Construction of semisymmetric entropic quasigroups

Abstract

Elliptic curves are non-singular cubic plane-curves over a field containing at least one point and naturally form a group, where the associative property is based on Cayley-Bacharach theorem. A semisymmetric entropic (medial) quasigroup (SEQ) is a non-commutative generalization of this principle. This is a magma satisfying the pair of identities: and. Every SEQ is associated (not necessarily uniquely) with an Abelian group. In this paper, we give methods for constructing SEQs based on their corresponding Abelian groups. In particular, we show that SEQs are central quasigroups.