You are here
Homealternative definition of a quasigroup
Primary tabs
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), $\backslash$ (left division), and $/$ (right division), such that the following are satisfied:

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

(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. 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 welldefined. 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 $d\cdot a=(b/a)\cdot a=b$. ∎
Mathematics Subject Classification
20N05 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections