Inmathematics, especially inabstract algebra, aquasigroup is analgebraic structure that resembles agroup in the sense that "division" is always possible. Quasigroups differ from groups mainly in that theassociative andidentity element properties are optional. In fact, a nonempty associative quasigroup is a group.^[1][2]

A quasigroup that has an identity element is called aloop.

There are at least two structurally equivalent formal definitions of quasigroup:

• One defines a quasigroup as a set with onebinary operation.
• The other, fromuniversal algebra, defines a quasigroup as having three primitive operations.

Thehomomorphicimage of a quasigroup that is defined with a single binary operation, however, need not be a quasigroup, in contrast to a quasigroup as having three primitive operations.^[3] We begin with the first definition.

Aquasigroup(Q, ∗) is a non-emptysetQ with a binary operation∗ (that is, amagma, indicating that a quasigroup has to satisfy the closure property), obeying theLatin square property. This states that, for eacha andb inQ, there exist unique elementsx andy inQ such that both$${\displaystyle a\ast x=b}$$ $${\displaystyle y\ast a=b}$$  hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, orCayley table. This property ensures that the Cayley table of a finite quasigroup, and, in particular, a finite group, is aLatin square.) The requirement thatx andy be unique can be replaced by the requirement that the magma becancellative.^[4][a]

The unique solutions to these equations are writtenx =a \b andy =b /a. The operations '\' and '/' are called, respectively,left division andright division. With regard to the Cayley table, the first equation (left division) means that theb entry in thea row is in thex column while the second equation (right division) means that theb entry in thea column is in they row.

Theempty set equipped with theempty binary operation satisfies this definition of a quasigroup. Some authors accept the empty quasigroup but others explicitly exclude it.^[5][6]

Given somealgebraic structure, anidentity is an equation in which all variables are tacitlyuniversally quantified, and in which alloperations are among the primitive operations proper to the structure. Algebraic structures that satisfy axioms that are given solely by identities are called avariety. Many standard results inuniversal algebra hold only for varieties. Quasigroups form a variety if left and right division are taken as primitive.

Aright-quasigroup(Q, ∗, /) is a type(2, 2) algebra that satisfy both identities:$${\displaystyle y=(y/x)\ast x}$$ $${\displaystyle y=(y\ast x)/x}$$ 

Aleft-quasigroup(Q, ∗, \) is a type(2, 2) algebra that satisfy both identities:$${\displaystyle y=x\ast (x\mathrm{\setminus }y)}$$ $${\displaystyle y=x\mathrm{\setminus }(x\ast y)}$$ 

Aquasigroup(Q, ∗, \, /) is a type(2, 2, 2) algebra (i.e., equipped with three binary operations) that satisfy the identities:^[b]$${\displaystyle y=(y/x)\ast x}$$ $${\displaystyle y=(y\ast x)/x}$$ $${\displaystyle y=x\ast (x\mathrm{\setminus }y)}$$ $${\displaystyle y=x\mathrm{\setminus }(x\ast y)}$$ 

In other words: Multiplication and division in either order, one after the other, on the same side by the same element, have no net effect.

Hence if(Q, ∗) is a quasigroup according to the definition of the previous section, then(Q, ∗, \, /) is the same quasigroup in the sense of universal algebra. And vice versa: if(Q, ∗, \, /) is a quasigroup according to the sense of universal algebra, then(Q, ∗) is a quasigroup according to the first definition.

Aloop is a quasigroup with anidentity element; that is, an element,e, such that

x ∗e =x ande ∗x =x for allx inQ.

It follows that the identity element,e, is unique, and that every element ofQ has uniqueleft andright inverses (which need not be the same).

A quasigroup with anidempotent element is called apique ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given anabelian group,(A, +), taking its subtraction operation as quasigroup multiplication yields a pique(A, −) with the group identity (zero) turned into a "pointed idempotent". (That is, there is aprincipal isotopy(x,y,z) ↦ (x, −y,z).)

A loop that is associative is a group. A group can have a strictly nonassociative pique isotope, but it cannot have a strictly nonassociative loop isotope.

There are weaker associativity properties that have been given special names.

For instance, aBol loop is a loop that satisfies either:

x ∗ (y ∗ (x ∗z)) = (x ∗ (y ∗x)) ∗z     for eachx,y andz inQ (aleft Bol loop),

or else

((z ∗x) ∗y) ∗x =z ∗ ((x ∗y) ∗x)     for eachx,y andz inQ (aright Bol loop).

A loop that is both a left and right Bol loop is aMoufang loop. This is equivalent to any one of the following single Moufang identities holding for allx,y,z:

x ∗ (y ∗ (x ∗z)) = ((x ∗y) ∗x) ∗z

z ∗ (x ∗ (y ∗x)) = ((z ∗x) ∗y) ∗x

(x ∗y) ∗ (z ∗x) =x ∗ ((y ∗z) ∗x)

(x ∗y) ∗ (z ∗x) = (x ∗ (y ∗z)) ∗x.

According to Jonathan D. H. Smith, "loops" were named after theChicago Loop, as their originators were studying quasigroups in Chicago at the time.^[9]

Smith (2007) names the following important properties and subclasses:

A quasigroup issemisymmetric if any of the following equivalent identities hold for allx,y:^[c]

x ∗y =y /x

y ∗x =x \y

x = (y ∗x) ∗y

x =y ∗ (x ∗y).

Although this class may seem special, every quasigroupQ induces a semisymmetric quasigroupQΔ on the direct product cubeQ³ via the following operation:

(x₁,x₂,x₃) ⋅ (y₁,y₂,y₃) = (y₃ /x₂,y₁ \x₃,x₁ ∗y₂) = (x₂ //y₃,x₃ \\y₁,x₁ ∗y₂),

where "//" and "\\" are theconjugate division operations given byy //x =x /y andy \\x =x \y.

A quasigroup may exhibit semisymmetrictriality.^[10]

A narrower class is atotally symmetric quasigroup (sometimes abbreviatedTS-quasigroup) in which allconjugates coincide as one operation:x ∗y =x /y =x \y. Another way to define (the same notion of) totally symmetric quasigroup is as a semisymmetric quasigroup that is commutative, i.e.x ∗y =y ∗x.

Idempotent total symmetric quasigroups are precisely (i.e. in a bijection with)Steiner triples, so such a quasigroup is also called aSteiner quasigroup, and sometimes the latter is even abbreviated assquag. The termsloop refers to an analogue for loops, namely, totally symmetric loops that satisfyx ∗x = 1 instead ofx ∗x =x. Without idempotency, total symmetric quasigroups correspond to the geometric notion ofextended Steiner triple, also called Generalized Elliptic Cubic Curve (GECC).

A quasigroup(Q, ∗) is calledweakly totally anti-symmetric if for allc,x,y ∈Q, the following implication holds.^[11]

(c ∗x) ∗y = (c ∗y) ∗x implies thatx =y.

A quasigroup(Q, ∗) is calledtotally anti-symmetric if, in addition, for allx,y ∈Q, the following implication holds:^[11]

x ∗y =y ∗x implies thatx =y.

This property is required, for example, in theDamm algorithm.

• Everygroup is a loop, becausea ∗x =bif and only ifx =a⁻¹ ∗b, andy ∗a =b if and only ify =b ∗a⁻¹.
• TheintegersZ (or therationalsQ or therealsR) withsubtraction (−) form a quasigroup. These quasigroups are not loops because there is no identity element (0 is a right identity becausea − 0 =a, but not a left identity because, in general,0 −a ≠a).
• The nonzero rationalsQ^× (or the nonzero realsR^×) withdivision (÷) form a quasigroup.
• Anyvector space over afield ofcharacteristic not equal to 2 forms anidempotent,commutative quasigroup under the operationx ∗y = (x +y) / 2.
• EverySteiner triple system defines anidempotent,commutative quasigroup:a ∗b is the third element of the triple containinga andb. These quasigroups also satisfy(x ∗y) ∗y =x for allx andy in the quasigroup. These quasigroups are known asSteiner quasigroups.^[12]
• The set{±1, ±i, ±j, ±k} whereii = jj = kk = +1 and with all other products as in thequaternion group forms a nonassociative loop of order 8. Seehyperbolic quaternions for its application. (The hyperbolic quaternions themselves donot form a loop or quasigroup.)
• The nonzerooctonions form a nonassociative loop under multiplication. The octonions are a special type of loop known as aMoufang loop.
• An associative quasigroup is either empty or is a group, since if there is at least one element, theinvertibility of the quasigroup binary operation combined with associativity implies the existence of an identity element, which then implies the existence of inverse elements, thus satisfying all three requirements of a group.
• The following construction is due toHans Zassenhaus. On the underlying set of the four-dimensionalvector spaceF⁴ over the 3-elementGalois fieldF =Z/3Z define

  (x₁,x₂,x₃,x₄) ∗ (y₁,y₂,y₃,y₄) = (x₁,x₂,x₃,x₄) + (y₁,y₂,y₃,y₄) + (0, 0, 0, (x₃ −y₃)(x₁y₂ −x₂y₁)).

Then,(F⁴, ∗) is acommutativeMoufang loop that is not a group.^[13]

• More generally, the nonzero elements of anydivision algebra form a quasigroup with the operation of multiplication in the algebra.

In the remainder of the article we shall denote quasigroupmultiplication simply by juxtaposition.

Quasigroups have thecancellation property: ifab =ac, thenb =c. This follows from the uniqueness of left division ofab orac bya. Similarly, ifba =ca, thenb =c.

The Latin square property of quasigroups implies that, given any two of the three variables inxy =z, the third variable is uniquely determined.

The definition of a quasigroup can be treated as conditions on the left and rightmultiplication operatorsL_x,R_x :Q →Q, defined by

L_x(y) =xy

R_x(y) =yx

The definition says that both mappings arebijections fromQ to itself. A magmaQ is a quasigroup precisely when all these operators, for everyx inQ, are bijective. The inverse mappings are left and right division, that is,

L⁻¹
_x(y) =x \y

R⁻¹
_x(y) =y /x

In this notation the identities among the quasigroup's multiplication and division operations (stated in the section onuniversal algebra) are

L_xL⁻¹
_x =id           corresponding to          x(x \y) =y

L⁻¹
_xL_x =id           corresponding to          x \ (xy) =y

R_xR⁻¹
_x =id           corresponding to           (y /x)x =y

R⁻¹
_xR_x =id           corresponding to           (yx) /x =y

whereid denotes the identity mapping onQ.

The multiplication table of a finite quasigroup is aLatin square: ann ×n table filled withn different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.

Conversely, every Latin square can be taken as the multiplication table of a quasigroup in many ways: the border row (containing the column headers) and the border column (containing the row headers) can each be any permutation of the elements. SeeSmall Latin squares and quasigroups.

For acountably infinite quasigroupQ, it is possible to imagine an infinite array in which every row and every column corresponds to some elementq ofQ, and where the elementa ∗b is in the row corresponding toa and the column responding tob. In this situation too, the Latin square property says that each row and each column of the infinite array will contain every possible value precisely once.

For anuncountably infinite quasigroup, such as the group of non-zeroreal numbers under multiplication, the Latin square property still holds, although the name is somewhat unsatisfactory, as it is not possible to produce the array of combinations to which the above idea of an infinite array extends since the real numbers cannot all be written in asequence. (This is somewhat misleading however, as the reals can be written in a sequence of length⁠${\displaystyle \mathfrak{c}}$ ⁠, assuming thewell-ordering theorem.)

The binary operation of a quasigroup isinvertible in the sense that bothL_x andR_x, theleft and right multiplication operators, are bijective, and henceinvertible.

Every loop element has a unique left and right inverse given by

x^λ =e /x          x^λx =e

x^ρ =x \e          xx^ρ =e

A loop is said to have (two-sided)inverses ifx^λ =x^ρ for allx. In this case the inverse element is usually denoted byx⁻¹.

There are some stronger notions of inverses in loops that are often useful:

• A loop has theleft inverse property ifx^λ(xy) =y for allx andy. Equivalently,L⁻¹
  _x =L_x^λ orx \y =x^λy.
• A loop has theright inverse property if(yx)x^ρ =y for allx and y. Equivalently,R⁻¹
  _x =R_x^ρ ory /x =yx^ρ.
• A loop has theantiautomorphic inverse property if(xy)^λ =y^λx^λ or, equivalently, if(xy)^ρ =y^ρx^ρ.
• A loop has theweak inverse property when(xy)z =e if and only ifx(yz) =e. This may be stated in terms of inverses via(xy)^λx =y^λ or equivalentlyx(yx)^ρ =y^ρ.

A loop has theinverse property if it has both the left and right inverse properties. Inverse property loops also have the antiautomorphic and weak inverse properties. In fact, any loop that satisfies any two of the above four identities has the inverse property and therefore satisfies all four.

Any loop that satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses.

A quasigroup or loophomomorphism is amapf :Q →P between two quasigroups such thatf(xy) =f(x)f(y). Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).

LetQ andP be quasigroups. Aquasigroup homotopy fromQ toP is a triple(α,β,γ) of maps fromQ toP such that

α(x)β(y) =γ(xy)

for allx,y inQ. A quasigroup homomorphism is just a homotopy for which the three maps are equal.

Anisotopy is a homotopy for which each of the three maps(α,β,γ) is abijection. Two quasigroups areisotopic if there is an isotopy between them. In terms of Latin squares, an isotopy(α,β,γ) is given by a permutation of rowsα, a permutation of columnsβ, and a permutation on the underlying element setγ.

Anautotopy is an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup forms a group with theautomorphism group as a subgroup.

Every quasigroup is isotopic to a loop. If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup that is isotopic to a group need not be a group. For example, the quasigroup onR with multiplication given by(x,y) ↦ (x +y)/2 is isotopic to the additive group(R, +), but is not itself a group as it has no identity element. Everymedial quasigroup is isotopic to anabelian group by theBruck–Toyoda theorem.

Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation ∗ (i.e.,x ∗y =z) we can form five new operations:x oy :=y ∗x (theopposite operation),/ and\, and their opposites. That makes a total of six quasigroup operations, which are called theconjugates orparastrophes of ∗. Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).

If the setQ has two quasigroup operations, ∗ and ·, and one of them is isotopic to a conjugate of the other, the operations are said to beisostrophic to each other. There are also many other names for this relation of "isostrophe", e.g.,paratopy.

Ann-ary quasigroup is a set with ann-ary operation,(Q,f) withf :Qⁿ →Q, such that the equationf(x₁, ...,x_n) =y has a unique solution for any one variable if all the othern variables are specified arbitrarily.Polyadic ormultiary meansn-ary for some nonnegative integern.

A 0-ary, ornullary, quasigroup is just a constant element ofQ. A 1-ary, orunary, quasigroup is a bijection ofQ to itself. Abinary, or 2-ary, quasigroup is an ordinary quasigroup.

An example of a multiary quasigroup is an iterated group operation,y =x₁ ·x₂ · ··· ·x_n; it is not necessary to use parentheses to specify the order of operations because the group is associative. One can also form a multiary quasigroup by carrying out any sequence of the same or different group or quasigroup operations, if the order of operations is specified.

There exist multiary quasigroups that cannot be represented in any of these ways. Ann-ary quasigroup isirreducible if its operation cannot be factored into the composition of two operations in the following way:

f(x₁, ...,x_n) =g(x₁, ...,x_i−1,h(x_i, ...,x_j),x_j+1, ...,x_n),

where1 ≤i <j ≤n and(i,j) ≠ (1,n). Finite irreduciblen-ary quasigroups exist for alln > 2; seeAkivis & Goldberg (2001) for details.

Ann-ary quasigroup with ann-ary version ofassociativity is called ann-ary group.

The number of isomorphism classes of small quasigroups (sequenceA057991 in theOEIS) and loops (sequenceA057771 in theOEIS) is given here:^[14]

• Division ring – a ring in which every non-zero element has a multiplicative inverse
• Semigroup – an algebraic structure consisting of a set together with an associative binary operation
• Monoid – a semigroup with an identity element
• Planar ternary ring – has an additive and multiplicative loop structure
• Problems in loop theory and quasigroup theory
• Mathematics of Sudoku

• quasigroups
• "Quasi-group",Encyclopedia of Mathematics,EMS Press, 2001 [1994]