Geometry induced by flocks

Using the vectors and symmetry of affine geometry induced by the ternary quasigroup satisfying the para-associative laws, we found the conditions under which such quasigroup becomes a ternary group. The obtained results also give a simple characterization of semiabelian n-ary groups.


Introduction
Ternary heaps (introduced by Prüfer [10]) have interesting applications to projective geometry [1], affine geometry [2], theory of nets (webs), theory of knots and even to the differential geometry [7]. A special case of ternary heaps defined on a group were considered by Certaine [3]. Vakerelov considered in [13] the affine geometry defined by ternary groups that are heaps satisfying some additional conditions. S.A. Rusakov extended this concept to the case of n-ary groups (see [11,12]). His research was continued by his students and co-workers (see for example [8,9]). Dudek noted in [5] that the proofs of the results obtained by Rusakov and his co-workers can be much shorter and clearer if we use other much simpler algebras called flocks instead of n-ary groups (see also [6]).
In this short note, we continue this line of research. We give some short and simple proofs of some important results previously proven for affine geometry induced by an n-ary group.

Preliminaries
We use standard terminology and notation (cf. [4,12]). We just recall that an n-ary group (n-group) (G, f ) is semiabelian if f (x, a, . . . , a, y) = f (x, (n−2) a , y) = f (y, (n−2) a , x) for all x, y ∈ G and any fixed a ∈ G. Starting with the observation that in the affine geometry of the plane Rusakov (cf. [11] or [12]) extended this concept to n-groups in his way that points a, b, c, d of an n-group c , d), or using the free covering group for an n-group (G, f ) a, c, d, b forms a parallelogram if and only if a · c −1 · d = b. The operation [x, y, z] = x · y −1 · z is idempotent and satisfies the following para-associative law: Any ternary groupoid (G, [ ]) satisfying (1) is called para-associative or a semiheap (cf. [7]). A paraassociative quasigroup is called a flock.
holds for all x, y, z ∈ G. A semiabelian flock is a ternary group and conversely, any semiabelian ternary group is a flock. But there are flocks that are not ternary groups and ternary groups that are not flocks.
If a is a fixed element of a flock (G, [ ]), then (G.·), where x · y = [x, a, y], is a group called the retract of (G, [ ]) and denoted by ret a (G, [ ]).
All retracts of a given flock are isomorphic (cf. [5]). A flock is a ternary group if and only if it has a commutative retract.
Thus, all flocks of orders p, p 2 and pq, where p > q are prime integers such that ( p − 1, q) = 1, are ternary groups. A minimal flock which is not a ternary group has six elements and is defined on the symmetric group S 3 .
Retracts of isomorphic flocks are isomorphic. The converse statement is not true. For example, the two flocks , are not isomorphic (the first is idempotent, the second has no idempotents), but their retracts are isomorphic.
We are going to need the following two theorems proved in [5].
is satisfied.

Geometry of flocks
The relation ≡ defined on the set of all intervals of a flock (G, [ ]) by is an equivalence. Equivalence classes of this relation can be interpreted as vectors, i.e.
The set of such defined vectors is denoted by V (G). Intervals belonging to the same equivalence class are parallel.
Another concept of parallel intervals defined by flocks was proposed in [5] (see also [4]). In [5], following Rusakov's definition for n-groups, two intervals Below, we prove several elementary properties of parallelograms defined by flocks. These results were previously proved for n-groups. The proofs for n-groups are long and hard to read. For flocks, they are much shorter and simpler. at least one tetragon a, b, c, d , b, a, d, c , c, d, a, b or d, c, b, a is a parallelogram, then the other three remaining tetragons are also parallelograms.
shows that d, c, b, a is a parallelogram, too.
Thus, the following fact is obvious.
This addition is well defined.
Hence, the addition of vectors is well defined. Such defined addition is associative because for c, d], e, f ] . Moreover, − → aa is the zero vector, and − − → ab = − → ba for all a, b ∈ G. Consequently, (V (G), +) is a group. Such defined addition of vectors is a generalization of Rusakov's addition of vectors defined on semiabelian n-groups. For vectors of a semiabelian n-group (G, f ) Rusakov defined (cf. [11] or [12]) the addition of vectors by the formula where b , a). For flocks, this formula has the form , which is valid for all flocks, in the Russakov's definition of addition of vectors the assumption that an n-group is semiabelian can be omitted. Consequently, for flocks the addition of vectors can be defined by (8)

Proposition 3.4 The flock (G, [ ]) is semiabelian if and only if for all
The above result means that affine geometry induced by n-groups (or flocks) is not commutative. It is commutative only in the case when it is induced by a semiabelian n-group (respectively, by a flock that is a ternary group).
Further, for the sake of clarity, the expression [[x, y, z], u, v] will be written as [x, y, z, u, v].

Theorem 3.5 A flock (G, [ ]) is semiabelian if and only if
for any points x, y, z, u ∈ G.
Proof Since the equation given in the above theorem is equivalent to Substituting y = z, we will see that this flock is semiabelian.

Theorem 3.6 A flock (G, [ ]) is semiabelian if and only if
holds for all x, y, z, u, v, w ∈ G.

Theorem 3.8 A flock (G, [ ]) is semiabelian if and only if
for any points x, y, z, u, w ∈ G such that x, y, z, w is a parallelogram. Thus, Therefore, Since x, y, z, w is a parallelogram, w = [x, y, z]. Hence, This means that (13) can be written in the form  Yet another version was proposed by Kulazhenko (see for example [8]). In his version the addition of vectors is defined by (9) [8] where this lemma is used) is wrong.
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