A Representation of a Point Symmetric 2-Structure by a Quasi-Domain |
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Authors: | Helmut Karzel Jarosław Kosiorek Andrzej Matraś |
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Institution: | 1. Zentrum Mathematik, T.U. München, 80290, Munich, Germany 2. Faculty of Mathematics and Computer Science, University of Warmia and Mazury, S?oneczna 54, 10-710, Olsztyn, Poland
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Abstract: | In a symmetric 2-structure ${\Sigma =(P,\mathfrak{G}_1,\mathfrak{G}_2,\mathfrak{K})}$ we fix a chain ${E \in \mathfrak{K}}$ and define on E two binary operations “+” and “·”. Then (E,+) is a K-loop and for ${E^* := E {\setminus}\{o\}}$ , (E *,·) is a Bol loop. If ${\Sigma}$ is even point symmetric then (E,+ ,·) is a quasidomain and one has the set ${Aff(E,+,\cdot) := \{a^+\circ b^\bullet | a \in E, b \in E^*\}}$ of affine permutations. From Aff(E, +, ·) one can reproduce via a “chain derivation” the point symmetric 2-structure ${\Sigma}$ . |
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