engleski

Geometry of pentagonal quasigroups

Pentagonal quasigroups are idempotent medial quasigroups satisfying the additional identity of pentagonality, (ab · a)b · a = b. Basic example is C(q) = (C, ∗), where ∗ is binary operation on C defined by a ∗ b = (1 − q)a + qb for a, b ∈ C, and q is a solution of the equation q 4 −3q 3 + 4q 2 −2q + 1 = 0. Using this example as motivation, some geometrical concepts, such as parallelogram, midpoint of a segment, regular pentagon and regular decagon, are defined in a general pentagonal quasigroup. These concepts and their mutual relations are studied and presented in C(q) and in some finite pentagonal quasigroups of order 5 and 11. Using only algebraic identities which hold in pentagonal quasigroups many generalizations of theorems of the Euclidean plane can be proved in a general pentagonal quasigroup.

medial quasigroup; parallelogram; midpoint; regular pentagon; regular decagon

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