Vectors and Transfers in Hexagonal Quasigroup

Hexagonal quasigroup is idempotent, medial and semisymmetric quasigroup. In this article we define and study vectors, sum of vectors and transfers. The main result is the theorem on isomorphism between the group of vectors, group of transfers and the Abelian group from the characterization theorem of the hexagonal quasigroups. 1. Hexagonal quasigroup Hexagonal quasigroups are defined in article [3]. Definition 1.1. A quasigroup (Q, ·) is called hexagonal if it is idempotent, medial and semisymmetric; i.e. if its elements a, b, c, d satisfy a · a = a (a · b) · (c · d) = (a · c) · (b · d) a · (b · a) = (a · b) · a = b. When it doesn’t cause confusion, we can omit the sign ”·”, e.g. instead of (a · b) · (c · d) we shall write ab · cd. Theorem 1.2. In any hexagonal quasigroup (Q, ·) the identities a · bc = ab · ac and ab · c = ac · bc hold for all a, b, c ∈ Q. The equalities ab = c, bc = a and ca = b are equivalent. The basic example of a hexagonal quasigroup studied in [3] is the following. 2000 Mathematics Subject Classification. 20N05.