Free ultra-groups, generators and relations
In this paper, we intend to define an ultra-group by its presentation. The attitude of the presentation for a group was the key for us to investigate in this area. Instead of writing whole elements of an ultra-group, we denote it by its generators and the relations among those generators. A general...
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irk-123456789-1884962023-03-03T01:27:14Z Free ultra-groups, generators and relations Tolue, B. Zolfaghari, P. Moghaddasi, Gh. In this paper, we intend to define an ultra-group by its presentation. The attitude of the presentation for a group was the key for us to investigate in this area. Instead of writing whole elements of an ultra-group, we denote it by its generators and the relations among those generators. A general computational approach for finitely presented ultra-groups by quotient ultra-groups and subultra-groups is described and some examples are presented. It is the way that can clarify the structure of an ultra-group quicker than having just a list of elements. 2019 Article Free ultra-groups, generators and relations / B. Tolue, P. Zolfaghari, Gh. Moghaddasi // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 308–316. — Бібліогр.: 4 назв. — англ. 1726-3255 2010 MSC: 20F05, 08C05. http://dspace.nbuv.gov.ua/handle/123456789/188496 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper, we intend to define an ultra-group by its presentation. The attitude of the presentation for a group was the key for us to investigate in this area. Instead of writing whole elements of an ultra-group, we denote it by its generators and the relations among those generators. A general computational approach for finitely presented ultra-groups by quotient ultra-groups and subultra-groups is described and some examples are presented. It is the way that can clarify the structure of an ultra-group quicker than having just a list of elements. |
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Tolue, B. Zolfaghari, P. Moghaddasi, Gh. |
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Tolue, B. Zolfaghari, P. Moghaddasi, Gh. Free ultra-groups, generators and relations Algebra and Discrete Mathematics |
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Tolue, B. Zolfaghari, P. Moghaddasi, Gh. |
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Tolue, B. |
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Free ultra-groups, generators and relations |
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Free ultra-groups, generators and relations |
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Free ultra-groups, generators and relations |
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Free ultra-groups, generators and relations |
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Free ultra-groups, generators and relations |
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free ultra-groups, generators and relations |
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Інститут прикладної математики і механіки НАН України |
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Free ultra-groups, generators and relations / B. Tolue, P. Zolfaghari, Gh. Moghaddasi // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 308–316. — Бібліогр.: 4 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT tolueb freeultragroupsgeneratorsandrelations AT zolfagharip freeultragroupsgeneratorsandrelations AT moghaddasigh freeultragroupsgeneratorsandrelations |
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“adm-n4” — 2020/1/24 — 13:02 — page 308 — #158
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 28 (2019). Number 2, pp. 308–316
c© Journal “Algebra and Discrete Mathematics”
Free ultra-groups, generators and relations
Behnaz Tolue, Parvaneh Zolfaghari,
and Gholamreza Moghaddasi
Communicated by I. Ya. Subbotin
Abstract. In this paper, we intend to define an ultra-group
by its presentation. The attitude of the presentation for a group
was the key for us to investigate in this area. Instead of writing
whole elements of an ultra-group, we denote it by its generators
and the relations among those generators. A general computational
approach for finitely presented ultra-groups by quotient ultra-groups
and subultra-groups is described and some examples are presented.
It is the way that can clarify the structure of an ultra-group quicker
than having just a list of elements.
Introduction
In universal algebra, an algebra is a set together with a collection
of operations on it. The need for such a definition was noted by several
mathematicians such as Whitehead in 1898, and later by Noether, the
credit for realizing this goal goes to Birkhoff in 1933. S. Burris, H. P.
Sankapanavar developed the most general and fundamental notions of
universal algebra. Moreover, Free algebras are discussed in great detail by
them [1].
In [3] the new concept of an ultra-group was presented. It is an algebraic
structure which was introduced vastly in Definition 2.
In the group theory, one method of defining a group is by its presen-
tation. We specify a set X of generators such that every element of the
group can be written as a product of powers of some of these generators,
2010 MSC: 20F05, 08C05.
Key words and phrases: ultra-groups, presentation, free ultra-groups.
“adm-n4” — 2020/1/24 — 13:02 — page 309 — #159
B. Tolue, P. Zolfaghari, Gh. Moghaddasi 309
and a set R of relations among those generators. We say the group G has
the finitely presented if there is a finite presentation 〈X|R〉 with G ∼= F/N
where F is the free group on X and N is the normal closure of R in F
(see [2, Chapter I, Section 9.]).
In this paper, our aim is to provide such a strong capability in the
category of ultra-groups. More precisely, we intend to write a presentation
for an ultra-groups instead of their elements and Cayley operation tables.
The reference [3] include the results about an ultra-group, such as subultra-
group, homomorphism and isomorphism theorems. Let us recall some of
the basic definitions.
LetH be a subgroup of the groupG andM a subset ofG. If |M∩Hg| =
1 for all g ∈ G, then G = HM . For the group G which satisfies the above
conditions, we have MH ⊆ G = HM . Therefore, for every element
mh ∈MH there exists h′ ∈ H and m′ ∈M such that mh = h′m′.
Definition 1. Let H be a subgroup of a multiplicative group G. A
subset M of G is called (right unitary) complementary set with respect
to subgroup H, if for any elements m ∈ M and h ∈ H there exist the
unique elements h
′
∈ H and m
′
∈ M such that mh = h
′
m
′
and e ∈ M .
We denote h′ and m′ by mh and mh, respectively.
Similarly for any elements m1,m2 ∈ M there exist unique elements
[m1,m2] ∈M and (m1,m2)h ∈ H such that m1m2 = (m1,m2)h[m1,m2]. For
every element a ∈M , there exists a−1 belonging to G. As G = HM , there
is a(−1) ∈ H and a[−1] ∈M such that a−1 = a(−1)a[−1].
Definition 2. A (right) ultra-group HM is a complementary set of sub-
group H over group G with a binary operation α : HM × HM → HM
and unary operation βh : HM → HM defined by α((m1,m2)) := [m1,m2]
and βh(m) := mh for all h ∈ H.
A (left) ultra-group MH is defined similarly via (left unitary) comple-
mentary set. In this text we concentrate on the right ultra-group.
Throughout this paper, we denote a right ultra-group over the subgroup
H of the group G by HM and consider its binary operation and unary
operation by α and βh, respectively.
Although we have an associative property for the groups, but this
property is not valid for the binary operation α of the ultra-groups. There-
fore, we convent α(a, b, c) = α(α(a, b), c), where a, b, c are the elements
of the ultra-group HM and α is its first binary operation over it. For a
positive integer n, [. . . [x1, x2], x3], . . . , xn]
︸ ︷︷ ︸
n−1 times α
is n− 1 times iteration of the
“adm-n4” — 2020/1/24 — 13:02 — page 310 — #160
310 Free ultra-groups, generators and relations
binary operation α for n elements of ultra-group. In this paper we denote
it by αn(x1, x2, . . . , xn), where xi ∈ HM . The structural and categorical
properties of the ultra-groups discussed in [3,4]. Here, we have recalled
just some necessary notions which are useful in this research, for more
details one can see those references.
By mimicking the techniques of obtaining a presentation for a group,
we require the free ultra-group which is constructed on the set of generators
of an ultra-group. By the free ultra-group we mean the free object in
the concrete category of ultra-groups. In [4], we proved the existence of
the free object in the ultra-groups category and its structure has been
described. In this research, we generalized the Van Dyck ’s Theorem for
the ultra-groups and consequently the presentation for an ultra-group is
defined.
1. Preliminaries
Let HM be an ultra-group of the subgroup H over the group G. The
non-empty subset Y of elements of HM is called a generator set of HM
provided that m = αs(y1, y2, . . . , ys), for yi ∈ Y and the positive integer
s. It is clear that every ultra-group has a generator set. We can consider
the underlying set of an ultra-group as its generator set, in the worst
conditions.
The normal subultra-group was discussed vastly in [3, Definition 2.8].
The normal subultra-group generated by a set T ⊆ HM is the intersection
of all normal subultra-groups of HM that contains T .
If S is a normal subultra-group of HM , then the quotient ultra-group
HM/S is an ultra-group over the subgroup H1 = {h[S, e] : h ∈ H} of
the group G1 = {h[S, a] : h ∈ H, a ∈ HM} = H[S,HM ], where [ , ]
denotes the binary operation of HM , (see [3] for the notations). One can
verify that G1 is a group with a binary operation ∗, which is defined by
h1[S, a1] ∗ h2[S, a2] = h1h2[[S, a1], [S, a2]] = h1h2[S, [a1, a2]], hi ∈ H and
ai ∈ HM , i = 1, 2. Note that, the normality of S in HM implies the
second equality (see [3, Lemma 2.5 (ii)]). With the same notations here,
we conclude the following result.
Lemma 1. The map π: HM → HM/S is an ultra-group epimorphism.
Proof. The map π with the rule a 7→ [S, a], satisfies the definition of
ultra-groups homomorphism (see [3, Definition 2.5]). By the normal
subultra-groups property π([a1, a2]) = [S, [a1, a2]] = [[S, a1], [S, a2]] =
“adm-n4” — 2020/1/24 — 13:02 — page 311 — #161
B. Tolue, P. Zolfaghari, Gh. Moghaddasi 311
[π(a1), π(a2)]. Moreover,
π(ah) = [S, ah] = [S, a]hS = [S, a]ϕ(h) = π(a)ϕ(h),
where ϕ : H → H1. Hence the assertion is clear.
We are going to prepare the tools to generalized the Van Dyck’s
Theorem in the group theory for the ultra-groups.
Theorem 1. Let f : H1
M1 → H2
M2 be an ultra-group homomorphism
and Si is a normal subultra-group of Hi
Mi, i = 1, 2 such that f(S1) be the
proper subultra-group of S2. Then f induce the ultra-group homomorphism
f : H1
M1/S1 → H2
M2/S2.
Proof. Consider the composition of ultra-group homomorphisms
H1
M1 → H2
M2 → H2
M2/S2.
Clearly, S1 ⊂ Ker(πf) ⊆ f−1(S2), where π: H2
M2 → H2
M2/S2. Define
the map f :H1
M1/S1 → H2
M2/S2. By the rule [S1, a] 7→ [S2, f(a)], where
a ∈ H1
M1. Assume a, a1, a2 ∈ H1
M1 and h ∈ H1. The map f is an
ultra-group homomorphism, because
f([[S1, a1], [S1, a2]]) = f([[S1, [a1, a2]]) = [S2, f([a1, a2])]
= [S2, [f(a1), f(a2)]] = [f([S1, a1]), f([S1, a2])],
and also,
f([S1, a]
hS1) = f([S1, a
h] = [S2, f(a
h)] = [S2, (f(a))
ψ(h)]
= [S2, f(a)]
ψ(h)S2 = f([S1, a])
ϕ(hS1),
where ϕ is the group homomorphism between the two subgroups of
which two ultra-groups H1
M1/S1 and H2
M2/S2 are constructed (see the
argument before Lemma 1) and ψ : H1 → H2 is the group homomorphism
which is extracted from the ultra-group homomorphism f .
2. The presentation of an ultra-group
Let F be a free group on the non-empty set X (see [2] for more details).
We know every subgroup of a free group is itself a free group. Choose K
one of the subgroups of F . Constructing all the ultra-groups of a subgroup
over a group has been vastly discussed in [3]. Suppose (W (X), α, βk) is
the ultra-group of the subgroup K over the free group F , where α and βk
“adm-n4” — 2020/1/24 — 13:02 — page 312 — #162
312 Free ultra-groups, generators and relations
are binary and unary operations, for all k ∈ K. Let w1, w2 ∈W (X). Since
W (X) ⊆ F elements of W (X) are all reduced words. The binary operation
on the free group F is just juxtaposition of two reduced words. Therefore,
since w1w2 ∈ F and F = KW (X) we deduce w1w2 = (w1,w2)k [w1, w2],
where (w1,w2)k ∈ K and [w1, w2] ∈ W (X). It is not hard to see that
α(w1, w2) = [w1, w2] by an ultra-group definition. Furthermore, since
W (X)K ⊆ F = KW (X) we have wk = wkwk. Thus βk(w) = wk, for
w ∈ W (X) and all k ∈ K. We call W (X) the free ultra-group on the
non-empty set Y ⊆ X, where Y is the set of all one letter word such that
the words of W (X) is obtained. We observed that W (X) is a free object
in the category of ultra-groups (see [4] for more details). In the following
we denote the free ultra-group by W (Y ).
By considering the ultra-group HM = 〈Y 〉 over the subgroup H of the
group G, there is a generating set X for the group G such that Y ⊆ X.
Moreover, note that G = HM and F = KW (Y ), where F is the free group
on X, K is the free subgroup of F on the set X − Y and W (Y ) is free
ultra-group on Y . Since F is the free group on the set X, the group G is
homomorphic image of F . Thus there exists a unique group epimorphism
ϕ : F → G such that ϕi = f , where i : X → F and f : X → G are
inclusion maps. Now restrict i, f on Y and ϕ on W (Y ). Let i|Y = i′,
f |Y = f ′ and ϕ|W (Y ) = ψ. We have the following diagram.
Y
HM
W (Y )
i′
f ′
ψ
In the following, we use the same notations of the above argument.
Lemma 2. For every w ∈ F ∩W (Y ), ψ(w) = [[. . . [w1, w2], w3], . . . , wn],
where w = w1w2 . . . wn.
Proof. By definition of the binary operation of the free ultra-group, we
deduce that ϕ(ab) = (ϕ(a),ϕ(b))k [ϕ(a), ϕ(b)], where a, b ∈ F and k ∈ K.
Since ψ = ϕ|W (Y ) the assertion is clear.
The above discussion deduce the following theorem.
Theorem 2. Every ultra-group is homomorphic image of a free-ultra
group.
“adm-n4” — 2020/1/24 — 13:02 — page 313 — #163
B. Tolue, P. Zolfaghari, Gh. Moghaddasi 313
Proof. Let HM be an ultra-group with the generating set Y and W (Y ) a
free ultra-group which is constructed on Y . Since W (Y ) is a free object in
the category of ultra-groups, there exists an ultra-groups homomorphism
ψ :W (Y ) → HM such that
ψ(y1y2 . . . ym) = ϕ(y1y2 . . . ym)
= ϕ(y1)ϕ(y2) . . . ϕ(ym)
= ϕ(i(y1))ϕ(i(y2)) . . . ϕ(i(ym))
= [[. . . [f(y1), f(y2)], f(y3)], . . . , f(ym)]
= [[. . . [y1, y2], y3], . . . , ys] = αm(y1y2 . . . ym)
and ϕ : F → G is the group homomorphism, where F is the free group
which is constructed on the generating set ofG, (See [4, Theorem 3.1]). Sup-
pose x ∈ HM . Clearly x can be written of the form [[. . . [y1, y2], y3], . . . , ys],
for yi ∈ Y . Now, consider the word y1y2 . . . ys ∈ W (Y ) which maps to
x by the ultra-group homomorphism ψ. This shows ψ is an ultra-group
epimorphism.
We continue with the same notations as in the proof of Theorem 2
and the argument before that. By the first isomorphism theorem of ultra-
groups, W (Y )/Ker(ψ) is an ultra-group isomorphic to HM , where Ker(ψ)
is the set,
{ykyl . . . yt ∈W (Y ) : ψ(ykyl . . . yt) = [[. . . [yk, yl], ym], . . . , yt] = e},
and e is the identity element of the group of which HM is constructed
on (see [3, Definition 2.6, Theorem 2.3] for more details). Therefore, in
order to describe HM up to isomorphism we need only specify Y, W (Y )
and Ker(ψ). By [2, Theorem 7.8] W (Y ) is determined up to ultra-group
isomorphism by Y and Ker(ψ) is determined by any subset that generates
it as a subgroup of W (Y ). If y1y2 . . . ys ∈W (Y ) is a generator of Ker(ψ),
then under the ultra-group epimorphism ψ :W (Y ) → HM , y1y2 . . . ys 7→
[[. . . [y1, y2], y3], . . . , ys] = e ∈ HM . The equation
[[. . . [y1, y2], y3], . . . , ys] = e
in HM is called a relation on the generators yi. Conversely, suppose we
are given a set Y and a set T of reduced words on the elements of Y
(see [4, Section 3.]). Does there exists an ultra-group (HM,α, βh) such
that is generated by Y and all the relations [[. . . [y1, y2], y3], . . . , ys] = e,
where y1y2 . . . ys ∈ T and [[. . . [y1, y2], y3], . . . , ys] = αs(y1, y2, . . . , ys)?
“adm-n4” — 2020/1/24 — 13:02 — page 314 — #164
314 Free ultra-groups, generators and relations
The answer is positive. We construct such an ultra-group as follows.
Let W (Y ) be the free ultra-group on Y and N the normal subultra-
group of W (Y ) generated by T . Let HM be the quotient ultra-group
W (Y )/N and identifying Y with its image under the map Y ⊂W (Y )
π
−→
W (Y )/N . Every coset [N,w] ∈W (Y )/N correspondence to the element
m = αs(y1, y2, y3, . . . , ys), where w = y1y2 . . . ys is a reduced word in
W (Y ). Thus we recognize HM by this method, which implies HM is
generated by Y . If w = y
l1
y
l2
. . . y
ls
∈ T , then y
l1
y
l2
. . . y
ls
∈ N , since N
is generated by T . This fact deduce that [N, y
l1
y
l2
. . . y
ls
] = N as N is
a normal subultra-group and so αs(y
l1
y
l2
. . . y
ls
) = e by considering the
map π. Hence, HM is the ultra-group which asked in the above question.
Similar to the definition of the presentation for the groups [2, Defi-
nition 9.4], and according to the above discussion we have the following
significant definition.
Definition 3. If X be a finite set and Y a finite set of (reduced) words
on X, then an ultra-group HM is said to be the ultra-group defined by the
generators x ∈ X and relations w = e (w ∈ Y ) provided HM ∼=W (X)/N ,
whereW (X) is the free ultra-group on X andN the normal subultra-group
of W (X) generated by Y . One says that 〈X|Y 〉 is a finite presentation
of HM .
We are ready to present the Van Dyck’s Theorem for ultra-groups.
The proof follows by Theorems 1, 2 and is very similar to the proof of
Theorem 9.5 in [2].
Theorem 3. Let X be a set, Y a set of reduced words on X and H1
M1 the
ultra-group defined by generators x ∈ X and relations w = e (w ∈ Y ). If
H2
M2 is any ultra-group such that H2
M2 = 〈X〉 and satisfies all relations
w = e (w ∈ Y ), then there is an epimorphism H1
M1 → H2
M2.
Proof. If W (X) is the free ultra-group on X, then the inclusion map
X → H2
M2 induces the ultra-group epimorphism ψ : W (X) → H2
M2
by Theorem 2. Since H2
M2 satisfies the relations w = e (w ∈ Y ),
Y ⊂ Kerψ. Consequently, the normal subgroup N generated by Y in
W (X) is contained in Kerψ. By Theorem 1 follows an epimorphism
H1
M1
∼= W (X)/N →H2
M2/{e} ∼= H2
M2.
We can associate to a given group different ultra-groups. In the follow-
ing, we are going to find the presentations for the distinct ultra-groups
which are assign to the dihedral group Dn.
“adm-n4” — 2020/1/24 — 13:02 — page 315 — #165
B. Tolue, P. Zolfaghari, Gh. Moghaddasi 315
Let Dn = 〈a, b : an = b2 = e, ab = a−1〉 be the dihedral group of order
2n and H = 〈ad〉 its subgroup, where d|n. We know all ultra-groups over
the subgroup H of the group Dn are isomorphic to
{e, a, a2, . . . , ad−1, b, ab, . . . , ad−1b}.
We call it dihedral ultra-group over H and denote it by HDn.
Recall that the order of the ultra-group HM is the number of the
elements in its underlying set which is equal to |G|/|H| and we denote it
by |HM |.
Example 1. Let HM be an ultra-group with generators x, y and the
relations [. . . [x, x], x], . . . , x]
︸ ︷︷ ︸
d times x
= e and [y, y] = e, where [ , ] denotes the
first binary operation of the ultra-group HM . Since HDn, the dihedral ultra-
group of order 2d, is generated by a, b satisfies the relations, by Theorem
3 we have the ultra-group epimorphism ϕ : HM → HDn. Therefore,
|HM | > |HDn| = 2d.
Now, consider the free ultra-group W (X) on the set X = {x, y} and
its normal subultra-group S generated by [. . . [x, x], x, . . . , x]
︸ ︷︷ ︸
d times x
and [y, y].
All the elements of W (X)/S are the form
[
S, [[. . . [x, x], x], . . .], x]
︸ ︷︷ ︸
i times x
, y]
]
,
where 0 6 i 6 d − 1. Since, we can show that every element in W (X)
by [[. . . [x, x], x], . . .], x]
︸ ︷︷ ︸
i times x
, y] such that 0 6 i 6 d − 1. Thus |HM | 6 2d
which implies that ϕ is an isomorphism and HDn has the presentation
〈x, y : [. . . [x, x], x, . . . , x]
︸ ︷︷ ︸
d times x
= [y, y] = e〉.
If we change the subgroup of Dn to K = 〈ad, b〉, then the ultra-group
over the subgroup K of Dn is KDn = {e, ab, a2b, . . . , ad−1b}, where d|n.
The following table is its binary operation α.
Example 2. Suppose HM is an ultra-group with generators x and y
and relations [x, x] = [y, y] = [x, [x, [x, y]]] = [[[x, y], y], x] = e. By the
Table 1 of the binary operation α of KDn, it is clear that ab and a3b
are generators of KDn and satisfy the relations of HM . Therefore, by
Theorem 3 we have the ultra-group epimorphism ϕ : HM → KDn and
|HM | > d. Now, consider the free ultra-group W (X) on the setX = {x, y}
“adm-n4” — 2020/1/24 — 13:02 — page 316 — #166
316 Free ultra-groups, generators and relations
Table 1.
α e ab a2b a3b · · · ad−1b
e e ab a2b a3b · · · ad−2b
ab ab e ab a2b · · · ad−1b
a2b a2b ad−1b e ab · · · ad−3b
a3b a3b ad−2b ad−3b e · · · ad−4b
...
...
...
...
...
...
...
ad−1b ad−1b a2b a3b e · · · e
and its normal subultra-group S generated by [x, x], [y, y], [x, [x, [x, y]]]
and [[[x, y], y], x]. Thus every element of W (X)/S is of the form [S,w]
such that w = [[x, y], x], . . . , x]
︸ ︷︷ ︸
at most d positions
. Thus |HM | 6 d which implies that ϕ
is an isomorphism and KDn has the presentation 〈x, y : [x, x] = [y, y] =
[x, [x, [x, y]]] = [[[x, y], y], x] = e〉.
References
[1] S. Burris, H. P. Sankapanavar, A Course in Universal Algebra, Springer, 1981.
[2] T. W. Hungerford, Algebra, Springer, New York, One of the standard graduate
algebra texts, 1974.
[3] Gh. Moghaddasi, B. Tolue and P. Zolfaghari, On the structure of the ultra-groups
over a finite group, U.P.B. Sei. Bull., Series A. Vol. 78 2, 2016, pp.173-184.
[4] B. Tolue, Gh. Moghaddasi and P. Zolfaghari, On the category of ultra-groups,
Hacettepe Journal of Mathematics and Statistics, Vol. 46 3, 2017, pp.1-11.
Contact information
Gh. Moghaddasi,
B. Tolue
Department of Pure Mathematics, Hakim
Sabzevari University, Sabzevar, Iran
E-Mail(s): r.moghadasi@hsu.ac.ir,
b.tolue@gmail.com
Web-page(s): http://profs.hsu.ac.ir/tolue/
P. Zolfaghari Department of Pure Mathematics, Farhangian
University, Mashhad, Iran
E-Mail(s): p.zolfaghari@cfu.ac.ir
Received by the editors: 10.06.2017
and in final form 26.09.2017.
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