Semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups
In this paper new criteria of semi-commutativity and results on self-coincidence of an arbitrary point P in the terms of properties of vectors of n-ary groups are obtained.
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| Опубліковано в: : | Algebra and Discrete Mathematics |
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| Дата: | 2010 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Інститут прикладної математики і механіки НАН України
2010
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups / Yu.I. Kulazhenko// Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 96–105. — Бібліогр.: 9 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860026654048387072 |
|---|---|
| author | Kulazhenko, Yu.I. |
| author_facet | Kulazhenko, Yu.I. |
| citation_txt | Semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups / Yu.I. Kulazhenko// Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 96–105. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | In this paper new criteria of semi-commutativity and results on self-coincidence of an arbitrary point P in the terms of properties of vectors of n-ary groups are obtained.
|
| first_indexed | 2025-12-07T16:50:28Z |
| format | Article |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 9 (2010). Number 2. pp. 96 – 105
c© Journal “Algebra and Discrete Mathematics”
Semi-commutativity criteria and self-coincidence
elements expressed by vectors properties of n-ary
groups
Yu. I. Kulazhenko
Communicated by Kirichenko V.V.
Abstract. In this paper new criteria of semi-commutativity
and results on self-coincidence of an arbitrary point P in the terms
of properties of vectors of n-ary groups are obtained.
It is well known that the most important tool for investigation of n-ary
groups and for development of their applications is the concept of semi-
commutativity. In this connection see for example [1, 2, 3, 4, 5, 6, 7, 8].
In the paper [9] P.A. Alexandrov introduced the concept of self-
coincidence for geometric figures. He used this concept to construct differ-
ent types of groups.
The results by S.A. Rusakov [5] and P.S. Alexandrov [9] allowed to
introduce the concept of self-coincidence of points (of elements) of an
n-ary group G.
Finding of new semi-commutativity criteria of n-ary groups as well
as the study of self-coincidence of some elements of geometric figures
constructed on the basis of an n-ary group is a very topical problem in
our opinion.
The results presented in the paper are connected with the above-
mentioned field of investigation. It should be noted that vector equalities
which are presented in our theorems not only describe semi-commutativity
criteria of an n-ary group G = 〈X, ( ),[−2] 〉 but establish the fact of
self-coincidence of an arbitrary point p ∈ X as well.
Recall that an n-ary group G is said to be semi-abelian if the equality
(x1x
n−1
2 xn) = (xnx
n−1
2 x1)
2000 Mathematics Subject Classification: ????.
Key words and phrases: ????.
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Yu. I. Kulazhenko 97
holds for any sequence xn1 ∈ Xn. Further for the elements of an n-ary
group G = 〈X, ( ),[−2] 〉 we use the term a point.
A point
Sa(b) = (ab[−2]
2n−4
b a)
is called a point that is symmetric with a point b relatively a point a. The
sequence of k elements of X is called a k-gon of G. A tetragon 〈a, b, c, d〉
of an n-ary group G is called a parallelogram of G if
(ab[−2]
2n−4
b c) = d.
Let’s say that a point p ∈ X self-coincides if there is a sequence of
symmetries of this point relatively other points of X, in the result of which
this point maps into itself.
An ordered pair 〈a, b〉 of points a, b ∈ X is called a directed segment
of an n-ary group G and it is denoted by ab.
If a, b, c, d ∈ X, then the directed segments ab and cd are called to be
equal and they write ab = cd if the tetragon 〈a, c, d, b〉 is a parallelogram
of G.
Let V be the set of all directed segments of an n-ary group G. According
to Proposition 1 in the paper [5] the binary relation = on the set V is a
relation of equivalence and partitions the set V into disjoint classes. The
class generated by the directed segment ab has the following form:
K(ab) = {uv | uv ∈ V , uv = ab}.
A vector
−→
ab of an n-ary group G is a class K(ab), i.e.
−→
ab = K(ab).
Other notations, definitions and results used in the paper can be found
in the following papers [4, 5, 6, 7, 8].
Now let us introduce the obtained results.
Theorem 1. Let a, b, c, p be arbitrary points of X and d ∈ X be a
point such that the tetragon 〈a, b, c, d〉 is a parallelogram of G. An n-ary
group G is semi-abelian if and only if the following equality holds:
−→pa+
−−−−→
Sa(p)b+
−−−−−−→
Sb(Sap))c+
−−−−−−−−−−→
Sc(Sb(Sa(p)))d =
−→
0 . (1)
Proof. 1. Let G be a semi-abelian n-ary group. Let’s establish the validity
of (1).
Taking into account Theorem 8 in [8], Definition 4 in [5], Proposition 1
in [8], Equality 3.28 in [4], and the fact that for any x ∈ X sequences
x[−2]2n−4
x and x[−2]2n−4
x x are neutral 2(n− 1)-sequences the following can
be obtained:
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98 Semi-commutativity criteria
−→pa+
−−−−→
Sa(p)b =
−−−−−−−−−−−−−−−−−−→
p(a(Sa(p))
[−2] Sa(p) . . .
︸ ︷︷ ︸
2n−4
b) =
=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(a(ap[−2]2n−4
p a)[−2] (ap[−2]2n−4
p a) . . .
︸ ︷︷ ︸
2n−4
b) =
=
−−−−−−−−−−−−−−−−−→
p(aa[−2]2n−4
a pa[−2]2n−4
a b) =
−−−−−−−−−−→
p(pa[−2]2n−4
a b). (2)
Taking into account (2) one can obtain
−→pa+
−−−−→
Sa(p)b+
−−−−−−−→
Sb(Sa(p))c =
−−−−−−−−−−→
p(pa[−2]2n−4
a b) +
−−−−−−−−−−−→
Sb(ap
[−2]2n−4
p a)c =
=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
p((pa[−2]2n−4
a b)(Sb(ap
[−2]2n−4
p a))[−2] Sb(ap
[−2]2n−4
p a) . . .
︸ ︷︷ ︸
2n−4
c) =
= p((a[−2]2n−4
a b)(b(ap[−2]2n−4
p a)[−2] (ap[−2]2n−4
p a) . . .
︸ ︷︷ ︸
2n−4
b)[−2]
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
(b(ap[−2]2n−4
p a)[−2] (ap[−2]2n−4
p a) . . .
︸ ︷︷ ︸
2n−4
b) . . .
︸ ︷︷ ︸
2n−4
c) =
= p((pa[−2]2n−4
a b)(ba[−2]2n−4
a pa[−2]2n−4
a b)[−2]
−−−−−−−−−−−−−−−−−−−−−→
(ba[−2]2n−4
a pa[−2]2n−4
a b) . . .
︸ ︷︷ ︸
2n−4
c) =
=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(pa[−2]2n−4
a bb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c) =
−−−−−−−−−→
p(ab[−2]
2n−4
b c). (3)
Now taking into account Definition 4 in [5], Equality 3.28 in [4],
Proposition 1 in [8] we have
Sc(Sb(Sa(p))) = Sc(Sb(ap
[−2]2n−4
p a)) =
= Sc(b(ap
[−2]2n−4
p a)[−2] (ap[−2]2n−4
p a) . . .
︸ ︷︷ ︸
2n−4
b) =
= Sc(ba
[−2]2n−4
a pa[−2]2n−4
a b) =
= (c(ba[−2]2n−4
a pa[−2]2n−4
a b)[−2] (ba[−2]2n−4
a pa[−2]2n−4
a b) . . .
︸ ︷︷ ︸
2n−4
c) =
= (cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c). (4)
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Yu. I. Kulazhenko 99
Taking into consideration (3) and (4), the fact that the tetragon 〈a, b, c, d〉
is a parallelogram of G and that G is a semi-abelian group one can obtain
−→pa+
−−−−→
Sa(p)b+
−−−−−−−→
Sb(Sa(p))c+
−−−−−−−−−−→
Sc(Sb(Sa(p)))d =
=
−−−−−−−−−→
p(ab[−2]
2n−4
b c) +
−−−−−−−−−−−−−−−−−−−−−−−−−→
(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c)d =
= p((ab[−2]
2n−4
b c)(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c)[−2]
−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c) . . .
︸ ︷︷ ︸
2n−4
d) =
=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(ab[−2]
2n−4
b cc[−2]2n−4
c ba[−2]2n−4
a pa[−2]2n−4
a bc[−2]2n−4
c d) =
=
−−−−−−−−−−−−−−−−−→
p(pa[−2]2n−4
a bc[−2]2n−4
c d) =
−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(pa[−2]2n−4
a bc[−2]2n−4
c (ab[−2]
2n−4
b c)) =
=
−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(pa[−2]2n−4
a bc[−2]2n−4
c (cb[−2]
2n−4
b a)) = −→pp =
−→
0 .
Thus we proved the equality (1).
2. Now we suppose that (1) is true. We shall prove that G is semi-
abelian.
From (1) we have
−→pa+
−−−−→
Sa(p)b+
−−−−−−−→
Sb(Sa(p))c = −
−−−−−−−−−−→
Sc(Sb(Sa(p)))d
and so
−→pa+
−−−−→
Sa(p)b+
−−−−−−−→
Sb(Sa(p))c =
−−−−−−−−−−→
dSc(Sb(Sa(p))).
Therefore from (3) and (4) we have
−−−−−−−−−→
p(ab[−2]
2n−4
b c) =
−−−−−−−−−−−−−−−−−−−−−−−−−→
d(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c). (5)
From (5) on the basis of Definition 2 in [5] we conclude that the tetragon
〈p, d, (cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c), (ab[−2]
2n−4
b c)〉
is a parallelogram of G, so the equality
(pd[−2]
2n−4
d (cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c)) = (ab[−2]
2n−4
b c). (6)
holds.
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100 Semi-commutativity criteria
Since by the hypothesis the tetragon 〈a, b, c, d〉 is a parallelogram of
G the equality
(ab[−2]
2n−4
b c) = d. (7)
is valid.
In view of (7) we obtain from (6) that
(p(ab[−2]
2n−4
b c)[−2] (ab[−2]
2n−4
b c) . . .
︸ ︷︷ ︸
2n−4
(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c)) =
= (ab[−2]
2n−4
b c)
and hence
(pc[−2]2n−4
c ba[−2]2n−4
a (cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c)) = (ab[−2]
2n−4
b c).
Therefore
(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c) = (ab[−2]
2n−4
b cp[−2]2n−4
p ab[−2]
2n−4
b c),
so
(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b cc[−2]2n−4
c ba[−2]2n−4
a p) = (ab[−2]
2n−4
b c).
Then
(cb[−2]
2n−4
b a) = (ab[−2]
2n−4
b c). (8)
Since a, b, c are arbitrary points of X then on the basis of Proposition 4
in [7] and (8) we conclude that G is a semi-abelian n-ary group.
The proof is complete.
Theorem 2. Let a, b, c, d, p be arbitrary points of X. An n-ary group
G is semi-abelian if and only if the following equality holds:
−→pa+
−−−−→
Sa(p)b+
−−−−−−−→
Sb(Sa(p))c+
−−−−−−−−−−→
Sc(Sb(Sa(p)))d+
+
−−−−−−−−−−−−−−−−−−−−−−→
Sd(Sc(Sb(Sa(p))))(dc
[−2]2n−4
c b)+
+
−−−−−−−−−−−−−−−−−−−−−−−→
S
(dc[−2]2n−4
c b)
(Sd(Sc(Sb(Sa(p))))) =
−→
0 . (9)
Proof. 1. Let G be a semi-abelian n-ary group. We shall show that
Equality (9) is true. In order to prove this we sequentially summarize
vectors mentioned in (9) taking into account Theorem 8 in [8], Definition
4 in [5], Equality 3.28 in [4], Proposition 1 in [8], and the fact that for
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Yu. I. Kulazhenko 101
any x ∈ X the sequences x[−2]2n−4
x x and xx[−2]2n−4
x are neutral 2(n− 1)-
sequences.
So we have
−→pa+
−−−−→
Sa(p)b =
−−−−−−−−−−−−−−−−−→
p(a(Sa(p))
[−2] Sa(p) . . .
︸ ︷︷ ︸
2n−4
) =
=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(a(ap[−2]2n−4
p a)[−2] (ap[−2]2n−4
p a)) . . .
︸ ︷︷ ︸
2n−4
b) =
=
−−−−−−−−−−−−−−−−−→
p(aa[−2]2n−4
a pa[−2]2n−4
a b) =
−−−−−−−−−−→
p(pa[−2]2n−4
a b);
−→pa+
−−−−→
Sa(p)b+
−−−−−−−→
Sb(Sa(p))c =
−−−−−−−−−−→
p(pa[−2]2n−4
a b) +
−−−−−−−→
Sb(Sa(p))c =
=
−−−−−−−−−−→
p(pa[−2]2n−4
a b) +
−−−−−−−−−−−−−−−−−−→
(b(Sa(p))
[−2] Sa(p) . . .
︸ ︷︷ ︸
2n−4
b)c =
=
−−−−−−−−−−→
p(pa[−2]2n−4
a b) +
−−−−−−−−−−−−−−−−−→
(ba[−2]2n−4
a pa[−2]2n−4
a b)c =
= p((pa[−2]2n−4
a b)(ba[−2]2n−4
a pa[−2]2n−4
a b)[−2]
−−−−−−−−−−−−−−−−−−−−−→
(ba[−2]2n−4
a pa[−2]2n−4
a b) . . .
︸ ︷︷ ︸
2n−4
c) =
=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(pa[−2]2n−4
a bb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c) =
−−−−−−−−−→
p(ab[−2]
2n−4
b c). (10)
Taking into account Definition 4 in [5], Equality 3.28 in [4], and
Proposition 1 in [8] we have
Sc(Sb(Sa(p))) = Sc(Sb(ap
[−2]2n−4
p a)) =
= Sc(b(ap
[−2]2n−4
p a)[−2] (ap[−2]2n−4
p a) . . .
︸ ︷︷ ︸
2n−4
b) =
= Sc(ba
[−2]2n−4
a pa[−2]2n−4
a b) =
= (c(ba[−2]2n−4
a pa[−2]2n−4
a b)[−2] (ba[−2]2n−4
a pa[−2]2n−4
a b) . . .
︸ ︷︷ ︸
2n−4
c) =
= (cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c). (11)
Hence in view of (10) and (11) we obtain
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102 Semi-commutativity criteria
−→pa+
−−−−→
Sa(p)b+
−−−−−−−→
Sb(Sa(p))c+
−−−−−−−−−−→
Sc(Sb(Sa(p)))d =
=
−−−−−−−−−→
p(ab[−2]
2n−4
b c) +
−−−−−−−−−−−−−−−−−−−−−−−−−→
(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c)d =
p((ab[−2]
2n−4
b c)(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c)[−2]
−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c) . . .
︸ ︷︷ ︸
2n−4
d) =
=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(ab[−2]
2n−4
b cc[−2]2n−4
c ba[−2]2n−4
a pa[−2]2n−4
a bc[−2]2n−4
c d) =
=
−−−−−−−−−−−−−−−−−→
p(pa[−2]2n−4
a bc[−2]2n−4
c d). (12)
Taking into consideration (11) and the previous arguments we have
Sd(Sc(Sb(Sa(p)))) =
= (d(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c)[−2]
(cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b c) . . .
︸ ︷︷ ︸
2n−4
d) =
=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
(dc[−2]2n−4
c ba[−2]2n−4
a pa[−2]2n−4
a bc[−2]2n−4
c d). (13)
Taking into account (12) and (13) we have
−→pa+
−−−−→
Sa(p)b+
−−−−−−−→
Sb(Sa(p))c+
−−−−−−−−−−→
Sc(Sb(Sa(p)))d+
+
−−−−−−−−−−−−−−−−−−−−−−→
Sd(Sc(Sb(Sa(p))))(dc
[−2]2n−4
c b) =
−−−−−−−−−−−−−−−−−→
p(pa[−2]2n−4
a bc[−2]2n−4
c d)+
+ (dc[−2]2n−4
c ba[−2]2n−4
a pa[−2]2n−4
a bc[−2]2n−4
c d)[−2]
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
(dc[−2]2n−4
c ba[−2]2n−4
a pa[−2]2n−4
a bc[−2]2n−4
c d) . . .
︸ ︷︷ ︸
2n−4
(dc[−2]2n−4
c b)) =
= p(pa[−2]2n−4
a bc[−2]2n−4
c dd[−2]
2n−4
d cb[−2]
2n−4
b ap[−2]2n−4
p
−−−−−−−−−−−−−−−−−−−−−−−→
ab[−2]
2n−4
b cd[−2]
2n−4
d dc[−2]2n−4
c b) = −→pa. (14)
But G is semi-ablelian and so in view of (13) we have
S
(dc[−2]2n−4
c b)
(Sd(Sc(Sb(Sa(p))))) =
= ((dc[−2]2n−4
c b)(dc[−2]2n−4
c ba[−2]2n−4
a pa[−2]2n−4
a bc[−2]2n−4
c d)[−2]
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Yu. I. Kulazhenko 103
(dc[−2]2n−4
c ba[−2]2n−4
a pa[−2]2n−4
a bc[−2]2n−4
c d) . . .
︸ ︷︷ ︸
2n−4
(dc[−2]2n−4
c b)) =
= ((bc[−2]2n−4
c d)d[−2]
2n−4
d cb[−2]
2n−4
b ap[−2]2n−4
p ab[−2]
2n−4
b cd[−2]
2n−4
d
dc[−2]2n−4
c b) = (ap[−2]2n−4
p a). (15)
Finally using (14) and (15) we obtain
−→pa+
−−−−→
Sa(p)b+
−−−−−−−→
Sb(Sa(p))c+
+
−−−−−−−−−−→
Sc(Sb(Sa(p)))d+
−−−−−−−−−−−−−−−−−−−−−−→
Sd(Sc(Sb(Sa(p))))(dc
[−2]2n−4
c b)+
+
−−−−−−−−−−−−−−−−−−−−−−−−→
S
(dc[−2]2n−4
c b)
(Sd(Sc(Sb(Sa(p)))))a = −→pa+
−−−−−−−−−−→
(ap[−2]2n−4
p a)a =
=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(a(ap[−2]2n−4
p )[−2] (ap[−2]2n−4
p a) . . .
︸ ︷︷ ︸
2n−4
a) =
=
−−−−−−−−−−−−−−−−−−→
p(aa[−2]2n−4
a pa[−2]2n−4
a a) = −→pp =
−→
0 .
Consequently we have proved that (9) holds.
2. Suppose that (9) is true. We shall prove that G is a semi-abelian
group.
Since in the previous arguments the property of semi-commutativity
of G was used only in (15) we can conclude that (14) holds.
From (9) we obtain
−→pa+
−−−−−−−−−−−−−−−−−−−−−−−−→
S
(dc[−2]2n−4
c b)
(Sd(Sc(Sb(Sa(p)))))a =
−→
0
so taking into account (13) we have
−→pa+
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
S
(dc[−2]2n−4
c b)
(dc[−2]2n−4
c ba[−2]2n−4
a pa[−2]2n−4
a bc[−2]2n−4
c d)a =
−→
0 ,
so
−→pa+ ((dc[−2]2n−4
c b)(dc[−2]2n−4
c ba[−2]2n−4
a pa[−2]2n−4
a bc[−2]2n−4
c d)[−2]
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
(dc[−2]2n−4
c ba[−2]2n−4
a pa[−2]2n−4
a bc[−2]2n−4
c d) . . .
︸ ︷︷ ︸
2n−4
(dc[−2]2n−4
c b))a =
−→
0 ,
and hence
−→pa+ ((dc[−2]2n−4
c b)d[−2]
2n−4
d cb[−2]
2n−4
b ap[−2]2n−4
p
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104 Semi-commutativity criteria
−−−−−−−−−−−−−−−−−−−−−−−−→
ab[−2]
2n−4
b cd[−2]
2n−4
d dc[−2]2n−4
c b)a =
−→
0 ,
or
−→pa+
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
((dc[−2]2n−4
c b)d[−2]
2n−4
d cb[−2]
2n−4
b ap[−2]2n−4
p a)a =
−→
0 .
Then taking into consideration Theorem 8 in [8] we have
p(a(dc[−2]2n−4
c bd[−2]
2n−4
d cb[−2]
2n−4
b ap[−2]2n−4
p a)[−2]
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
(dc[−2]2n−4
c bd[−2]
2n−4
d cb[−2]
2n−4
b ap[−2]2n−4
p a) . . .
︸ ︷︷ ︸
2n−4
a) =
−→
0 ,
so
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(aa[−2]2n−4
a pa[−2]2n−4
a bc[−2]2n−4
c db[−2]
2n−4
b cd[−2]
2n−4
d a) =
−→
0 ,
and hence
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(pa[−2]2n−4
a bc[−2]2n−4
c db[−2]
2n−4
b cd[−2]
2n−4
d a) =
−→
0 .
Since
−→
0 = −→pp holds for any p ∈ X, then
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
p(pa[−2]2n−4
a bc[−2]2n−4
c db[−2]
2n−4
b cd[−2]
2n−4
d a) = −→pp.
From this equality it can be concluded that
(pa[−2]2n−4
a bc[−2]2n−4
c db[−2]
2n−4
b cd[−2]
2n−4
d a) = p.
Let’s multiply both parts of this equality by the expression ap[−2]2n−4
p
from the left, and by the expression a[−2]2n−4
a dc[−2]2n−4
c b from the right.
Then
(ap[−2]2n−4
p pa[−2]2n−4
a (bc[−2]2n−4
c d)b[−2]
2n−4
b cd[−2]
2n−4
d aa[−2]2n−4
a
dc[−2]2n−4
c b) = (ap[−2]2n−4
p pa[−2]2n−4
a dc[−2]2n−4
c b).
Hence taking into account the neutrality of the sequences x[−2]2n−4
x x and
xx[−2]2n−4
x for any x ∈ X we obtain
(bc[−2]2n−4
c d) = (dc[−2]2n−4
c b). (16)
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Yu. I. Kulazhenko 105
Taking into consideration the arbitrariness of points b, c, d ∈ X on the
basis of Proposition 4 in [7] and (16) we conclude that G is a semi-abelian
n-ary group.
The proof is complete.
References
[1] W. Dornte. Untersuchungn uber einen verallgemeinerten Gruppenbergriff // Math.
Z. — 1928. — Bd. 29. — S. 1–19.
[2] H. Prüfer. Theorie der abelshen Gruppen I. Grundeigenschaften, Math. Z. — 1924.
— Bd. 20. — S. 165–187.
[3] E.L. Post. Polyadic groups // Trans. Amer. Math. Soc. — 1940. — Vol. 48, N 2. —
P. 208–350.
[4] S.A. Rusakov. Algebraic n-ary systems: Sylow theory of n-ary groups. Minsk:
Belaruskaya navuka, 1992. — 264 p.
[5] S.A. Rusakov. Some applications of the theory of n-ary groups. Minsk: Belaruskaya
navuka, 1998. — P.182.
[6] Yu.I. Kulazhenko. Congruence motion of elements of n-ary groups. Some questions
of algebra and applied mathematics: Compilation of scientific proceedings / Ed.
by T.I. Vasiljeva, Gomel, 2002. — P. 66–71.
[7] Yu.I. Kulazhenko. Construction of figures of affine geometry on n-ary groups //
Questions of algebra and applied mathematics: Compilation of scientific proceed-
ings / Ed. by S.A. Rusakov, Gomel, 1995. — P. 65–82.
[8] Yu.I. Kulazhenko. Parallelograms’ geometry / Questions of algebra and applied
mathematics: Compilation of scientific proceedings / Ed. by S.A. Rusakov, Gomel,
1995. — P. 47–64.
[9] P.S. Alexandrov. Introduction into the theory of groups. M.: Nauka, 1980. — 144 p.
Received by the editors: ????
and in final form ????.
Yu. I. Kulazhenko
|
| id | nasplib_isofts_kiev_ua-123456789-154576 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T16:50:28Z |
| publishDate | 2010 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Kulazhenko, Yu.I. 2019-06-15T16:34:59Z 2019-06-15T16:34:59Z 2010 Semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups / Yu.I. Kulazhenko// Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 96–105. — Бібліогр.: 9 назв. — англ. 1726-3255 https://nasplib.isofts.kiev.ua/handle/123456789/154576 In this paper new criteria of semi-commutativity and results on self-coincidence of an arbitrary point P in the terms of properties of vectors of n-ary groups are obtained. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups Article published earlier |
| spellingShingle | Semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups Kulazhenko, Yu.I. |
| title | Semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups |
| title_full | Semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups |
| title_fullStr | Semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups |
| title_full_unstemmed | Semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups |
| title_short | Semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups |
| title_sort | semi-commutativity criteria and self-coincidence elements expressed by vectors properties of n-ary groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/154576 |
| work_keys_str_mv | AT kulazhenkoyui semicommutativitycriteriaandselfcoincidenceelementsexpressedbyvectorspropertiesofnarygroups |