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# {{math|1=''x'' ∗ ''y'' = ''y'' ∗ ''x'' ⇒ ''x'' = ''y''}},
and it is called weak totally anti-symmetric if only the first implication holds. Damm proved that the existence of a totally anti-symmetric quasigroup of order {{math|''n''}} is equivalent to the existence of a weak totally anti-symmetric quasigroup of order {{math|''n''}}. For the Damm algorithm with the check equation
{{math|1=(...((0 ∗ ''x<sub>m</sub>'') ∗ ''x''<sub>''m''−1</sub>) ∗ ...) ∗ ''x''<sub>0</sub> = 0}},
a weak totally anti-symmetric quasigroup with the property
{{math|1=''x'' ∗ ''x'' = 0}}
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