nLab
quasigroup

Idea

A quasigroup is a generalization of a group without the associativity law or identity element. A quasigroup with identity is called a loop.

Note that, in the absence of associativity, it's not enough (even for a loop) to say that every element has an inverse element (on either side); instead, you must say that division is always possible. This is because the definition $x/y=x{y}^{-1}$ won't work right without associativity.

Some consider the concept of quasigroup to be an example of centipede mathematics and uninteresting due to their lack of deep applications. For example, one mathematician has written:

The meeting was dominated by algebraic loop theory. It occured to me that as a way to use your intellectual resources this was very akin in significance to doing a difficult sudoku, a thought that was made very ironic when one speaker started making loops out of what were essentially sudoku squares.

Nonetheless it can be instructive to ponder these concepts, and there are some nontrivial examples.

Definitions

The usual definition is this:

Definition

A quasigroup is a set $G$ equipped with a binary operation $G\times G\to G$ (which we will write with concatenation) such that:

• for all $x$ and $y$, there exist unique $l$ and $r$ such that $lx=y$ and $xr=y$.

Then $l$ is called the left quotient $y/x$ ($y$ divided by $x$, $y$ over $x$) and $r$ is called the right quotient $x\backslash y$ ($x$ dividing $y$, $x$ under $y$).

Note that we must specify, in the definition, that $l$ and $r$ are unique; without associativity, we cannot prove this.

As with the inverse elements of a group, we can make the quotients into operations so that all axioms are equations:

Definition

A quasigroup is a set $G$ equipped with three binary operations (product, left quotient, and right quotient) such that these equations always hold:

• $(x/y)y=x$,
• $x(x\backslash y)=y$,
• $(xy)/y=x$,
• $x\backslash (xy)=y$.

Thus quasigroups are described by a Lawvere theory and can therefore be internalized into any cartesian monoidal category.

In any case:

Definition

A loop is a quasigroup with an identity element.

Loops are also described by a Lawvere theory.

Note that, even in a loop, left and right inverses need not agree. See the discussion on the English Wikipedia for convenient inverse properties.

For good measure, here is another special kind of quasigroup:

Definition

A group is an associative loop.

Examples

• Any group is a loop, of course.
• Any abelian group is a quasigroup in two other ways: the product switches places with one of the quotients. (The other quotient remains a quotient.)
• The nonzero elements of a (not necessarily associative) division algebra (such as the octonions) form a quasigroup; this fact is basically the definition of ‘division algebra’.

Applications

Quasigroups have applications to the study of Latin square?s (basically the ‘sudoku squares’ from the quotation above).