alternative definition of a quasigroup

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.

Definition. A quasigroup is a set $Q$ with three binary operations $\cdot$ (multiplication), $\backslash \backslash$ (left division), and $/$ (right division), such that the following are satisfied:

• $(Q,\cdot )$ is a groupoid (not in the category theoretic sense)

• (left division identities) for all $a,b\in Q$, $a\backslash \backslash (a\cdot b)=b$ and $a\cdot (a\backslash \backslash b)=b$

• (right division identities) for all $a,b\in Q$, $(a\cdot b)/b=a$ and $(a/b)\cdot b=a$

Proposition 1.

The two definitions of a quasigroup are equivalent.

Proof.

Suppose $Q$ is a quasigroup using the definition given in the parent entry (http://planetmath.org/LoopAndQuasigroup). Define $\backslash \backslash$ on $Q$ as follows: for $a,b\in Q$, set $a\backslash \backslash b:=c$ where $c$ is the unique element such that $a\cdot c=b$. Because $c$ is unique, $\backslash \backslash$ is well-defined. Now, let $x=a\cdot b$ and $y=a\backslash \backslash x$. Since $a\cdot y=x=a\cdot b$, and $y$ is uniquely determined, this forces $y=b$. Next, let $x=a\backslash \backslash b$, then $a\cdot x=b$, or $a\cdot (a\backslash \backslash b)=b$. 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 $c=a\backslash \backslash b$ and $d=b/a$. Then $a\cdot c=a\cdot (a\backslash \backslash b)=b$ and $d\cdot a=(b/a)\cdot a=b$.∎