nLab quasigroup

Idea

A quasigroup is a binary algebraic structure (a magma) in which one-sided multiplication is a bijection in that all equations of the form

$$\begin{aligned} a \cdot x & = b \\ x \cdot a & = b \end{aligned}$$

have a unique solution for $x$.

The notion of quasigroup is hence a generalization of the notion of group, in that it does not require the associativity law nor the existence of an identity element.

If a quasigroup does have a two-sided identity element then it is called a loop (French la boucle, Russian лупа) and a Moufang loop if some further relations are satisfied.

Note that, in the absence of associativity, it is 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, see more at historical notes on quasigroups.

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 $l x = y$ and $x r = y$.

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:

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

Also, without the right quotient we have left quasigroups, and without the left quotient the right quasigroups. Thus quasigroups are described by a Lawvere theory and can therefore be internalized into any cartesian monoidal category. There are weaker structures, say left and right quasigroups in which either $\backslash$ or $/$ is well defined.

Properties

Every left quasigroup $G$ has a function $id_L:G\to G$ defined as $id_L(x)=x\backslash x$ for all $x$ in $G$, and every right quasigroup $G$ has a function $id_R:G\to G$ defined as $id_R(x)=x/x$ for all $x$ in $G$. Every left quasigroup additionally has a function $inv_L:G\to G$ defined as $inv_L(x)=x\backslash id_L(x)$ for all $x$ in $G$, and every right quasigroup has a function $inv_R:G\to G$ defined as $inv_R(x)=id_L(x)/x$ for all $x$ in $G$.

A quasigroup is a possibly empty left loop if $id_L(x) = id_L(y)$ for all $x,y$ in $G$, and it is a possibly empty right loop if $id_R(x) = id_R(y)$ for all $x,y$ in $G$. A quasigroup is a possibly empty loop if additionally $id_L(x) = id_R(y)$ for all $x,y$ in $G$, and it is an invertible quasigroup if $inv_L(x) = inv_R(x)$ for all $x$ in $G$. If the quasigroup is commutative, then the left and right quotients are opposite magmas of each other.

The left (resp. right) quasigroups that are also possibly empty left (right) loops can be explicitly defined through the left (right) quotient operation itself, with multiplication being defined from the left (right) quotient.

Let us define a left quotient on a set $G$ as a binary operation $(-)\backslash(-):G\times G\to G$ and functions $id_L(a) = a\backslash a$ and $inv_L(a) = a\backslash id_L(a)$ such that:

• For all $a$ and $b$ in $G$, $id_L(a)=id_L(b)$
• For all $a$ in $G$, $inv_L(a)\backslash id_L(a)=a$

For any element $a$ in $G$, the element $id_L(a) = a\backslash a$ is called a right identity element, and the element $inv_L(a) = a\backslash id_L(a)$ is called the right inverse element of $a$. For all elements $a$ and $b$ in $G$, left multiplication of $a$ and $b$ is defined as $a \cdot_L b = inv_L(a) \backslash b$.

The multiplication operation is associative if for all $a$, $b$, and $c$ in $G$, $(a\backslash b)\backslash c= b\backslash (a \cdot_L c)$, unital if there exists an element $1$ in $G$ such that for all $a$ in $G$, $id_L(a) = 1$, and commutative if for all $a$ and $b$ in $G$, $a \cdot_L b = b \cdot_L a$.

A right quotient on a set $G$ is a binary operation $(-)/(-):G\times G\to G$ such that:

• For all $a$ and $b$ in $G$, $a/a=b/b$
• For all $a$ in $G$, $(a/a)/((a/a)/a)=a$

For any element $a$ in $G$, the element $a/a$ is called a left identity element, and the element $(a/a)/a$ is called the left inverse element of $a$. For all elements $a$ and $b$, right multiplication of $a$ and $b$ is defined as $a/((b/b)/b)$.

The multiplication operation is associative if for all $a$, $b$, and $c$ in $G$, $a/(b/c)=(a/((c/c)/c)/b$, unital if there exists an element $1$ in $G$ such that for all $a$ in $G$, $a/a=1$, and commutative if for all $a$ and $b$ in $G$, $a/((b/b)/b) = b/((a/a)/a)$.

A quasigroup that is left invertible and right invertible can be defined as a set with a left and right quotient $(G,\backslash,/)$ where left and right multiplications are equal (i.e. $a/((b/b)/b) = (a\backslash (a\backslash a))\backslash b$) for all $a$ and $b$ in $G$. If additionally, there exists a element $1$ in $G$ such that $a\backslash a = 1$ and $a/a = 1$, then the quasigroup is a loop. If the condition is relaxed to the requirements that left and right identity elements are equal (i.e. $a/a = a \backslash a$) for all $a$ in $G$ and the element $1$ is not required to be in $G$, then the loop might possibly be empty. If the associativity requirement is added to the left and right quotients of the quasigroup, then it becomes an associative quasigroup, where equality of left and right identity and inverse elements can be derived.

Examples

• Any group is an associative quasigroup with identity elements.
• Every associative quasigroup, every nonassociative group, and every loop is a quasigroup.
• Every invertible quasigroup is a quasigroup.
• 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.)
• H-spaces are (homotopy-) loops — this is because the shearing maps $(x,y)\mapsto (x,x y)$ and $(x,y)\mapsto (x y, y)$ are equivalences. This generalizes the octonion examples. Note that a loop space is always equivalent to a group, hence not all homotopy loops are loop spaces. In particular …
• 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

There are interesting subvarieties of quasigroups (which are still not associative). Also, left racks (and quandles in particular) are precisely left distributive left quasigroups, with abundance of recent applications in the study of knots and links. Finite racks have been studied in the connection to classification of finite dimensional pointed Hopf algebras. Local augmented Lie racks appeared as integration objects in the local integration theory of Leibniz algebras.

TS-quasigroups are related to Steiner triple systems.

Cayley multiplication tables of finite quasigroups are Latin squares (basically the ‘sudoku squares’ from the quotation here).

As a sample of centipede mathematics, we have the following result on smooth quasigroups, i.e., quasigroups internal to the category of smooth manifolds:

Proposition

The tangent bundle of a smooth quasigroup $Q$ is trivial.

Proof

Suppose WLOG that $Q$ is inhabited by an element $x$, and let $V = T_x(Q)$ be the tangent space at $x$. Define a map

$$\phi \colon Q \times V \to T Q: (y, v) \mapsto (d (K \circ L_y))(v)$$

where $L_y: Q \to Q$ is the smooth map $z \mapsto y z$ and $K: Q \to Q$ is the map $z \mapsto z/x$. The map $\phi$ commutes with the bundle projections $\pi_Q: Q \times V \to Q$, $\pi: T Q \to Q$. The map $K$ has an inverse $J: z \mapsto z x$ and each map $L_y$ has an inverse $M_y: z \mapsto y \backslash z$. We may therefore write down an inverse to $\phi$:

$$T Q \to Q \times V: w \mapsto (\pi(w), d(M_{\pi(w)} \circ J))(w)).$$

This shows $T Q$ is isomorphic to the product bundle $Q \times V$.

Related concepts

• loop (algebra), Moufang loop

• n-ary quasigroup

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

Literature

• eom: quasi-group, Webs, geometry of, Net (in differential geometry)
• R.H. Bruck, A survey of binary systems, Springer-Verlag 1958
• Kenneth Kunen, Quasigroups, loops, and associative laws, J. Algebra 185 (1) (1996), pp. 194–204
• Momo Bangoura, Bigèbres quasi-Lie et boucles de Lie, Bull. Belg. Math. Soc. Simon Stevin 16:4 (2009), 593-616 euclid arXiv:math.SG/0607662; Quasi-bigèbres de Lie et cohomologie d’algèbre de Lie, arxiv/1006.0677
• Lev Vasilʹevich Sabinin, Smooth quasigroups and loops: forty-five years of incredible growth, Commentationes Mathematicae Universitatis Carolinae 41 (2000), No. 2, 377–400 cdml pdf