Geometry of flocks and n-ary groups
Semiabelinan flocks and n-ary groups are characterized by the properties of parallelograms and vectors of the affine geometry defined by these flocks and n-ary groups.
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irk-123456789-1884772023-03-03T01:27:00Z Geometry of flocks and n-ary groups Dog, S. Semiabelinan flocks and n-ary groups are characterized by the properties of parallelograms and vectors of the affine geometry defined by these flocks and n-ary groups. 2019 Article Geometry of flocks and n-ary groups / S. Dog // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 1. — С. 60–74. — Бібліогр.: 25 назв. — англ. 1726-3255 2010 MSC: 20N15, 51A25, 51D15 http://dspace.nbuv.gov.ua/handle/123456789/188477 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Semiabelinan flocks and n-ary groups are characterized by the properties of parallelograms and vectors of the affine geometry defined by these flocks and n-ary groups. |
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Geometry of flocks and n-ary groups |
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Geometry of flocks and n-ary groups |
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Geometry of flocks and n-ary groups |
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Geometry of flocks and n-ary groups |
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Geometry of flocks and n-ary groups |
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geometry of flocks and n-ary groups |
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Geometry of flocks and n-ary groups / S. Dog // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 1. — С. 60–74. — Бібліогр.: 25 назв. — англ. |
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“adm-n3” — 2019/10/20 — 9:35 — page 60 — #62
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 28 (2019). Number 1, pp. 60–74
c© Journal “Algebra and Discrete Mathematics”
Geometry of flocks and n-ary groups
Sonia Dog
Communicated by I. V. Protasov
Abstract. Semiabelinan flocks and n-ary groups are charac-
terized by the properties of parallelograms and vectors of the affine
geometry defined by these flocks and n-ary groups.
1. Introduction
If in the standard (affine) geometry is fixed point O, then any point
P of this geometry is uniquely determined by the vector ~p =
−→
OP , and
conversely, any vector
−→
OP uniquely determines the point P. Moreover,
any interval AB is interpreted as the vector ~a−~b or as the vector ~b− ~a.
In the first case,
AB = CD ⇐⇒ ~a−~b+ ~d = ~c,
or, in the other words
AB = CD ⇐⇒ f(a, b, d) = c,
where each vector ~v is treated as an element v of a commutative group
(G,+). The operation f has the form f(x, y, z) = x− y + z. Groups (also
non-commutative) with a ternary operation defined in such a way were
considered by J. Certaine [3] as a special case of ternary heaps investigated
by H. Prüfer [18]. Ternary heaps have interesting applications to projective
geometry [1], affine geometry [2], theory of nets (webs), theory of knots
and even to the differential geometry [24], [25].
2010 MSC: 20N15, 51A25, 51D15.
Key words and phrases: n-ary group, flock, symmetry, affine geometry.
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S. Dog 61
All affine geometries may be treated as geometries defined by some
n-ary relations (see, for example, [23]). The class of affine geometries
defined by n-ary groups, which are a natural generalization of the notion
of groups, was introduced by S.A. Rusakov (see [20], [21]) and in detail
described by Yu.I. Kulazhenko.
Below, using methods proposed by W.A. Dudek in his fundamental
paper [7], we give very short and elegant proofs of various Kulazhenko’s
results.
2. Preliminaries
We will use the standard notation: the sequence xi, . . . , xj will be
denoted as x
j
i (for j < i it is the empty symbol). In the case xi+1 = . . . =
xi+k = x instead of xi+k
i+1 we will write
(k)
x . Obviously
(0)
x is the empty
symbol. In this notation the formula
f(x1, . . . , xi, yi+1, . . . , yi+k, xi+k+1, . . . , xn),
where yi+1 = . . . = yi+k = y, will be written as f(xi1,
(k)
y , xni+k+1).
By an n-ary group (G, f) we mean (see [4]) a non-empty set G together
with one n-ary operation f : Gn → G satisfying for all i = 1, 2, . . . , n the
following two conditions:
10 the associative law:
f(f(xn1 ), x
2n−1
n+1 ) = f(xi−1
1 , f(xn+i−1
i ), x2n−1
n+i )
20 for all x1, x2, . . . , xn, b ∈ G there exits a unique xi ∈ G such that
f(xi−1
1 , xi, x
n
i+1) = b.
Such n-ary group may be considered also as an algebra (G, f, g) with
one associative n-ary operation f and one unary operation g satisfying
some identities (see, for example, [5], [6], [8] or [9]). In particular, an n-ary
group may be treated as an algebra (G, f, [−2]) with one associative n-ary
operation f and one unary operation [−2] : x 7→ x[−2] such that
f(x[−2],
(n−2)
x , f(
(n−1)
x , y)) = f(f(y,
(n−1)
x ),
(n−2)
x , x[−2]) = y, (1)
is true for all x, y ∈ G (see [19]).
Applying associativity to (1) we obtain
f(f(x[−2],
(n−1)
x ),
(n−2)
x , y) = f(y,
(n−2)
x , f(
(n−1)
x , x[−2])) = y, (2)
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62 Geometry of flocks and n-ary groups
which together with results proved in [9] and [5] shows that
f(x[−2],
(n−1)
x ) = f(
(n−1)
x , x[−2]) = x, (3)
where x denotes the skew element to x (see [4], [5] or [9]). In general x 6= x,
but the situation when x = x or x = y for x 6= y also is possible (see [8]).
Moreover, in any n-ary group (G, f) with n > 3 we have
f(x,
(i−3)
y , y,
(n−i)
y , z) = f(x,
(j−3)
y , y,
(n−j)
y , z) (4)
for all x, y, z ∈ G and 3 6 i, j 6 n.
An n-ary operation f defined on G is semiabelian if
f(x1, x
n−1
2 , xn) = f(xn, x
n−1
2 , x1)
for all x1, . . . , xn ∈ G.
One can prove (for details see [5]) that for n > 3 an n-ary group (G, f)
is semiabelian if and only if there exists a ∈ G such that for all x, y ∈ G
holds f(z,
(n−2)
a , y) = f(z,
(n−2)
a , y), or equivalently,
f(z, a,
(n−3)
a , y) = f(z, a,
(n−3)
a , y). (5)
A nonempty set G with one ternary operation [ · , · , · ] satisfying the
para-associative law
[[x, y, z], u, w] = [x, [u, z, y], w] = [x, y, [z, u, w]]
and such that for all a, b, c ∈ G there are uniquely determined x, y, z ∈ G
such that
[x, a, b] = [a, y, b] = [a, b, z] = c (6)
is called a flock (see [7] or [10]). Obviously, a semiabelian flock is a
semiabelian ternary group. So, a flock (G, [ · , · , · ]) is semiabelian if and
only if there exists a ∈ G such that [x, a, z] = [z, a, x] for all x, z ∈ G.
Properties of flocks are similar to properties of ternary groups.
Further, we will use the following lemmas proved in [7].
Lemma 2.1. In any flock (G, [ · , · , · ]) for each x ∈ G there exists x such
that
[x, x, y] = [x, x, y] = [y, x, x] = [y, x, x] = y
for all y ∈ G.
Lemma 2.2. In any flock x = x and [x, y, z] = [x, y, z].
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S. Dog 63
By the Post’s Coset Theorem (see [17]), for any n-ary group (G, f) there
exists a binary group (G#, ·) such that G ⊂ G# and f(xn1 ) = x1 ·x2 ·. . .·xn
for x1, . . . , xn ∈ G. Since in this group x = x2−n for all x ∈ G, then
[x, y, z] = f(x, y,
(n−3)
y , z) (7)
is an idempotent para-associative ternary operation. So, if (G, f) is an
n-ary group then (G, [ · , · , · ]) with an operation defined by (7) is an
idempotent ternary flock. We will say that this flock is induced by an
n-group (G, f). From Post’s Theorem it follows that the operation of this
flock can be presented in the form [x, y, z] = x · y−1 · z, where (G#, ·) is
the covering group of the corresponding an n-ary group (G, f).
The following obvious lemma plays an important role in the proofs of
our results presented in this paper.
Lemma 2.3. An n-ary group (G, f) is semiabelian if and only if the flock
(G, [ · , · , · ]) defined by (7) is semiabelian.
Further, for simplicity, instead of [. . . [[x1, x2, x3], x4, x5], . . . , x2k, x2k+1]
we will write [x1, x2, . . . , x2k+1]. Since the operation [ · , · , · ] is para-
associative we also have
[x1, x2, . . . , x2k+1] = [x1, x2, [x3x4, [. . . [x2k−1, x2k, x2k+1] . . . ]]].
3. Parallelograms
Generalizing the idea presented by W. Szmielew (see [23]) S.A. Rusakov
considered in [22] the affine geometry as the geometry induced by n-ary
groups. In his generalization elements of an n-ary group (G, f) are points.
The ordered pair of two points a, b ∈ G is called an interval and is denoted
by 〈a, b〉. The set of four points a, b, c, d ∈ G such that 〈a, b〉, 〈b, c〉, 〈c, d〉
and 〈d, a〉 are intervals is called a quadrangle. Intervals 〈a, b〉, 〈b, c〉, 〈c, d〉
and 〈d, a〉 are sides of this quadrangle. Intervals 〈a, c〉 and 〈b, d〉 are its
diagonals.
It is easy to see that the relation ≡ defined on the set of all intervals
by
〈a, b〉 ≡ 〈c, d〉 ⇐⇒ f(a, b[−2],
(2n−4)
b , d) = c (8)
is an equivalence. In view of (3) this relation can be rewritten in the form
〈a, b〉 ≡ 〈c, d〉 ⇐⇒ f(a, b,
(n−3)
b , d) = c.
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64 Geometry of flocks and n-ary groups
The equivalence class of 〈a, b〉 is interpreted as a vector
−→
ab. Such defined
vectors form some vector space (see [22]), where the addition of vectors
can be defined (see [11]) by
−→
ab +
−→
cd = −→ag =
−→
hd,
where g = f(b, c[−2],
(2n−4)
c , d) and h = f(c, b[−2],
(2n−4)
b , a), or equivalently
(see [13]) g = f(b, c,
(n−3)
c , d) and h = f(c, b,
(n−3)
b , a).
According to [22] four points a, b, c, d ∈ G form a parallelogram if
f(a, b[−2],
(2n−4)
b , c) = d. (9)
As a simple consequence of (4) and (7) we obtain the following two
lemmas.
Lemma 3.1. For an n-ary group (G, f), where n > 3, the following
conditions are equivalent:
(i) elements a, b, c, d ∈ G form a parallelogram,
(ii) f(a, b,
(n−3)
b , c) = d,
(iii) [a, b, c] = d.
Lemma 3.2. For an n-ary group (G, f), where n > 3, the following
conditions are equivalent:
(i) intervals 〈a, b〉 and 〈c, d〉 are equivalent,
(ii) f(a, b,
(n−3)
b , d) = c,
(iii) [a, b, d] = c.
Now let (G, [ · , · , · ]) be an arbitrary flock and a, b, c, d ∈ G. Then the
relation
〈a, b〉 ≡ 〈c, d〉 ⇐⇒ [a, b, d] = c
is an equivalence. Similarly as in the case of n-ary groups, the equivalence
class of 〈a, b〉 can be interpreted as a vector
−→
ab. Consequently,
−→
ab =
−→
cd ⇐⇒ [a, b, d] = c. (10)
The addition of such vectors is defined by
−→
ab +
−→
cd =
−−−−−→
a[b, c, d] =
−−−−−→
[c, b, a]d. (11)
Thus points a, b, c, d ∈ G form a parallelogram if [a, b, c] = d.
“adm-n3” — 2019/10/20 — 9:35 — page 65 — #67
S. Dog 65
Therefore for n > 2 the affine geometry introduced by Rusakov and
investigated by Kulazhenko is a special case of the affine geometry induced
by flocks (see [7]). Namely, for n > 2, the affine geometry induced by
an n-ary group (G, f) coincides with the affine geometry induced by an
idempotent flock defined by (7). So, all Kulazhenko’s results on parallelo-
grams proved in [11] and [13] are a simple consequence of Dudek’s results
from [7].
4. Vectors of semiabelian flocks
In this section we characterize semialelian flocks by the properties of
vectors of the corresponding geometry.
Lemma 4.1. A flock (G, [ · , · , · ]) is semiabelian if and only if
z = [x, u, z, u, y, x, u, y, u]
is true for all x, y, z, u ∈ G.
Proof. A semiabelian flock is a ternary group, hence the operation [ · , · , · ]
is associative. Thus, by Lemma 2.1
[x, u, z, u, y, x, u, y, u] = [x, u, z, u, u, x, y, y, u] = [x, u, z, x, u]
= [x, x, z, u, u] = z.
Conversely, if z = [x, u, z, u, y, x, u, y, u] for all x, y, z, u ∈ G, then
multiplying this equation on the right by u, y, u, x we obtain
[z, u, y, u, x] = [x, u, z, u, y],
which for z = u gives [y, u, x] = [x, u, y]. This, by Lemma 2.2, means that
[y, v, x] = [x, v, y] for all x, y, v ∈ G. So, (G, [ · , · , · ]) is a semiabelian
flock.
Lemma 4.2. A flock (G, [ · , · , · ]) is semiabelian if and only if
z = [x, y, u, x, z, v, y, u, v]
holds for all x, y, z, u, v ∈ G.
Proof. The proof of the necessity is the same as in the previous lemma.
To prove the sufficiency it is sufficient to multiple this equation on the
right by v, u, y, v. Then, after reduction, putting x = z = v, we can see
that (G, [ · , · , · ]) is semiabelian.
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66 Geometry of flocks and n-ary groups
In the same way we can prove
Lemma 4.3. A flock (G, [ · , · , · ]) is semiabelian if and only if
u = [x, y, z, x, y, x, u, z, x],
or equivalently
u = [x, y, z, x, u, z, y]
for all x, y, z, u ∈ G.
In the case when a flock is defined by (7), as a simple consequence of
the above lemmas we obtain the Kulazhenko’s result proved in [16].
Theorem 4.4. (Kulazhenko) For n > 2 an n-ary group (G, f) is semia-
belian if and only if one of the following equivalent identities is satisfied
(a) z = f(x, u[−2],
(2n−4)
u , z, u[−2],
(2n−4)
u , y, x[−2],
(2n−4)
x x, u, y[−2],
(2n−4)
y , u),
(b) z = f(x, y[−2],
(2n−4)
y , u, x[−2],
(2n−4)
x , z, v[−2],
(2n−4)
v v, y, u[−2],
(2n−4)
u , v).
Theorem 4.5. A flock (G, [ · , · , · ]) is semiabelian if and only if for any
pairs (xi, yi) of elements of G, where 1 6 i 6 t and t > 2 is an odd natural
number, the identity
[x1, x2, x3, x4, . . . , xt−2, xt−1, xt, x1, y1, y2, x2, x3, y3,
. . . , yt−1, xt−1, xt, yt−1, yt−2, . . . , y6, y5, y4, y3, y2] = y1,
(12)
is valid.
Proof. Applying Lemma 2.1 we can see that (12) holds in any semiabelian
flock.
Conversely, if (12) holds in the flock (G, [ · , · , · ]), then putting y1 =
[x, y, z], xt = z and x for other xi and yj we obtain
[x, x, x, x, . . . , x, x, z, x, [x, y, z], x, x, x, x, x, x, z, x, x, . . . , x, x, x, x, x]
= [x, y, z].
The left side of this equation, after application of Lemma 2.1, can be re-
ducedto the form [z, x, [x, y, z], z, x]. Later, applying the para-associativity
of the operation [ · , · , · ] and Lemma 2.1, we obtain
[z, x, [x, y, z], z, x] = [[z, x, [x, y, z]], z, x] = [[[z, x, x, ]y, z], z, x]
= [[z, y, z], z, x] = [z, y, [z, z, x]] = [z, y, x].
This proves that a flock (G, [ · , · , · ]) is semiabelian.
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S. Dog 67
Theorem 4.6. A flock (G, [ · , · , · ]) is semiabelian if and only if
−−−−−−−−−−→
[x, y, z][u, v, w] = −→xu−−→yv +−→zw (13)
holds for all x, y, z, u, v, w ∈ G.
Proof. Let (G, [ · , · , · ]) be a semiabelian flock. Then it is a ternary group
and
−→xu−−→yv +−→zw = −→xu+−→vy +−→zw =
−−−−−→
x[u, v, y] +−→zw =
−−−−−−−−−→
x[u, v, y, z, w].
Consider the quadrangle 〈[x, y, z], x, [u, v, y, z, w], [u, v, w]〉. Since, as it is
not difficult to verify, [[x, y, z], x, [u, v, y, z, w]] = [u, v, w], this quadrangle
is a parallelogram (Lemma 3.1). Thus
−−−−−−−−−−→
[x, y, z][u, v, w] =
−−−−−−−−−→
x[u, v, y, z, w].
This proves (13).
Conversely, if (13) holds for all x, y, z, u, v, w ∈ G, then
−−−−−−−−−−→
[x, y, z][u, v, w] = −→xu−−→yv +−→zw =
−−−−−−−−−→
x[u, v, y, z, w],
i.e., the quadrangle 〈[x, y, z], x, [u, v, y, z, w], [u, v, w]〉 is a parallelogram.
Thus [[x, y, z], x, [u, v, y, z, w]] = [u, v, w]. Multiplying this identity on the
right by w, u we obtain the identity from Lemma 4.2. Hence this flock is
semiabelian.
Theorem 4.7. A flock (G, [ · , · , · ]) is semiabelian if and only if for any
pairs (xi, yi) of elements of G, where 1 6 i 6 t and t > 2 is an odd natural
number, the identity
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
[x1, x2, x3, x4, . . . , xt−1, xt][y1, y2, y3, y4, . . . , yt−1, yt]
= −−→x1y1 −
−−→x2y2 +
−−→x3y3 − . . .−−−−−−→xt−1yt−1 +
−−→xtyt
(14)
is satisfied.
Proof. By Theorem 4.6 in a flock (G, [ · , · , · ]) we have
−−→x1y1 −
−−→x2y2 +
−−→x3y3 =
−−−−−−−−−−−−−−→
[x1, x2, x3][y1, y2, y3].
Thus
−−→x1y1 −
−−→x2y2 +
−−→x3y3 −
−−→x4y4 +
−−→x5y4
=
−−−−−−−−−−−−−−→
[x1, x2, x3][y1, y2, y3]−
−−→x4y4 +
−−→x5y5
=
−−−−−−−−−−−−−−−−−−−−−−−−−→
[x1, x2, x3, x4, x5][y1, y2, y3, y4, y5],
and so on. This proves (14).
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68 Geometry of flocks and n-ary groups
Conversely, if (14) holds in a flock (G, [ · , · , · ]), then
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
[x1, x2, x3, x4, . . . , xt][y1, y2, y3, y4, . . . , yt]
= −−→x1y1 −
−−→x2y2 +
−−→x3y3 − . . .−−−−−−→xt−1yt−1 +
−−→xtyt
= −−→x1y1 +
−−→y2x2 +
−−→x3y3 − . . .+−−−−−→yt−1xt−1 +
−−→xtyt
=
−−−−−−−−→
x1[y1, y2, x2] +
−−→x3y3 + . . .+−−−−−→yt−1xt−1 +
−−→xtyt
=
−−−−−−−−−−−−−→
x1[y1, y2, x2, x3, y3] + . . .+−−−−−→yt−1xt−1 +
−−→xtyt
=
−−−−−−−−−−−−−−−−−−−→
x1[y1, y2, x2, x3, y3, y4, x4] + . . .+−−−−−→yt−1xt−1 +
−−→xtyt
= . . . =
−−−−−−−−−−−−−−−−−−−−−−−−−−→
x1[y1, y2, x2, x3, y3, y4, x4, . . . , xt, yt].
This means that
〈[x1, x2, x3, . . . , xt], x1, [y1, y2, x2, x3, y3, y4, x4 . . . , xt, yt],
[y1, y2, y3. . . . , yt]〉
is a parallelogram. So, by Lemma 3.1,
[[x1, x2, x3, . . . , xt], x1, [y1, y2, x2, x3, y3, y4, x4, . . . , xt, yt]]
= [y1, y2, y3, . . . , yt].
Multiplying this identity by yt, yt−1, yt−2, yt−3, . . . , y3, y2 and applying
Lemma 2.1, we obtain (12). Hence, by Theorem 4.5, this flock is semiabelian.
In the case when xi = x (resp. yi = y) for all i = 1, 2, . . . , t we obtain
Corollary 4.8. A flock (G, [ · , · , · ]) is semiabelian if and only if for all
elements x, y1, y2, . . . , yt ∈ G, where t > 2 is an odd natural number, the
identity
−−−−−−−−−−−−−−−−−−−→
x[y1, y2, y3, y4, . . . , yt−1, yt] =
−→xy1 −
−→xy2 +
−→xy3 − . . .−−−−→xyt−1 +
−→xyt
is satisfied.
Corollary 4.9. A flock (G, [ · , · , · ]) is semiabelian if and only if for all
elements x1, x2, . . . , xt, y ∈ G, where t > 2 is an odd natural number, the
identity
−−−−−−−−−−−−−−−−−−−−→
[x1, x2, x3, x4, . . . , xt−1, xt]y = −→x1y −
−→x2y +
−→x3y − . . .−−−−→xt−1y +
−→xty
is satisfied.
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S. Dog 69
5. Symmetry and semiabelianism
According to Rusakov (see [20] or [22]) two elements a and c of an n-ary
group (G, f) are called symmetric if and only if there exists a uniquely
determined point x ∈ G such that
f(f(a, x[−2],
(n−2)
x ),
(n−2)
x , c) = x.
Thus, in view of the above results, for n > 3 this definition can be
formulated in the form:
Definition 5.1. Two elements a and c of an n-ary group (G, f) are
symmetric if and only if there exists one and only one x ∈ G such that
f(a, x,
(n−3)
x , c) = x. (15)
Thus for symmetric elements a and c there exists uniquely determined
element x ∈ G and the symmetry Sx such that Sx(a) = c. Since in (15)
the element c is uniquely determined by a and x, then using the same
method as in [5] and [9] one can prove that the symmetry Sx has the
form:
Sx(a) = f(x, a,
(n−3)
a , x).
In the case of flocks (see [7]) points a, c ∈ G are symmetric if and only
if there exists a uniquely determined x ∈ G such that
[a, x, c] = x.
In this case Sx(a) = [x, a, x].
Theorem 5.2. A flock (G, [ · , · , · ]) is semiabelian if and only if
−→ux+
−−−−→
Sx(u)y +
−−−−−−→
SySx(u)z +
−−−−−−−−→
SzSySx(u)w =
−→
0 (16)
for any points x, y, z, u, w ∈ G such that 〈x, y, z, w〉 is a parallelogram.
Proof. Observe first that
−→
0 = −→xx for any x ∈ G. By Lemma 2.2 we also
have
SySx(u) = [y, x, u, x, y] = [y, x, u, x, y]
and
SzSySx(u) = [z, y, x, u, x, y, z] = [z, y, x, u, x, y, z].
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70 Geometry of flocks and n-ary groups
Thus
−→ux+
−−−−→
Sx(u)y +
−−−−−−→
SySx(u)z +
−−−−−−−−→
SzSySx(u)w
=
−−−−−−−−−→
u[x, Sx(u), y] +
−−−−−−→
SySx(u)z +
−−−−−−−−→
SzSySx(u)w
=
−−−−−−−−−−→
u[x, [x, u, x], y] +
−−−−−−→
SySx(u)z +
−−−−−−−−→
SzSySx(u)w
=
−−−−−→
u[u, x, y] +
−−−−−−→
SySx(u)z +
−−−−−−−−→
SzSySx(u)w
=
−−−−−→
u[x, y, z] +
−−−−−−−−→
SzSySx(u)w
=
−−−−−→
u[x, y, z] +
−−−−−−−−−−−−−→
[z, y, x, u, x, y, z]w =
−−−−−−−−−→
u[u, x, y, z, w].
So,
−→ux+
−−−−→
Sx(u)y +
−−−−−−→
SySx(u)z +
−−−−−−−−→
SzSySx(u)w =
−−−−−−−−−→
u[u, x, y, z, w].
But w = [x, y, z] because the quadrangle 〈x, y, z, w〉 is a parallelogram.
Hence
−→ux+
−−−−→
Sx(u)y +
−−−−−−→
SySx(u)z +
−−−−−−−−→
SzSySx(u)w =
−−−−−−−−−−−−→
u[u, x, y, z, x, y, z].
This means that the condition (16) can be written in the form
−−−−−−−−−−−−→
u[u, x, y, z, x, y, z] = −→zz,
which, by (10), is equivalent to
[x, y, z, x, y] = z,
and consequently, to [x, y, z] = [z, y, x]. This completes the proof.
Theorem 5.3. A flock (G, [ · , · , · ]) is semiabelian if and only if
−→ux+
−−−−→
Sx(u)y +
−−−−−−→
SySx(u)z +
−−−−−−−−→
SzSySx(u)w +
−−−−−−−−−−→
SwSzSySx(u)v
+
−−−−−−−−−−−−→
SvSwSzSySx(u)x =
−→
0
(17)
for any points x, y, z, u, w ∈ G and v = [w, z, y].
Proof. As in the previous proof,
−→ux+
−−−−→
Sx(u)y +
−−−−−−→
SySx(u)z +
−−−−−−−−→
SzSySx(u)w =
−−−−−−−−−→
u[u, x, y, z, w].
Since −−−−−−−−−−→
SwSzSySx(u)v =
−−−−−−−−−−−−−−−−−→
[w, z, y, x, u, x, y, z, w]v,
we have
−→ux+
−−−−→
Sx(u)y +
−−−−−−→
SySx(u)z +
−−−−−−−−→
SzSySx(u)w +
−−−−−−−−−−→
SwSzSySx(u)v = −→ux
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S. Dog 71
because
−−−−−−−−−→
u[u, x, y, z, w] +
−−−−−−−−−−−−−−−−−→
[w, z, y, x, u, x, y, z, w]v
=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
u[[u, x, y, z, w], [w, z, y, x, u, x, y, z, w], v]
and
[[u, x, y, z, w], [w, z, y, x, u, x, y, z, w], v]
= [[u, x, y, z, w], [w, z, y, x, u, x, y, z, w], v]
= [[[u, x, y, z, w]w, z], [w, z, y, x, u, x, y], v]
= [[u, x, y, ], [w, z, y, x, u, x, y], v]
= [[[u, x, y, ]y, x], [w, z, y, x, u], v] = [u, [w, z, y, x, u], v]
= [[u, u, x], [w, z, y], v] = [x, [w, z, y], v] = x
for v = [w, z, y] (Lemma 2.1).
Similarly,
−−−−−−−−−−−−→
SvSwSzSySx(u)x =
−−−−−−−−−−−−−−−−−−−−→
[v, w, z, y, x, u, x, y, z, w, v]x
=
−−−−−−−−−−−−−−−−−→
[w, z, y, w, z, y, x, u, x]x,
because
[v, w, z, y, x, u, x, y, z, w, v] = [[w, z, y], w, z, y, x, u, x, y, z, w, [w, z, y]]
= [[w, z, y], w, z, y, x, u, x, y, [z, w, [w, z, y]]] = [w, z, y, w, z, y, x, u, x].
Consequently, −→ux+
−−−−−−−−−−−−→
SvSwSzSySx(u)x =
−−−−−−−−−−−−−→
[w, z, y, w, z, y, x]x, by (11).
Thus (17) has the form
−−−−−−−−−−−−−→
[w, z, y, w, z, y, x]x = −→xx,
which, by (10), is equivalent to
[w, z, y, w, z, y, x] = x,
i.e., to [w, z, y] = [y, z, w], which completes the proof.
Theorem 5.4. A flock (G, [ · , · , · ]) is semiabelian if and only if
2−→xy = −→zu+
−−−−−−−→
Sx(z)Sy(u)
for any points x, y, z, u ∈ G.
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72 Geometry of flocks and n-ary groups
Proof. According to (11) we have
2−→xy = −→xy +−→xy =
−−−−−→
x[y, x, y]
and
−→zu+
−−−−−−−→
Sx(z)Sy(u) =
−→xu+
−−−−−−−−−−→
[x, z, x][y, u, y] =
−−−−−−−−−−−−−→
[x, z, x, u, z][y, u, y].
So, by (10), the equation mentioned in our theorem can be written as
[x, [y, x, y], [y, u, y]] = [x, z, x, u, z],
i.e., as
[x, y, x, u, y] = [x, z, x, u, z].
The last equation is valid in semiabelian flocks only.
Theorem 5.5. A flock (G, [ · , · , · ]) is semiabelian if and only if
2(−→xy +−→zu) = 2−→xy + 2−→zu
for any points x, y, z, u ∈ G.
Proof. Since
2(−→xy +−→zu) =
−−−−−−−−−−−−→
x[y, z, u, x, y, z, u]
and
2−→xy + 2−→zu =
−−−−−−−−−−−−→
x[y, x, u, z, u, z, u],
the equation given in the above theorem is equivalent to
−−−−−−−−−−−−→
x[y, z, u, x, y, z, u] =
−−−−−−−−−−−−→
x[y, x, u, z, u, z, u],
i.e., to
[x[y, z, u, x, y, z, u][[y, x, y], z, [u, z, u]]] = x.
The last equation can be written as
[x[y, z, u, x, y, z, u][y, x, y, z, u, z, u]] = x,
which, in view of Lemma 2.1 and (6), means that
[y, z, u, x, y, z, u] = [y, x, y, z, u, z, u],
i.e., to [y, z, u, x, y] = [y, x, y, z, u]. This equation holds only in semiabelian
flocks.
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S. Dog 73
6. Conclusion
Our results are valid for arbitrary flocks and generalize various results
proved by S.A. Rusakov and Yu.I. Kulazhenko for n-ary groups with
n > 3. Moreover, in the case idempotent flocks, i.e., flocks with the
property x = x, our results coincide with the corresponding results proved
for n-ary groups. It is a consequence of (7) and Lemma 2.3. Namely, in the
case of idempotent flocks, our Lemma 4.1 coincides with the Kulazhenko’s
Proposition from [16], Lemma 4.2 with Lemma from [16], and Theorem 4.6
with the main theorem of [16]. His results are presented in very complicated
form (see our Theorem 4.4). Theorems 4.5, 4.7 and Corollary 4.8 (also 4.9)
generalize Kulazhenko’s results from [15]. In the case of idempotent flocks
these results are identical. Results of Section 5 generalize Kulazhenko’s
results from [12].
Another consequence of our results are short and more elegant proofs.
For example, the original proof of Theorem 5.2 (for n-ary groups) has in
[12] three printed pages of rather complicated transformations; the proof
of Theorem 5.3 has four pages. Also the original proofs of Theorems 5.4
and 5.5 presented in [13] are much longer.
References
[1] R.Baer, Zur Einführung des Scharbegriffs, J. Reine Angew. Math. 160 (1929),
199-207.
[2] D.Brǎnzei, Structures affines et opérations ternaires, An. Şti. Univ. Iaşi, Sect. I a
Mat. (N.S.) 23 (1977), 33-38.
[3] J.Certaine, The ternary operation (abc) = ab
−1
c of a group, Bull. Amer. Math.
Soc. 49 (1943), 869-877.
[4] W.Dörnte, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Math. Z.
29 (1928), 1-19.
[5] W.A.Dudek, Remarks on n-groups, Demonstratio Math. 13 (1980), 165-181.
[6] W.A.Dudek, Varieties of polyadic groups, Filomat (Niš) 9 (1995), 657-674.
[7] W.A.Dudek, Ternary quasigroups connected with the affine geometry, Algebras,
Groups and Geometries 16 (1999), 329-354.
[8] W.A.Dudek, On some old and new problems in n-ary groups, Quasigroups and
Related Systems 8 (2001), 15-36.
[9] W.A.Dudek, B.Gleichgewicht and K.Głazek, A note on the axioms of n-groups,
Colloquia Math. Soc. J. Bolyai, 29. Universal Algebra, Esztergom (Hungary) 1977,
195-202.
[10] A.Knoebel, Flocks, groups and heaps, joined with semilattices, Quasigroups and
Related Systems 24 (2016), 43-66.
“adm-n3” — 2019/10/20 — 9:35 — page 74 — #76
74 Geometry of flocks and n-ary groups
[11] Yu.I.Kulazhenko, Geometry of parallelograms, (Russian), in Vopr. Algeb. and Prik.
Mat., Izdat. Belorus. Gos. Univ. Transp., Gomel 1995, 47-64.
[12] Yu.I.Kulazhenko, Semi-commutativity criteria and self-coincidence of elements
expressed by vector properties of n-ary groups, Algebra Discrete Math. 9 (2010),
no.2. 98-107.
[13] Yu.I.Kulazhenko, Geometry of semiabelian n-ary groups, Quasigroups Related
Systems 19 (2011), 265-278.
[14] Yu.I.Kulazhenko, Vectors and semiabelian criteria of n-ary groups, (Russian),
Probl. Phys. Math. Tekh. 1(6) (2011), 65-68.
[15] Yu.I.Kulazhenko, Sequences of vectors and criteria of semmi-commutativity of
n-ary groups, Southeast Asian Bull. Math. 37 (2013), 745-752.
[16] Yu.I.Kulazhenko, Semiabelianness and properties of vectors on n-ary groups, (Rus-
sian), Tr. Inst. Mat. Mekh. 20 (2014), no.1, 142-147.
[17] E.L.Post, Polyadic groups, Trans. Amer. Math. Soc. 48 (1940), 208-350.
[18] H.Prüfer, Theorie der Abelschen Gruppen, Math. Z. 20 (1924), 166-187.
[19] S.A.Rusakov, A definition of n-ary group, (Russian), Doklady AN BSSR (Minsk)
23 (1972), 965-967.
[20] S.A.Rusakov, Existence of n-ary rs-groups (Russian), Voprosy Algebry 6 (1992),
89-92.
[21] S.A.Rusakov, Vectors of n-ary groups. Linear operations and their properties
(Russian), in Vopr. Algeb. and Prik. Mat. Izdat. Belorus. Gos. Univ. Transp.,
Gomel 1995, 10-30.
[22] S.A.Rusakov, Some applications of n-ary groups, (Russian), Belaruskaya navuka,
Minsk, 1998.
[23] W.Szmielew, Theory of n-ary equivalences and its application to geometry, Disser-
tationes Math. 191 (1980).
[24] A.K.Sushkevich, The theory of generalized groups, (Russian), Kharkov, Kiev, 1937.
[25] V.V.Vagner, Theory of generalized heaps and generalized groups, (Russian), Mat.
Sb. 32(74) (1953), 545-632.
Contact information
S. Dog 22 Pervomayskaya str, 39600 Kremenchuk,
Ukraine
E-Mail(s): soniadog2@gmail.com
Received by the editors: 13.12.2017.
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