Symmetries in Hexagonal Quasigroups

When it doesn’t cause confusion, we can omit the sign “·”, e.g. instead of (a · b) · (c · d) we may write ab · cd. In this article, Q will always be a hexagonal quasigroup. The basic example of hexagonal quasigroup is formed by the points of Euclidean plane, with the operation · such that the points a, b and a · b form a positively oriented regular triangle. This structure was used for all the illustrations in this article. Motivated by this example, Volenec in [3] and [4] introduced some geometric terms to any hexagonal quasigroup. Some of these terms can be defined in any idempotent medial quasigroup (see [2]) or even medial quasigroup (see [1]). The elements of hexagonal quasigroup are called points, and pairs of points are called segments.