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alternative definition of a quasigroup


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In the parent entry, a quasigroup is defined as a set, together with a binary operation on it satisfying two formulas, both of which using existential quantifiers.  In this entry, we give an alternative, but equivalent, definition of a quasigroup using only universally quantified formulas.  In other words, the class of quasigroups is an equational class.

\textbf{Definition}.  A \emph{quasigroup} is a set $Q$ with three binary operations $\cdot$ (multiplication), $\backslash$ (\emph{left division}), and $/$ (\emph{right division}), such that the following are satisfied:
\item $(Q,\cdot)$ is a groupoid (not in the category theoretic sense)
\item (left division identities) for all $a,b\in Q$, $a \backslash (a \cdot b)=b$ and $a\cdot (a \backslash b) = b$
\item (right division identities) for all $a,b\in Q$, $(a \cdot b)/ b=a$ and $(a/b) \cdot b = a$

\begin{prop} The two definitions of a quasigroup are equivalent. \end{prop}
Suppose $Q$ is a quasigroup using the definition given in the \PMlinkname{parent entry}{LoopAndQuasigroup}.  Define $\backslash$ on $Q$ as follows: for $a,b\in Q$, set $a\backslash b:=c$ where $c$ is the unique element such that $a\cdot c = b$.  Because $c$ is unique, $\backslash$ is well-defined.  Now, let $x = a\cdot b$ and $y = a\backslash x$.  Since $a\cdot y = x = a \cdot b$, and $y$ is uniquely determined, this forces $y=b$.  Next, let $x=a\backslash b$, then $a \cdot x =b$, or $a \cdot (a\backslash b) = b$.  Similarly, define $/$ on $Q$ so that $a/b$ is the unique element $d$ such that $d\cdot b=a$.  The verification of the two right division identities is left for the reader.

Conversely, let $Q$ be a quasigroup as defined in this entry.  For any $a,b\in Q$, let $c=a\backslash b$ and $d=b/a$.  Then $a \cdot c = a \cdot (a \backslash b) = b$ and $d \cdot a = (b/a) \cdot a = b$.