nLab
quasigroup

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Context

Algebra

Group Theory

Contents

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

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

have a unique solution for xx.

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=xy 1x/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 GG equipped with a binary operation G×GGG \times G \to G (which we will write with concatenation) such that:

  • for all xx and yy, there exist unique ll and rr such that lx=yl x = y and xr=yx r = y.

Then ll is called the left quotient y/xy / x (yy divided by xx, yy over xx) and rr is called the right quotient x\yx \backslash y (xx dividing yy, xx under yy).

Note that we must specify, in the definition, that ll and rr 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 GG equipped with three binary operations (product, left quotient, and right quotient) such that these equations always hold:

  • (x/y)y=x(x / y) y = x,
  • x(x\y)=yx (x \backslash y) = y,
  • (xy)/y=x(x y) / y = x,
  • x\(xy)=yx \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 GG has a function id L:GGid_L:G\to G defined as id L(x)=x\xid_L(x)=x\backslash x for all xx in GG, and every right quasigroup GG has a function id R:GGid_R:G\to G defined as id R(x)=x\xid_R(x)=x\backslash x for all xx in GG. Every left quasigroup additionally has a function inv L:GGinv_L:G\to G defined as inv L(x)=x\id L(x)inv_L(x)=x\backslash id_L(x) for all xx in GG, and every right quasigroup has a function inv R:GGinv_R:G\to G defined as inv R(x)=id L(x)/xinv_R(x)=id_L(x)/x for all xx in GG.

A quasigroup is a possibly empty left loop if id L(x)=id L(y)id_L(x) = id_L(y) for all x,yx,y in GG, and it is a possibly empty right loop if id R(x)=id R(y)id_R(x) = id_R(y) for all x,yx,y in GG. A quasigroup is a possibly empty loop if additionally id L(x)=id R(y)id_L(x) = id_R(y) for all x,yx,y in GG, and it is an invertible quasigroup if inv L(x)=inv R(x)inv_L(x) = inv_R(x) for all xx in GG. 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 GG as a binary operation ()\():G×GG(-)\backslash(-):G\times G\to G and functions id L(a)=a\aid_L(a) = a\backslash a and inv L(a)=a\id L(a)inv_L(a) = a\backslash id_L(a) such that:

  • For all aa and bb in GG, id L(a)=id L(b)id_L(a)=id_L(b)
  • For all aa in GG, inv L(a)\id L(a)=ainv_L(a)\backslash id_L(a)=a

For any element aa in GG, the element id L(a)=a\aid_L(a) = a\backslash a is called a right identity element, and the element inv L(a)=a\id L(a)inv_L(a) = a\backslash id_L(a) is called the right inverse element of aa. For all elements aa and bb in GG, left multiplication of aa and bb is defined as a Lb=inv L(a)\ba \cdot_L b = inv_L(a) \backslash b.

The multiplication operation is associative if for all aa, bb, and cc in GG, (a\b)\c=b\(a Lc)(a\backslash b)\backslash c= b\backslash (a \cdot_L c), unital if there exists an element 11 in GG such that for all aa in GG, id L(a)=1id_L(a) = 1, and commutative if for all aa and bb in GG, a Lb=b Laa \cdot_L b = b \cdot_L a.

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

  • For all aa and bb in GG, a/a=b/ba/a=b/b
  • For all aa in GG, (a/a)/((a/a)/a)=a(a/a)/((a/a)/a)=a

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

The multiplication operation is associative if for all aa, bb, and cc in GG, a/(b/c)=(a/((c/c)/c)/ba/(b/c)=(a/((c/c)/c)/b, unital if there exists an element 11 in GG such that for all aa in GG, a/a=1a/a=1, and commutative if for all aa and bb in GG, a/((b/b)/b)=b/((a/a)/a)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,\,/)(G,\backslash,/) where left and right multiplications are equal (i.e. a/((b/b)/b)=(a\(a\a))\ba/((b/b)/b) = (a\backslash (a\backslash a))\backslash b) for all aa and bb in GG. If additionally, there exists a element 11 in GG such that a\a=1a\backslash a = 1 and a/a=1a/a = 1, then the quasigroup is a loop. If the comdition is relaxed to the requirements that left and right identity elements are equal (i.e. a/a=a\aa/a = a \backslash a) for all aa in GG and the element 11 is not required to be in GG, 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)(x,xy)(x,y)\mapsto (x,x y) and (x,y)(xy,y)(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 QQ is trivial.

Proof

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

ϕ:Q×VTQ:(y,v)(d(KL y))(v)\phi \colon Q \times V \to T Q: (y, v) \mapsto (d (K \circ L_y))(v)

where L y:QQL_y: Q \to Q is the smooth map zyzz \mapsto y z and K:QQK: Q \to Q is the map zz/xz \mapsto z/x. The map ϕ\phi commutes with the bundle projections π Q:Q×VQ\pi_Q: Q \times V \to Q, π:TQQ\pi: T Q \to Q. The map KK has an inverse J:zzxJ: z \mapsto z x and each map L yL_y has an inverse M y:zy\zM_y: z \mapsto y \backslash z. We may therefore write down an inverse to ϕ\phi:

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

This shows TQT Q is isomorphic to the product bundle Q×VQ \times V.

algebraic structureoidification
truth valuetransitive relation
magmamagmoid
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
(left,right) cancellative monoid(left,right) cancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
differential ring?differential ringoid?
nonassociative algebralinear magmoid
nonassociative unital algebraunital linear magmoid
nonunital algebralinear semicategory
associative unital algebralinear category
C-star algebraC-star category
differential algebradifferential algebroid
flexible algebraflexible linear magmoid
alternative algebraalternative linear magmoid
Lie algebraLie algebroid
strict monoidal categorystrict 2-category
strict 2-groupstrict 2-groupoid
monoidal poset?2-poset
monoidal groupoid?(2,1)-category
monoidal category2-category/bicategory
2-group2-groupoid/bigroupoid

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

Last revised on May 25, 2021 at 19:44:29. See the history of this page for a list of all contributions to it.