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Existence and Uniqueness of Centers of Regular Polygons in Some Subclasses of IM-Quasigroups

The concept of an IM-quasigroup, an idempotent medial quasigroup, is defined as a quasigroup whose elements satisfy the identities of idempotency and mediality. Motivated by the example C(q) some geometrical concepts can be defined in an IM-quasigroup. Many subclasses of IM-quasigroups have been defined and studied, such as GS-quasigroups, quadratical quasigroups, hexagonal quasigroups and pentagonal quasigroups. In this paper a special emphasis is put on hexagonal and pentagonal quasigroups. The notions of parallelogram, midpoint of a segment, regular triangle and regular hexagon are defined in hexagonal quasigroups. Some finite examples of hexagonal quasigroups are given and existence and uniqueness of the midpoint of a segment and of the centers of regular triangles and regular hexagons are studied. The notions of regular pentagon and regular decagon are defined in pentagonal quasigroups. Existence and uniqueness of their centers are studied in some finite examples of pentagonal quasigroups. The resuls are justified using characterization of these subclasses of IM-quasigroups via abelian groups which possess certain types of automorphisms.

IM-quasigroup ; hexagonal quasigroup ; pentagonal quasigroup ; midpoint ; regular polygon

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