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A quasigroup is a pair $(Q,/)$, where $/$ is a binary operation on $Q$, such that (1) for each $a,b\in Q$ there exists unique solutions to the equations $a/x=b$ and $y/a=b$.

Now I want to extract a class of quasigroups that captures characteristics from $(Q_+,/)$, where $Q_+$ is the set of positive rational numbers and $/$ is division. So far I have chosen the three properties below and want to know if they are independent or if any of them can be derived from the other two plus (1):

  1. $a/(b/c)=c/(b/a)$, for all $a,b,c\in Q$
  2. $(a/b)/c=(a/c)/b$, for all $a,b,c\in Q$
  3. $(a/b)/(c/d)=(d/c)/(b/a)$, for all $a,b,c,d\in Q$
Lehs
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2 Answers2

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3 comes from the first two.

$$(a/b)/(c/d)\\=d/(c/(a/b))\\=d/(b/(a/c))\\=(a/c)/(b/d)\\=(a/(b/d))/c\\=(d/(b/a))/c\\=(d/c)/(b/a)$$

Lehs
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Sean English
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Quasigroups constructed from the right division operation in a group ($x/y = xy^{-1}$) are called Ward quasigroups. They are characterized by the identity $(x/y)/(z/y)=x/z$, that is, if $(Q,/)$ is a quasigroup satisfying this identity, then there exists a group structure $(Q,\cdot)$ such that its right division operation is $/$.

The idea has been rediscovered several times. A good reference, including citations to older work, is

K.W. Johnson and P. Vojtěchovský, Right division in groups, Dedekind-Frobenius group matrices, and Ward quasigroups, Abh. Math. Semin. Univ. Hambg. 75 (2005), 121-136. https://doi.org/10.1007/BF02942039

Michael Kinyon
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