Study of Jordan quasigroups and their construction

Abstract

Jordan quasigroups are commutative quasigroups satisfying the identity $x^{2}(yx)=(x^{2}y)x$. 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, $x^{3}y=y^{3}x\ \forall \ x,y\in Q$ 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.

Keywords

• Jordan loops
• Jordan's identity
• Jordan quasigroup
• unipotent quasigroup and entropic quasigroup