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→Design: Adding definition of "totally anti-symmetric quasigroup". It may need a little rearranging. |
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== Design ==
Its essential part is a [[quasigroup]] of [[Order (group theory)|order]] 10 (i.e. having a 10×10 [[Latin square]] as [[Cayley table|operation table]]) with the special feature of being totally anti-symmetric. <ref name=dhmd /><ref name=damm2007 /><ref group=lower-roman name=BIS2003 /><ref group=lower-roman name=Chen2009 /> Damm revealed several methods to create such totally anti-symmetric quasigroups of order 10 and gave some examples in his doctoral dissertation.<ref name=dhmd /><ref group=lower-roman name=BIS2003 /> With this, Damm also disproved an old conjecture that totally anti-symmetric quasigroups of order 10 do not exist.<ref name=damm2003 />
A quasigroup {{math|(''Q'', ∗)}} is called totally anti-symmetric if for all {{math|''c'', ''x'', ''y'' ∈ ''Q''}}, the following implications hold:<ref name=damm2007 />
# {{math|1=(''c'' ∗ ''x'') ∗ ''y'' = (''c'' ∗ ''y'') ∗ ''x'' ⇒ ''x'' = ''y''}}
# {{math|1=''x'' ∗ ''y'' = ''y'' ∗ ''x'' ⇒ ''x'' = ''y''}}.
== Algorithm ==
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