# Questions tagged [quasigroups]

A quasigroup is a grouplike structure $(Q, \ast)$, that satisfies the Latin square property but need not have an identity element, nor need it be associative. It coincides with the notion of a divisible magma.

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### Quasi-Group represented by a graph which is not a Triangle-Free Graph locally

Can each of all quasi-groups be represented by a graph (latin square graph), which is not locally triangle free graph ? Quasi Group can be represented by Latin Square matrix, thus by a Latin ...
• 499
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### In a loop, if $(xy)x^r = y$ then $x(yx^r)=y$

Consider a loop $L$, that is, a quasigroup with an identity, and recall that a quasigroup $L$ is a set together with a binary operation such that, for every $a$ and $b$ in $L$, the equations $ax=b$ ...
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### If $H$ is a subloop of a finte loop $L$ and $N$ is a normal subloop of $L$, then $HN$ is a subloop of $L$.

To prove this is a subloop, I have to show that for $x, y \in HN$, the following are also in $HN$: (a) $xy$, (b) $L^{-1}_x(y)$ and (c) $R^{-1}_{x}(y)$. Here $L_x(y) = xy$ and $R_x(y)=yx$. We have to ...
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### Does every non-empty quasigroup have a left or right identity?

I know that some quasigroups are not loops, meaning they don't have a two-sided identity. But are there non-empty quasigroups that don't even have one-sided identities?
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### A simple question about rational numbers without a simple proof?

As in this question, study the quasigroup $(Q_+,/)$ of positive rational numbers under division. There are two obvious identities: $a/(b/c)=c/(b/a)$, for all $a,b,c\in Q_+$ $(a/b)/c=(a/c)/b$, for all ...
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### A familiar quasigroup - about independent axioms

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 ...
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