Study of Jordan quasigroups and their construction

ABSTRACT Jordan quasigroups are commutative quasigroups satisfying the identity . In this paper we discuss the basic properties of Jordan quasigroups and prove that (i) every commutative idempotent quasigroup is Jordan quasigroup, (ii) if a Jordan quasigroup Q is distributive then Q is idempotent, (iii) an idempotent entropic quasigroup satisfies Jordan's identity, (iv) a unipotent quasigroup Q satisfying Jordan's identity satisfies left nuclear square property, (vi) if a quasigroup satisfies LC identity, then alternativity ⇔ Jordan's identity, (vii) for a unipotent Jordan quasigroup Q, and (viii) every Steiner quasigroup is Jordan quasigroup. We also prove some properties of the autotopism of Jordan loops. Moreover, we construct an infinite family of nonassociative Jordan quasigroups whose smallest member is of order 6.


Introduction
A magma (Q, ·) is a quasigroup if for each a, b [ Q, the equations ax=b,ya=b have unique solutions x, y [ Q.
Jordan loops are among the least studied loops. Powers of elements and order of Jordan loops have been discussed in [3][4][5]. The identity of Jordan loops consists of two variables x and y. This is why the properties of Jordan loops are not extensively studied because of the weakness of the identity. In this paper we study Jordan quasigroups.
In Section 1, we construct some basic results regarding the Jordan quasigroups. We also prove different type of quasigroups which are necessarily Jordan quasigroups.
In Section 2, we study some properties of the autotopism of Jordan loops. Also we prove that commutative IP loop of exponent 3 is Jordan loop.
In Section 3, we construct an infinite family of nonassociative Jordan quasigroups whose smallest member is of order 6.
We will need the following definitions. Proof: Proof: Proof: From Jordan's identity, it is easy to see that a quasigroup which satisfies the left nuclear square property also satisfies Jordan's identity. But the converse is not true. However, the converse also holds if the Jordan quasigroup satisfying Jordan's identity also satisfies unipotency.
Theorem 2.4: A unipotent quasigroup L satisfying Jordan's identity satisfies the left nuclear square property.
Proof. By Jordan's identity z 2 (yz) = (z 2 y)z since L is unipotent then Theorem 2.5: For a Jordan quasigroup, if x 2 = y 2 , then Proof: Corollary 2.2: For a unipotent Jordan quasigroup J, The following four are left central (LC) identities for loops collected in [7]: These identities are equivalent for loops but not for quasigroups as for example the quasigroup in Example 2.1 satisfies the LC identity ((x(xy))z) = (x(x(yz))) but does not satisfy the LC identity ((xx)(yz) = (x(xy))z).
The following theorem establishes a relation among a special LC, Jordan and left alternative identities.
Theorem 2.6: If a quasigroup Q satisfies LC identity (xx)(yz) = (x.xy)z then Q is left alternative ⇔ Q satisfies Jordan's identity.
Remark 2.1: An LC-loop has always left alternative property [8] but this is not necessary for quasigroup as the quasigroup in the following example satisfies the LC identity ((x(xy))z) = (x(x(yz))) but it does not have left alternative property. A commutative quasigroup is trivially entropic but the converse does not hold always. The following theorem shows that the converse also holds if the entropic quasigroup is central square. Proof: from (3) and (4) x 2 (yx) = (x 2 y)x. A Proof. Let (L x , R x , L x R x ) be an autotopism then  Proof:

On Jordan loops
⇒ y(xy 2 ) = (xy)y 2 ⇒ (y 2 x)y = y 2 (xy) by commutativity interchanging x and y we get is autotopism, then L is alternative. Proof: Which is left alternativity and since commutative hence alternative.

Construction of Jordan quasigroups
We now construct an infinite family of nonassociative Jordan quasigroups whose smallest member is a quasigroup of order 6. We adopt the same procedure as done for the construction of nonassociative and noncommutative C-loops in [6]. Let G be a multiplicative group with neutral element 1, and A be an abelian group written additively with neutral element 0. Let m : G × G A be a mapping, we can define multiplication on G × A by (g, a)(h, b) = (gh, a + b + m(g, h)) The resulting groupoid is clearly a quasigroup. It will be denoted by (G, A, m). Additional properties of (G, A, m) can be enforced by additional requirements on μ [10-12].  (5) shows that this happens iff Equations (6) and (7) are satisfied.
We now use a particular (G, A, m) to construct the above-mentioned family of Jordan quasigroups.
Then (G, A, m) is a nonassociative Jordan quasigroup.
Proof. To show that J = (G, A, m) is a Jordan quasigroup, we verify Equation (6) because Equation (7) is obviously satisfied. Case 1. There is nothing to prove when g=1. Case 2. Assume that g=x then Equation (6) becomes Case 3. Assume that g = x 2 then Equation (6) becomes m(h,   Since there is no command in Loops package of GAP [9]

Disclosure statement
No potential conflict of interest was reported by the authors.