Dickson's theorem for Bol loops
Dickson characterized groups in terms of one-sided invertibility. In this note, we give comparable characterizations for Bol and Moufang loops.
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Інститут прикладної математики і механіки НАН України
2017
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| Цитувати: | Dickson's theorem for Bol loops / Y. Movsisyan // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 2. — С. 297-301. — Бібліогр.: 13 назв. — англ. |
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irk-123456789-1566392019-06-19T01:29:15Z Dickson's theorem for Bol loops Movsisyan, Y. Dickson characterized groups in terms of one-sided invertibility. In this note, we give comparable characterizations for Bol and Moufang loops. 2017 Article Dickson's theorem for Bol loops / Y. Movsisyan // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 2. — С. 297-301. — Бібліогр.: 13 назв. — англ. 1726-3255 2010 MSC:20N05. http://dspace.nbuv.gov.ua/handle/123456789/156639 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Dickson characterized groups in terms of one-sided invertibility. In this note, we give comparable characterizations for Bol and Moufang loops. |
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Movsisyan, Y. |
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Movsisyan, Y. Dickson's theorem for Bol loops Algebra and Discrete Mathematics |
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Movsisyan, Y. |
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Movsisyan, Y. |
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Dickson's theorem for Bol loops |
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Dickson's theorem for Bol loops |
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Dickson's theorem for Bol loops |
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Dickson's theorem for Bol loops |
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Dickson's theorem for Bol loops |
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dickson's theorem for bol loops |
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Інститут прикладної математики і механіки НАН України |
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2017 |
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Dickson's theorem for Bol loops / Y. Movsisyan // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 2. — С. 297-301. — Бібліогр.: 13 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT movsisyany dicksonstheoremforbolloops |
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2025-07-14T09:01:29Z |
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2025-07-14T09:01:29Z |
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1837612337375215616 |
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“adm-n4” 22:47 page #119
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 24 (2017). Number 2, pp. 297–301
© Journal “Algebra and Discrete Mathematics”
Dickson’s theorem for Bol loops∗
Yuri Movsisyan
Communicated by A. V. Zhuchok
Abstract. Dickson characterized groups in terms of one-
sided invertibility. In this note, we give comparable characterizations
for Bol and Moufang loops.
Introduction
Dickson [3] characterized groups as follows: a group is a semigroup Q
which possesses a left (right) unit, under which every element of Q is
left (right) invertible. In this note we prove the similar result for the left
(right) Bol loops ([1],[2],[5]–[12]). As a consequence we formulate a similar
result for the Moufang loops.
Definition 1. A groupoid Q(·) is called:
1) a right division groupoid (right quasigroup) if the equation a · x = b
has a solution (unique solution) x ∈ Q for every a, b ∈ Q;
2) a left division groupoid (left quasigroup) if the equation y · a = b
has a solution (unique solution) y ∈ Q for every a, b ∈ Q;
3) a quasigroup (division groupoid) if Q(·) is a right and left quasigroup
(right and left division groupoid);
∗This research was supported by the State Committee of Science of the Republic of
Armenia, grants: 10-3/1-41 and 15T-1A258.
2010 MSC: 20N05.
Key words and phrases: left (right) Bol identity, left (right) quasigroup, quasi-
group, loop, left (right) Moufang identity.
“adm-n4” 22:47 page #120
298 Dickson’s theorem for Bol loops
4) left cancellative if the equation a · x = b has no more than one
solution x ∈ Q for every a, b ∈ Q, i.e. for arbitrary a, x1, x2 ∈ Q:
a · x1 = a · x2 −→ x1 = x2;
5) right cancellative if the equation y · a = b has no more than one
solution x ∈ Q for every a, b ∈ Q, i.e. for arbitrary a, y1, y2 ∈ Q:
y1 · a = y2 · a −→ y1 = y2;
6) a loop if Q(·) is a quasigroup with a unit;
7) a left Bol loop if Q(·) is a loop satisfying the left Bol identity:
(x · yx)z = x(y · xz); (1)
8) a right Bol loop if Q(·) is a loop satisfying the right Bol identity:
z(xy · x) = (zx · y)x; (2)
9) a Moufang loop if Q(·) is a loop satisfying one of the following left
and right Moufang identities:
x(y · xz) = (xy · x)z, (3)
(zx · y)x = z(x · yx); (4)
10) right (left) alternative if Q(·) satisfies the right (left) alternative
identity:
y(x · x) = (y · x)x, (x · x)y = x(x · y);
11) alternative if Q(·) is a right and left alternative.
It is well known that the Moufang identities (3) and (4) are equivalent
in the class of loops ([1], [6], [9]). For applications of right (left) quasigroup
operations in geometry and topology (knot theory) see [4,8,10]. For right
(left) loops see [13].
1. The main result
Let us start at the following auxiliary result.
Lemma 1. 1) Every idempotent element of a left alternative and right
division groupoid Q(·) is a left unit of Q(·);
2) Every idempotent element of a right alternative and left division
groupoid Q(·) is a right unit of Q(·);
3) Every idempotent element of an alternative and division groupoid
is a unit of Q(·).
“adm-n4” 22:47 page #121
Yu. Movsisyan 299
Proof. 1) Let e ∈ Q be an idempotent element of Q(·), i.e. e · e = e. For
any b ∈ Q there exist x ∈ Q such that e · x = b, then by left alternativity:
e · b = e(e · x) = (e · e)x = e · x = b.
Theorem 1. 1) Let Q(·) be a left cancellative groupoid with a right unit,
satisfying the left Bol identity (1).
If every element of Q is a right invertible under one of the right units,
then Q(·) is a loop (left Bol loop).
2) Let Q(·) be a right cancellative groupoid with a left unit, satisfying
the right Bol identity (2).
If every element of Q is a left invertible under one of the left units,
then Q(·) is a loop (right Bol loop).
Proof. 1) Let e be a right unit of Q(·) and for every a ∈ Q there exist an
element a′
∈ Q that a · a′ = e. Let us denote: a′′ = (a′)′.
First, we prove that a′
· a = e. Namely, by the Bol left identity (1) we
have:
a′
· (a · a′) = a′
· e = a′,
a′
· a = a′
· (a · e) = a′
· (a · (a′
· a′′)) = (a′
· (a · a′)) · a′′ =
= (a′
· e) · a′′ = a′
· a′′ = e.
It is evident the uniqueness of the inverse element a′
∈ Q by left
cacellativity of Q(·). Hence, a′′ = a. Moreover, by the left Bol identity (1)
we obtain:
(x · x′x)z = x(x′
· xz) and (x · e)z = x(x′
· xz),
i.e.
x · z = x(x′
· xz).
According to the left cancellative property, we obtain: z = x′
· xz for
arbitrary x, z ∈ Q.
For any a, b ∈ Q the unique solution of a · x = b is the x = a′
· b.
Indeed, if x = a′
· b, then a · x = b, since a′′(a′
· b) = b and a′′ = a.
Let us consider the equation: y · a = b, where a, b ∈ Q. If y · a = b,
then (a · ya)a′ = ab · a′, i.e. a(y · aa′) = ab · a′ by the left Bol identity (1),
or ay = ab · a′, y = a′(ab · a′). And from y = a′(ab · a′) it follows that
y · a = b by the left Bol identity (1):
y · a = (a′
· (ab · a′)) · a = a′(ab · a′a) = a′(ab · e) = a′(ab) = b.
“adm-n4” 22:47 page #122
300 Dickson’s theorem for Bol loops
Hence, Q(·) is a quasigroup.
In order to prove that the right unit e ∈ Q is a unit of Q(·), first we
note that by setting y = e in the left Bol identity (1), we immediately
obtain the left alternative law: (x · x)z = x(x · z). Now, according to the
previous Lemma 1, every idempotent element of Q(·) is a left unit of Q(·).
In particular, the right unit e ∈ Q is a unit of Q(·).
2) The proof is dual to the proof of 1).
Corollary 1. 1) Let Q(·) be a left cancellative groupoid with a right unit,
satisfying the left Moufang identity (3).
If every element of Q is a right invertible under one of the right units,
then Q(·) is a loop (Moufang loop).
2) Let Q(·) be a right cancellative groupoid with a left unit, satisfying
the right Moufang identity (4).
If every element of Q is a left invertible under one of the left units,
then Q(·) is a loop (Moufang loop).
Proof. 1) Note, that by setting z = e (which is a right unit of Q(·)) in
the left Moufang identity (3) we obtain the flexible law: (xy)x = x(yx).
Hence, the identity (3) is converted to the left Bol identity (1). So we can
use the result 1) of the previous Theorem 1.
2) The proof is similar to the proof of 1).
Corollary 2. 1) Every left cancellative, alternative and division groupoid
with the left Bol identity (1) is a loop (Moufang loop).
2) Every right cancellative, alternative and division groupoid with the
right Bol identity (2) is a loop (Moufang loop).
Proof. 1) For any element a ∈ Q there exists an element ea ∈ Q such
that a · ea = a, since Q(·) is a division groupoid. Applying this equality
in the identity of right alternativity, we find:
a · e2
a
= (a · ea) · ea = a · ea,
i.e. ea is an idempotent element of Q(·): e2
a
= ea by left cancellativity
of Q(·).
Hence, Q(·) has a unit by the previous Lemma 1. Consequently, we
can use the result 1) of the previous theorem. Thus, Q(·) is a left Bol
loop. Then the right alternative low together with the left Bol identity
(1) implies that:
(x · yx)x = x(y · xx) = x(yx · x).
“adm-n4” 22:47 page #123
Yu. Movsisyan 301
Setting yx = z, we get the flexible law: (xz)x = x(zx), which implies that
the left Bol loop Q(·) is a Moufang loop.
2) The proof is similar to the proof of 1).
References
[1] V.D. Belousov, Foundations of The Theory of Quasigroups and Loops, Nauka,
Moscow, 1967 (Russian).
[2] G. Bol, Gewebe und Gruppen, Math. Ann., N.114, 1937, pp.414-432.
[3] L.E. Dickson, Definitions of a group and a field by independent postulates, Trans.
Amer. Math. Soc., N.6, 1905, pp.198-204.
[4] S.V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119(1) (1982),
pp.78-88.
[5] R. Moufang, Zur struktur von Alternativkörpern, Math. Ann., N.10, 1935, pp.416-
430.
[6] Yu.M. Movsisyan, Introduction of The Theory of Algebras with Hyperidentities,
Yerevan State University Press, Yerevan, 1986 (Russian).
[7] Yu.M. Movsisyan, Hyperidentitties in algebras and varieties, Uspekhi Mat. Nauk.,
53(1),1998,61–114; (Russian) English transl. in Russian Math. Surveys 53, 1998,
57–108.
[8] V. Pambuccian, Euclidean geometry problems rephrased in terms of midpoints and
point-reflections, Elem. Math., 60(2005), pp.19-24.
[9] H.O. Pflugfelder, Quasigroups and Loops, Introduction, Helderman Verlag Berlin,
1990.
[10] J. H. Przytycki, Knot and distributive homology: from arc colorings to Yang-Baxter
homology, in: New Ideas in Low Dimensional Topology, 2015, pp. 413-488.
[11] I.P. Shestakov, A.I. Shirshov, A.M. Slinko, K.A. Zhevlakov, Rings That are Nearly
Associative, Nauka, Moscow, 1978 (Russian); English transl. Academic Press, New
York, 1982.
[12] L.A. Skornyakov, Right alternative fields, Izvestia Akad. Nauk SSSR, Ser. Mat.,
N.15, 1951, pp.177-184.
[13] J.D.H. Smith, A.B. Romanowska, Post-Modern Algerbra, Wiley, New York, 1999.
Contact information
Yu. Movsisyan Yerevan State University, Alex Manoogian 1,
Yerevan 0025, Armenia
E-Mail(s): movsisyan@ysu.am
Web-page(s): www.ysu.am
Received by the editors: 15.10.2016
and in final form 15.02.2017.
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