engleski

Geometry of ARO–quasigroups

In this presentation a new class of idempotent medial quasigroups will be introduced, the so– called ARO–quasigroups. A quasigroup will be called ARO–quasigroup if it satisfies the identities of idempotency and mediality, i.e. we have the identities aa = a, ab · cd = ac · bd, and besides that if the identity ab · b = ba · a is also valid. Some examples of ARO–quasigroups will be given as well. These quasigroups are interesting because of the possibility of defining affine–regular octagons and to study them by means of formal calculations in a quasigroup. The “geometrical” concepts of a parallelogram and midpoint will be introduced in a general ARO– quasigroup. Some results about the introduced geometric concepts will be proved and a number of statements about new points obtained from the vertices of an affine–regular octagon will also be studied.

ARO–quasigroup; affine–regular octagon

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