engleski

The relation between pentagonal and GS-quasigroups

Pentagonal quasigroups are idempotent medial quasigroups in which identity (ab · a)b · a = b holds. GS-quasigroups are idempotent medial quasigroups in which one of the mutually equivalent identities a(ab · c) · c = b, a · (a · bc)c = b hold. We show that in every pentagonal quasigroup we can define GS-quasigroup. Using that we define geometric concepts of GS- trapezium and affine regular pentagon in pentagonal quasigroups, concepts already defined and studied in GSquasigroups. Consequently, pentagonal quasigroups inherit the entire geometry of GS-quasigroups. Geometric representations of some theorems regarding mentioned concepts are given in the quasigroup C(q), where q is a solution of the equation q 4 − 3q 3 + 4q 2 − 2q + 1 = 0.

IM-quasigroup; pentagonal quasigroup; GS-quasigroup; GS-trapezium; affine regular pentagon

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