nLab loop (algebra)

This entry is about loops in algebra. For loops in topology see loop (topology).

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Idea

In algebra a loop is a quasigroup with (two-sided) identity element.

Definition

With multiplication, divisions and identities

A left loop is a unital magma $(G, (-)\cdot(-):G\times G\to G,1:G)$ equipped with a left division $(-)\backslash(-):G \times G \to G$ such that $x \cdot (x \backslash y) = y$ and $x \backslash (x \cdot y) = y$. A right loop is a unital magma $(G, (-)\cdot(-):G\times G\to G,1:G)$ equipped with a right division $(-)/(-):G \times G \to G$ such that $(x / y) \cdot y = x$ and $(x \cdot y) / y = x$. A two-sided loop or just a loop is a unital magma that is both a left loop and a right loop.

With multiplication, inverses, and identity

Equivalently, one could speak of left and right inverse elements instead of left and right division:

A left loop is a unital magma $(G, (-)\cdot(-):G\times G\to G,1:G)$ equipped with a left inverse $^{-1}(-):G \to G$ such that $x \cdot (^{-1}x \cdot y) = y$ and $^{-1}x \cdot (x \cdot y) = y$. A right loop is a unital magma $(G, (-)\cdot(-):G\times G\to G,1:G)$ equipped with a right inverse $(-)^{-1}:G \to G$ such that $(x \cdot y^{-1}) \cdot y = x$ and $(x \cdot y) \cdot y^{-1} = x$. A two-sided loop or just a loop is a unital magma that is both a left loop and a right loop. One then defines left division to be $x \backslash y = {^{-1}}x \cdot y$ and right division to be $x / y = x \cdot y^{-1}$.

With divisions and identity

There is another definition of a loop using only division and identity:

A left loop is a pointed magma $(G,\backslash,1)$ such that:

• For all $a$ and $b$ in $G$, $a\backslash a=1$
• For all $a$ in $G$, $(a\backslash 1)\backslash 1=a$

A right loop is a pointed magma $(G,/)$ such that:

• For all $a$ and $b$ in $G$, $a/a=1$
• For all $a$ in $G$, $1/(1/a)=a$

A loop is a left and right loop as defined above $(G,\backslash,/,1)$ such that $a/(1/b) = (a\backslash 1)\backslash b$ for all $a$ and $b$ in $G$.

Properties

Loops are 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. A loop with a two-sided inverse is a nonassociative group.

Examples

• Any group is a loop.

• Any nonassociative group is a loop.

• The nonzero elements of a (not necessarily associative) unital division algebra (such as the octonions) form a quasigroup; this fact is basically the definition of ‘division algebra’.

• code loops are loops which are central extensions of abelian groups (actually vector spaces over the finite field $\mathbb{F}_2$) by $\mathbb{Z}_2$.

• A Moufang loop is a loop.

Applications

Local analytic loops have interesting induced structure on the tangent space at the identity, generalizing the Lie algebra of a group, see Sabinin algebra. Sabinin algebras are closely related to the local study of affine connections on manifolds. They include some known important classes of nonassociative algebras, namely Lie algebras, Mal’cev algebras, Lie triple systems (related to the study of symmetric spaces), Bol algebras as simplest cases.

Related concepts

• Moufang loop

• possibly empty loop

• nonassociative group

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

• Kenneth Kunen, Quasigroups, loops, and associative laws, J. Algebra 185 (1) (1996), pp. 194–204

• Péter T. Nagy, Karl Strambach, Loops as invariant sections in groups, and their geometry, Canad. J. Math. 46(1994), 1027-1056 doi

• 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

See also:

• Wikipedia, Loops