engleski

Pentagonal quasigroups

A pentagonal quasigroup is an idempotent medial quasigroup which satisfies additional identity ((ab · a) · b) · a = b. Along with that identity, some other algebraic identities and their mutual relations are studied. The geometrical concepts of parallelogram, midpoint and regular pentagon can be defined in a general pentagonal quasigroup. The geometrical representation of these concepts and relations between them will be given in the quasigroup C(q), where q is a solution of the equation q 4 −3q 3 + 4q 2 −2q + 1 = 0. The characterization of pentagonal quasigroups using abelian groups with automorphism ϕ which satisfies ϕ 4 − 3ϕ 3 + 4ϕ 2 − 2ϕ + 1 = 0 is given. This characterization gives an algorithm for construction of some finite pentagonal quasigroups.

IM-quasigroup; pentagonality; regular pentagon

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