal subdigroups
(φ, µ), where φ is a group homomorphism and µ is an equivariant map
that preserves the distinguished fixed points.
Thus, in what follows we shall freely identify an element in a digroup,
x ∈ D, with a pair (a, α) = ρ(x) ∈ G× E; the operations on the digroup
then become
(a, α) ⊣ (b, β) = (ab, α) ;
(a, α) ⊢ (b, β) = (ab, a · β) .
Remark 3. When doing this identification we shall on occasion, as we
already did, write e = (e, ε) for the neutral element of the digroup; here e
refers to the neutral element of the group G, while ε denotes the preferred
fixed point of the G-set of bar units E. This is mostly for bookkeeping
convenience, because in fact the two elements might sometimes coincide;
in particular, this is the case if we start with a digroup D, so that G and
E are both subsets of D; indeed, in this case G ∩E = {e}, and therefore
e = ε.
Let us consider some examples to illustrate this correspondence:
Example 1. A basic class of digroups can be constructed as follows: Let
G be a Lie group, and V a representation space of G; then G×V becomes
a digroup, with the products defined as in (1) and (2). In particular, we
can take a linear Lie group G ⊂ GL(n,R), with the natural action of G
on R
n.
Example 2. Another simple class of digroups is the following: Let G
be any group, and let it act on itself by conjugation. Then D = G×G
becomes a digroup, with operations
(a, α) ⊣ (b, β) = (ab, α) ;
(a, α) ⊢ (b, β) =
(
ab, aβa−1
)
.
Notice that here, as a subset of D, the group of inverses is G× {e}, while
{e} × G corresponds to the halo, so the two factors play indeed quite
different roles, although here again e = ε.
2. Quotient digroups
2.1. Characterization of subdigroups and of normality
Theorem 2 gives the following very convenient characterization of
subdigroups:
F. Ongay, R. Velásquez, L. A. Wills-Toro 269
Proposition 2. Let D be a digroup with fixed element (e, ε). A subset
S ⊂ D is a subdigroup if and only if it is of the form S = R× T , where
R is a subgroup of G, and T ⊂ E is invariant under R and ε ∈ T .
Proof. Indeed, S is a subdigroup if ∀ x = (a, α) , y = (b, β) ∈ S, x ⊢ y =
(ab, a · β) ∈ S, and x ⊣ y = (ab, α) ∈ S.
As any digroup, we know that S will have the form S = R× T ; now
separate the above relations into coordinates: it follows that the group
part, R, must be a subgroup of G; the condition given by the operation
⊢ then requires that the R-orbit of the bar unit part of any element in S
be contained in T .
Finally, it is clear that we need to require (e, ε) ∈ R × T , and this
yields the desired result.
Following the reasoning in this proof, let us next compare the three
possible definitions of normality; a short and straightforward computation
gives that they can be expressed as follows, condition (3) being the one
we have chosen as definition of normality:
x ⊢ S ⊣ x−1 ⊆ S ⇐⇒
(
aba−1, a · β
)
∈ R× T, (3)
x ⊣ (S ⊢ x−1) ⊆ S ⇐⇒
(
aba−1, α
)
∈ R× T, (4)
(x ⊣ S) ⊢ x−1 ⊆ S ⇐⇒
(
aba−1, ε
)
∈ R× T, (5)
∀ x = (a, α) ∈ D, y = (b, β) ∈ S.
Again, splitting these conditions into coordinates, we see that in the
factor G it makes no difference which one of the different possibilities we
choose: the only option in the group factor for normality of the subdigroup
is that R ⊳ G, as expected. Thus, it is only on the bar units part that we
have to make choices.
But now consider condition (4). Since α is arbitrary, this implies that
T ⊇ E, that is T = E; hence this condition is too restrictive, because the
only normal subdigroups allowed by it would correspond in a one-to-one
fashion to the normal subgroups of G, and the quotients would be trivial
as digroups. Condition (5), on the other hand, imposes no restriction on
T (besides the R-invariance already implicit from the fact that S is a
subdigroup); we will see in a moment that in a definite sense this is too
flexible.
However, condition (3) imposes a non-trivial restriction, which we
record as:
Proposition 3. A subdigroup N = H ×K of a digroup D = G × E is
normal iff:
270 Normal subdigroups
1) H is a normal subgroup of G
2) K is an invariant subset of E under the action of the whole group G.
Proof. As said, it is immediate from the condition
∀x ∈ D, x ⊢ N ⊣ x−1 ⊆ N ⇐⇒
∀x = (a, α) ∈ D, y = (b, β) ∈ N,
(
aba−1, a · β
)
∈ H ×K.
Let us give some examples of subdigroups and normal subdigroups:
Example 3. Let D = G × G be a digroup as in Example 2. Let us
describe the subdigroups of D of the form N = H ×K, where K is also
a subgroup of G:
For a general digroup of this form, since H acts on K by conjugation
if and only if H is a subgroup of the normalizer of K, this is the only
restriction on N .
On the other hand, for N to be a normal subdigroup, K also needs
to be invariant under conjugation by the whole group, so both, H and K,
must be normal subgroups of G.
Example 4. Let G = S1 = U(1) be the circle group, and make it act on
the sphere S2 by rotations around the z-axis; to get a digroup choose as
fixed point one of the poles, say the south pole S.
Since G is abelian any subgroup is normal, and the G-orbits are the
parallels of the sphere, together with the two poles. Therefore, the normal
subdigroups are given by an arbitrary subgroup of G as the group factor,
together with the set theoretic union of the singleton {S} and an arbitrary
set of orbits, as the bar-units factor.
Examples of non-normal subdigroups are obtained by taking as sub-
digroup H ∼= Zn, the group of n-th roots of unity, and as K the union
of {S} and any orbit of H, such as a regular n-gon contained in a fixed
parallel.
2.2. Normal subdigroups and quotient digroups
We now start the analysis of the construction of quotient digroups,
by proving the assertion made after Definition 4:
Lemma 2. A subdigroup N is normal iff ∀x, x ⊢ N = N ⊣ x.
Any two of these sets either coincide or are disjoint.
F. Ongay, R. Velásquez, L. A. Wills-Toro 271
Proof. For the direct implication, note that x ⊢ N ⊣ x−1 ⊆ N implies
x ⊢ N = x ⊢ N ⊣ e = (x ⊢ N) ⊣ (x−1 ⊣ x)
= (x ⊢ N ⊣ x−1) ⊣ x ⊆ N ⊣ x (6)
To get the converse inclusion, observe that since for any x, e ⊣ x =
(x−1)−1, we have that
N ⊣ x = (x ⊢ x−1) ⊢ N ⊣ (e ⊣ x) = x ⊢ (x−1 ⊢ N ⊣ (x−1)−1) ⊆ x ⊢ N
The converse implication is even simpler, since x ⊢ N = N ⊣ x implies
x ⊢ N ⊣ x−1 = N ⊣ (x ⊣ x−1) = N ⊣ (x ⊢ x−1) = N ⊣ e = N.
For the last statement, if we write x = (a, α), then
x ⊢ N = { (ab, a · β) ; b ∈ H , β ∈ K } = aH ×K.
Since H ⊳ G, whenever two of these sets are distinct they are disjoint,
and so the assertion follows.
The previous lemma is the gist of the main difference between the
group and digroup cases, for it shows that the algebraic condition for
normality partitions in a non-trivial way the subset G × K ⊂ D, but
says nothing about its complement G× (E \K) (but in fact neither do
the other posibilities). As a consequence, this condition alone does not
suffice to uniquely determine suitable equivalence classes in D, on which
a digroup structure can be defined, and it turns out that such ‘good’
partitions of D are not unique. We now discuss this point, starting with
the following definition:
Definition 5. Let N = H × K be a normal subdigroup of a digroup
D = G×E. We say that a partition of E is admissible for the subdigroup
N , if it contains K and the group G/H acts on the partition; more
precisely, if we denote the equivalence class containing α ∈ E by [α], then
the partition satisfies H · [α] = {h · ξ ; h ∈ H , ξ ∈ [α]} = [α], and for all
a ∈ G and [α] in the partition [a · α] = a · [α], is also in the partition.
The first condition implies that the equivalence classes in such a
partition are composed by H-orbits, while the second ensures that G/H
acts on the equivalence classes. Examples of admissible partitions are K
together with all remaining H-orbits in E, or K and a partition of E \K
that is G-invariant, such as E \K itself.
272 Normal subdigroups
It is also clear that an admissible partition of E also partitions D,
namely, if we write [a] = aH, and let x ∈ D be identified with (a, α) ∈
G×E, the equivalence class containing x, say Ξ, will be the ‘rectangular
box’ in D [a] × [α], and we shall also say that such a partition of D is
admissible for N . Note that by hypothesis [e] = H and [ε] = K. Figure 1
might be helpful to visualize what we are doing; it exhibits in particular
the classes already discussed in Lemma 2.
✲
✻
N = H ×K = [e] × [ε]
❆
❆❆❑
GH
K
aH ×K = [a] × [ε]
✻
E
aH
aH × [α] = [a] × [α]✻ H × [α] = [e] × [α]
❍❍❨
[α]
Figure 1. An admissible partition for a normal subdigroup H ×K.
The important point is that the set of these equivalence classes can
be endowed with a structure of digroup:
Theorem 3. Let N = H ×K be a normal subdigroup of D = G× E.
Then, any partition of E admissible for N induces an admissible
partition of D, so that the set of equivalence classes inherits a structure
of digroup, with the operations defined as follows: Let Ξ = [a] × [α] and
Υ = [b] × [β] be two equivalence classes, then set
Ξ ⊢ Υ = [a][b] × [a] · [β] = [ab] × [a · β]; (7)
Ξ ⊣ Υ = [a][b] × [α] = [ab] × [α], . (8)
F. Ongay, R. Velásquez, L. A. Wills-Toro 273
The projection map
(a, α) 7→ [a] × [α]
is then a surjective morphism of digroups, whose kernel is N .
Conversely, any surjective morphism of digroups, whose kernel is N ,
Φ = φ× µ : D = G× E → D′ = G′ × E′,
induces an admissible partition of D by taking inverse images of elements
of E′.
Proof. Given a partition of D admissible for normal subdigroup N , denote
the quotient D̂ = Ĝ× Ê. It is then clear that the first equality in both (7)
and (8) is exactly what is required for the set of these equivalence classes
to be a digroup, according to Theorem 2, while the second equalities are
the natural way to relate the operations on the classes to the original
digroup operations.
We still need to check two things; first of all, that these products are
well defined. So, let a ∼ a′, b ∼ b′, and α ∼ α′, β ∼ β′; then, for the group
part, [ab] = [a′b′] is just the classical statement about normal subgroups,
and in particular Ĝ = G/H. As for the bar unit part, for the product ⊣
there is nothing to check, while for the ⊢ product, this holds true precisely
because of the way admissible partitions were defined.
Second, that Ĝ acts on Ê, so that (7) and (8) do define a quotient
digroup structure on the set of equivalence classes. As already said,
this is again by definition of admissible partition, for we have aH · [α] =
a·(H ·[α]) = a·[α], and because H ⊳ G, this is indeed an action. Moreover,
this also implies that the map induced on the bar units is equivariant,
thus showing that the projection is a digroup homomorphism.
To prove the last assertion, if Φ : D → D′ is a surjective digroup homo-
morphism, with ker Φ = N = H ×K, then for a given x′ = (a′, α′) ∈ D′,
Φ−1(x′) = φ−1(a′) × µ−1(α′),
so the partition of D obtained this way is indeed by rectangular boxes,
and we need to check that it is admissible.
Again, since H = kerφ ⊳ G, for the group part the assertion follows
immediately from the classical result, and hence, we only need to check
that the partition of E induced by µ is admissible. Now, K = µ−1(ε′), is
obviously in the partition, so consider any other bar-unit α′ ∈ E′, and
let α ∈ µ−1(α′); we need to show that for a ∈ G, we have the equality
274 Normal subdigroups
of sets a · µ−1(µ(α)) = µ−1(µ(a · α)), and by equivariance we have that
µ−1(µ(a · α)) = µ−1(φ(a) · µ(α)).
Thus, if ξ ∈ µ−1(µ(α)), then µ(a ·ξ) = φ(a) ·µ(ξ) = φ(a) ·µ(α), and we
get one contention. For the other contention, if ξ ∈ µ−1(φ(a) ·µ(α)), then
µ(a−1 · ξ) = φ(a−1) · µ(ξ) = µ(α), so that a−1 · ξ ∈ µ−1(µ(α)), and hence
ξ ∈ a · µ−1(α); this shows that the partition is admissible, as desired.
Remark 4. Notice that because in (7) and (8) a is arbitrary, these
constructions are consistent precisely because K is G-invariant, and not
just H-invariant, since this ensures that K is a fixed point for the action
on the quotient. This would not be the case had we chosen to require as
normality condition (5), (x ⊣ N) ⊢ x−1 ⊆ N , and this is an additional
justification to (3) as the proper definition of normality.
The previous theorem finally leads us to the following definition:
Definition 6. Let N = H ×K be a normal subdigroup of D = G× E.
An admissible quotient for N is a digroup Q, together with a surjective
digroup morphism ΨQ : D → Q, such that N = ker(ΨQ).
Let us give some examples of these quotients:
Example 5. Consider the digroups of Example 2, D = G×G; suppose
that H ⊳ K ⊳ G, and consider the subdigroup H × K. Then, if we
choose as the admissible partition of the set of bar units G the classes
induced by K, the quotient digroup constructed above can be identified
to G/H ×G/K.
Moreover, although the fact that G/K also has a group structure is
unimportant here, in this case the canonical projection πK : G → G/K
is equivariant, so that the map (πH , πK) is a digroup morphism, whose
kernel is precisely E = H ×K.
But there are other admissible quotients: For instance, the partition
of G, regarded as the set of bar units, {K , G \K }, is also admissible,
and the quotient digroup would then be G/H × { 0, 1 }, with the trivial
action of the group part on the bar units; but unless the order of K in G
is 2, this is a different quotient.
Example 6. Consider the digroup of Example 4, U(1) × S2; for the
normal subdigroup choose as H = Z
n, the subgroup of n-th roots of unity,
and as K the union {S}∪{N}∪E , where N is the north pole and E is any
parallel, other than the poles, such as the equator. Finally, as partition of
the complement take the individual H-orbits.
The quotient is then isomorphic to a product of the form S1×Ê, where
topologically Ê can be visualized as follows: first, contract the parallel E
F. Ongay, R. Velásquez, L. A. Wills-Toro 275
to its center; this leaves us with two smaller spheres touching at a point
on the axis connecting the two original poles; then, pinch the resulting
spheres until these three points get identified. The induced action of the
new circle group S1 ∼= U(1)/Zn on the quotient Ê can still be thought of
as being by rotations, but performed n times ‘faster’ than for the original
action, as the map E → Ê is generically an n-fold covering (technically,
it is a ramified covering).
3. The analogues of the Isomorphism Theorems
3.1. The Strong Kernel and the First Isomorphism Theorem
If we start with a fixed homomorphism Φ : D → D′, then we have
associated the normal subdigroupN = ker(Φ); but in contrast to the group
case, knowledge ofN alone does not allow us to recover the morphism, even
assuming it to be onto. This is so because, when we identify Φ = (φ, µ),
the subdigroup N does not keep track of the effect of the equivariant
map µ, as N one only involves the subset of the bar units K = µ−1(ε′).
Moreover, there is another admissible quotient naturally determined by Φ,
given by Φ−1(E′) = ker(φ) ×E (which is also a normal subdigroup of D);
but clearly this is still not enough to recover the morphism Φ, because
D/Φ−1(E′) ∼= G/ ker(φ).
To address this situation we proceed as follows:
First, the equivariance of µ determines a natural partition of E, with
classes µ−1(ξ′), ξ′ ∈ E′. Thus, it defines another admissible quotient that
can be associated to the normal subdigroup N , motivating the following:
Definition 7. If Φ = (φ, µ) is a morphism, we define its strong kernel
Sker(Φ), as the congruence induced by ker(φ) together with the admissible
partition induced by the equivariant map µ:
α ∼ β ⇐⇒ µ(α) = µ(β).
The partition given by the strong kernel is in fact the same as the
one given by x ∼ y iff Φ(x) = Φ(y). This is the standard way in universal
algebra to obtain a congruence from a morphism, and quotients via this
relation lead to isomorphism theorems. But in general with no regard
to other properties, such as those associated with the presence of a zero
object, in our case given by the special bar unit of the digroup. In a sense
what we are trying to do is measure the obstructions to the isomorphism
theorems when we attempt to define the quotient solely by the kernel
defined by the zero object. This was the motivation for our analysis of
276 Normal subdigroups
the partitions of the space of bar units, leading to notions of admissible
quotients and strong kernel.
In any case, Theorem 3 also suggests that for a fixed a normal subdi-
group N , the admissible quotient associated to the finest possible partition
plays a special role. The bar units of the corresponding quotient, obtained
from the individual H-orbits, are H ×H ·α, for α ∈ E \K, together with
N , which has to be the neutral element of the quotient digroup, and this
gives a sort of ‘semi-universal’ quotient. The whole idea is made precise
in the following technical lemma:
Lemma 3. Let N = H×K be a normal subdigroup of D = G×E. Then,
any partition of E \K as in Theorem 3, together with the corresponding
natural projection, determines an admissible quotient.
Conversely, any admissible quotient for N , ΨQ : D → Q, determines
a partition of E, so that the class of ε is K, and E \K is partitioned into
H-invariant sets determined by Sker(ΨQ).
Also, the digroup D′, defined on the partition determined by the bar
units of the form H×H ·α, for each α ∈ E\K, together with the special bar
unit K, is one of these admissible quotients for N ; one that is moreover
universal in the following sense:
For any admissible quotient Q associated to N there exists a unique
surjective digroup morphism ΦQ : D′ → Q such that the morphism ΨQ
factorizes through ΨD′:
ΨQ = ΦQ ◦ ΨD′ .
Proof. The first assertions, and in particular that D′ is an admissible
quotient have essentially been established; it only remains to show that
the mapping (a, α) 7→ [a] × [α] = aH ×H · α, α ∈ E \K, gives a digroup
morphism, but this follows immediately from the definition of the products
on the quotient, as given in Theorem 3.
Now, to prove the universal condition, assume Q = M × L is an
admissible quotient. If X ∈ Q, let x = (a, α) be such that X = ΨQ(x),
and define ΦQ(X) = ΨQ(x). We only need to check that ΦQ is well
defined.
But the digroup morphism ΨQ also splits into a group morphism and
an equivariant map: ΨQ(x) = (ψQ(a), µQ(α)). Thus, if (b, β) is in the
same class as (a, α), automatically ψQ(b) = ψQ(a), as ψQ is an ordinary
group projection, and hence we have just two possibilities left: either
α, β ∈ K or β = h · α, h ∈ H.
In the first case trivially µQ(β) = µQ(α), because by definition of a
morphism µQ(K) = { ε } ⊂ N ; in the second
µQ(β) = µQ(h · α) = ψQ(h) · µQ(α) = e · µQ(α) = µQ(α),
F. Ongay, R. Velásquez, L. A. Wills-Toro 277
and thus we reach the desired conclusion.
For convenience, let us introduce the following terminology:
Definition 8. Given a digroup morphism Φ = (φ, µ), the quotient digroup
D/Sker(Φ), is the admissible quotient associated to the normal subdigroup
N = ker(Φ) defined by the partition given by Sker(Φ), as in Definition 7.
Given a normal subdigroup N , we denote the quotient digroup con-
structed by the finest partition, as in Lemma 3, as D/N , and call it the
universal quotient of D mod N .
The two quotients might coincide, as shown by the following example;
but it is clear that in general they do not:
Example 7. Let H ⊳ G, and consider the digroup D = G × G as
before. The universal quotient of D determined by the normal subdigroup
N = H × H is D/N = G/H × G/H. This can also be described as
the admissible quotient D/Sker(Φ), associated to the digroup morphism
Φ = πH × πH , where πH : G → G/H is the natural projection.
We now state the analogue for digroups to the standard First Isomor-
phism Theorem; as consequence of the above discussion, the conclusion
has to be split into two parts:
Theorem 4. Each morphism Φ : D → D′ determines a normal sub-
digroup N = ker(Φ) and two admissible quotients for N , D/N and
D/Sker(Φ), so that:
1) There is a 1 to 1 correspondence between normal subdigroups N of
a digroup D and universal quotients D/N .
2) The digroup Φ(D) is a subdigroup of D′ isomorphic to D/Sker(Φ).
The proof is now immediate, but to conclude this part let us make
the following useful observation:
Remark 5. Given the normal subdigroup N ⊳ D, because the set (E\K)
is an H-set which we partition into its H-orbits, the universal quotient
D′ can be expressed in the following form:
D/N = (G/H) × ({K} ∪ (E \K)/H) .
Example 8. If D is a digroup, and H ⊳ G, there is clearly a minimal
normal subdigroup having H as its group factor, namely H × {ε}; cor-
respondingly, the maximal universal quotient having group part G/H is
(G/H) × (E/H).
278 Normal subdigroups
3.2. The lattice of subdigroups
and the Second and Third Isomorphism Theorems
To obtain the remaining isomorphism theorems we will regard them as
assertions concerning the universal quotients, as these are the non-trivial
admissible quotients independent of the additional datum of a specific
morphism.
First we define the join of two subdigroups S1, S2, denoted S1S2,
as the minimal subdigroup containing both subdigroups; this, together
with the intersection of subdigroups defines the lattice structure in the
subdigroups of a given digroup D. But in order to state the version for
digroups of the Second Isomorphism Theorem, a more explicit description
of the join is convenient; this is given in the next lemma:
Lemma 4. Let S1 = R1 × T1, S2 = R2 × T2, be two subdigroups of
D = G× E, and let S1S2 denote the minimal subdigroup containing both
S1 and S2.
Then S1S2 = R1R2 × (R1R2) · (T1 ∪ T2), where R1R2 is the subgroup
generated by R1 and R2 (that is, the standard join of the subgroups), and
(R1R2) · (T1 ∪ T2) denotes the union of R1R2-orbits of the elements of
T1 ∪ T2.
Proof. Again, we know that the join S1S2 will have a product structure
R× T . Since it is minimal, the group factor R is necessarily the standard
join R1R2 of the corresponding group parts.
But then, the bar units factor T must contain all the Ri-orbits of
elements of Ti, i = 1, 2, so it must include T1 ∪ T2, and must be R1R2-
invariant, and so must include the orbits under the action of R1R2, that
is (R1R2) · (T1 ∪ T2) ⊂ T .
Finally, sinceR1R2×(R1R2)·(T1∪T2) has the structure of a subdigroup,
and it has to be included in the join S1S2, by minimality this must be
the join of the subdigroups.
Now we can state the analogue for digroups of the Second Isomorphism
Theorem; notice that for the bar units factors the statement is weaker
than for the group parts:
Theorem 5. Let S be a subdigroup and N a normal subdigroup of the
digroup D. Then S ∩N is a normal subdigroup of S and we have, for the
corresponding universal quotients, a surjective digroup morphism:
S/(S ∩N) ։ SN/N,
that is an isomorphism on the group factors.
F. Ongay, R. Velásquez, L. A. Wills-Toro 279
In other words, SN/N is isomorphic to an admissible quotient for
the normal subdigroup S ∩ N of S, but in general not to the universal
quotient S/(S ∩N).
Proof. That S ∩N is normal in S is clear, the proof being identical to
the case of groups.
To prove the existence of the morphism, write the digroups as N =
H ×K, S = R×T , and observe first that for the group factors this is just
the classical theorem; therefore, we can express the quotients as digroups
of the form
SN/N = (RH/H) × Ẽ ; S/(S ∩N) = R/(R ∩H) × Ê,
and it now suffices to consider the bar units part. We want to construct an
equivariant projection µ : Ê → Ẽ, and to this end, we further decompose
these sets (according to the characterization of the universal quotients in
Theorem 4) as:
Ẽ = {K }∪{H ·α ; α ∈ RH ·(T \K) } = {K }∪{H ·α ; α ∈ H ·(T \K) },
where the last equality holds because H is normal and T \K is R-invariant,
and
Ê = {K ∩ T } ∪ { (H ∩R) · α ; α ∈ T \K }.
Now, consider the map
µ : (H ∩R) · α 7→ (H · α)
for α ∈ T \K, and µ({K∩T}) = {K}. It is well defined, for if β ∈ (H∩R)
is another representative, then there exists h ∈ H such that β = h · α;
but this means that H · α = H · β. Further, it is trivially a surjection.
Finally, the equivariance of the map also follows easily from its defini-
tion, since for r ∈ R the actions are, on the one side,
r(H ∩R) · ((H ∩R) · α) = (H ∩R) · (r · α),
while on the other,
rH · (H · α) = H · (r · α),
both equalities holding true by normality of H.
But as mentioned, the result is weaker than for groups: the map is not
in general an isomorphism, since it is not necessarily injective. This is due
to the fact that the orbits of the larger group H might contain more than
280 Normal subdigroups
one orbit of the smaller group R ∩H, and these will get identified in the
quotient. In other words, to have an isomorphism S/(S ∩N) ≈ SN/N we
would need the additional requirement that no two different H ∩R-orbits
in T \K are contained in the same H-orbit in H · (T \K). This can be
seen for instance in the following simple example:
Example 9. Consider the digroup D = Z6 × E, where E = P1 ∪ P6,
and where by Pn we denote the regular n-sided polygon (P1 being a
single point and P2 two points). Thus, here E can be viewed as a regular
hexagon together with its center, and the action is the natural one, by
rotations by 60◦ angles; the point P1 is needed to have a fixed point of
the action. Now, let N = Z2 × P1, and S = Z3 ×E, so that H ∩R = {e}.
Then, it is easy to see that SN = D and S ∩N = {e} × P1 = {e} is
the trivial subdigroup. Therefore,
SN/N = D/N ∼= Z3 × Ê,
where Ê = P1 ∪ P3 is a triangle, corresponding to the antipodal identifi-
cation of the vertices of the hexagon given by the action of Z2, together
with a point (corresponding to the projection of the center P1), while
obviously
S/(S ∩N) = S/{e} ∼= Z3 × E,
and so the quotient digroups are different.
Finally, the analogue of the Third Isomorphism Theorem is:
Theorem 6. Let N , S be normal subdigroups of D, such that N ⊂ S.
Then, for the corresponding universal quotients we have:
S/N is a normal subdigroup of D/N
and there is a digroup isomorphism
(D/N)/(S/N) → D/S.
Proof. Write N , S and D as in the previous theorem. To see that S/N is
normal in D/N we need to show that (T \K)/H is invariant under the
action of the group G/H. But this is clear, since the action is
(gH,H · α) 7→ gH · (H · α) = H · (g · α) ⊂ T \K,
because H is normal and both T andK are G-invariant. Thus, the quotient
(D/N)/(S/N) is well-defined.
F. Ongay, R. Velásquez, L. A. Wills-Toro 281
Moreover, according to Remark 5, this quotient can be written as
follows:
(D/N)/(S/N) = (G/H)/(R/H)×
(
{
{K} ∪ (T \K)/H
}
∪
(
((E \K)/H) \ ((T \K)/H)
)
/(R/H)
)
= (G/H)/(R/H)×
({
{K} ∪ (T \K)/H
}
∪ ((E \ T )/H) /(R/H)
)
,
because K ⊂ T , so that
(
((E \K)/H) \ ((T \K)/H)
)
/(R/H) = ((E \ T )/H) /(R/H)
)
.
Now, for the group parts the claim is again just the classical result,
and so it suffices to see what happens to the bar units. For these, obviously
we need to map
{
{K} ∪ (T \K)/H
}
7→ {T}, so it remains to define the
map on ((E \ T )/H)/(R/H). We do this by sending a class represented
by H · α, α ∈ E \ T , into the class R · α.
The map is well-defined, since if H ·α and H ·β are equivalent modulo
R/H, then rH · α = H · (r · α) = H · β, for some r ∈ R; but this is the
same as β = hr · α for some h ∈ H, and because H ⊂ R, this means that
R · β = R · α. Moreover, this is clearly a surjective map.
But now we can define an inverse map, by sending R · α 7→ (R/H) ·
(H ·α), and this is also well-defined, because if R ·α = R ·β, then ∃ r ∈ R
such that, β = r · α; therefore,
(R/H) ·(H ·β) = (R/H) ·(H ·r ·α) = r ·(R/H) ·(H ·α) = (R/H) ·(H ·α).
Finally, by construction, this map is clearly equivariant, and so we
have a digroup isomorphism, as claimed.
For the sake of completeness, let us conclude by giving an explicit
example illustrating the last theorem:
Example 10. With notations as in Example 9, let D = Z12 × (P1 ∪P12 ∪
P̂24), N = Z3 × P1, and S = Z6 × (P1 ∪ P12) (where we put the hat to
stress that the two Z12-orbits contained in P̂24 are not those in S). It is
clear that all the hypotheses of the theorem are satisfied.
Then it is easy to see that, for instance,
D/S ∼= Z2 × (P1 ∪ P̂4),
282 Normal subdigroups
where P1 now corresponds to the identification of the original P1 ∪P12 to
a point, while P̂4 corresponds to the quotient of P̂24 under the action of
Z6. Similarly, one has
D/N ∼= Z4 × (P1 ∪ P4 ∪ P̂8) ; S/N ∼= Z2 × (P1 ∪ P4).
Hence, for (D/N)/(S/N), the bar unit space of D/N is partitioned
into P1∪P4, which will be identified to a point, while P̂8 will be partitioned
into four Z2-orbits; thus we get again a point and a square, so that
(D/N)/(S/N) ∼= Z2 × (P1 ∪ P̂4),
and therefore both quotients are indeed isomorphic.
4. Some final remarks
There are several directions in which this work can be continued and
refined:
First, having a good control over the notion of normality and the
associated quotients, one can attempt a finer analysis of the structure of
digroups. To name just one possibility, it would be an interesting question
to decide if there is an analogue of the concept of a normal series and the
Jordan-Hölder Theorem for digroups.
Secondly, the interplay between the algebraic questions about digroups
and the more geometric questions about G-sets must give more insight in
both directions: In this work we saw, particularly through the examples,
such as Example 6, how some natural constructions coming from the alge-
braic side translate into geometric properties of the quotients. Obviously,
we can expect that much more could be obtained in this direction, and it
would be interesting to establish some general results; but the point we
want to stress now is that this is most certainly a two-way road, and so
there should be also some natural geometrical questions implying some
interesting algebraic properties. Suffice it to recall that, in fact, the very
notion of digroup arose from the geometrical problem of attempting to
define a structure furnishing integral manifolds for Leibniz algebras (see
e.g. [5] or [6]).
As an example, we mention finally the problem of extending the anal-
ysis to the so-called generalized digroups, which are defined by dropping
the requirement that inverses be bilateral, and which, for this reason, do
not require actions with fixed points. A straightforward generalization of
the constructions done here is probably not possible, but it seems certain
that at least some of the results can be recovered in this ampler context.
F. Ongay, R. Velásquez, L. A. Wills-Toro 283
Acknowledgements
R. Velásquez has the pleasure to thank CIMAT for its hospitality
during a research stay.
F. Ongay thanks the department of Geometry and Topology of the
University of Valencia (Spain), and especially Prof. J. Monterde, for his
friendship and mathematical insight.
All three authors wish to thank Prof. F. Guzman, of the
University of Binghamton (USA), for useful comments, and particularly
Prof. O.P. Salazar-Díaz, from the Universidad Nacional, Medellín campus,
whose contributions to this work are many and varied.
The authors wish to thank the anonymous referee for his (or her) very
helpful comments, which resulted in a markedly improved presentation of
the paper.
References
[1] J. Almeida, Finite Semigroups and Universal Algebra, Series in Algebra, Vol. 3,
World Scientific, 1994.
[2] S. Burris; H.P. Sankappanavar, A Course in Universal Algebra, The Millenium
Edition (electronic reprint).
[3] J.B. Fraleigh, A First Course in Abstract Algebra, 6th Edition, Addison-Wesley,
1999.
[4] H. Herrlich; G. E. Strecker, Category Theory, Allyn and Bacon, Boston, 1973.
[5] M.K. Kinyon, Leibniz algebras, Lie Racks, and Digroups, Journal of Lie Theory, 17
No. 4 (2007), 99-114.
[6] F. Ongay, On the Notion of Digroup Preprint CIMAT I-10-04 (MB) (2010).
[7] A.V. Zhuchok, Dimonoids and bar-units, Sib. Math. J. 56 (2015), no.5, 827-840.
Contact information
F. Ongay CIMAT, Jalisco S/N, Valenciana, Gto., C.P.
36240, México
E-Mail(s): ongay@cimat.mx
R. Velásquez Instituto de Matemáticas, Universidad de An-
tioquia, Medellín, Colombia
E-Mail(s): raul.velasquez@udea.edu.co
L.A. Wills-Toro Escuela de Matemáticas, Universidad Nacional
de Colombia, Medellín, Colombia
E-Mail(s): lawillst@unal.edu.co
Received by the editors: 06.12.2015
and in final form 04.04.2016.
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