Editing Quasigroup

{{Short description|Magma obeying the Latin square property}}
[[File:Magma to group4.svg|thumb|right|300px|Algebraic structures between [[Magma (algebra)|magmas]] and [[Group (mathematics)|groups]]: A ''quasigroup'' is a [[magma (algebra)|magma]] with the type of [[Division (mathematics)#Abstract algebra|divisibility]] given by the [[Quasigroup#Algebra|Latin square property]]. A [[Quasigroup#Loops|loop]] is a ''quasigroup'' with an [[identity element]].]]{{No footnotes|date=February 2024}}
In [[mathematics]], especially in [[abstract algebra]], a '''quasigroup''' is an [[algebraic structure]] that resembles a [[group (mathematics)|group]] in the sense that "[[division (mathematics)|division]]" is always possible. Quasigroups differ from groups mainly in that the [[associative]] and [[identity element]] properties are optional. In fact, a nonempty associative quasigroup is a group.<ref>[https://groupprops.subwiki.org/wiki/Nonempty_associative_quasigroup_equals_group Nonempty associative quasigroup equals group]</ref><ref>[https://planetmath.org/anassociativequasigroupisagroup an associative quasigroup is a group]</ref>

A quasigroup that has an identity element is called a '''loop'''.

{{Algebraic structures |Group}}

== Definitions ==
There are at least two structurally equivalent formal definitions of quasigroup:
* One defines a quasigroup as a set with one [[binary operation]].
* The other, from [[universal algebra]], defines a quasigroup as having three primitive operations.

The [[Homomorphism|homomorphic]] [[Image (mathematics)|image]] 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.{{sfn|ps=|Smith|2007|pp=3,26–27}} We begin with the first definition.

=== Algebra ===
A '''quasigroup''' {{math|(''Q'', ∗)}} is a non-empty [[Set (mathematics)|set]] {{mvar|Q}} with a binary operation {{math|∗}} (that is, a [[magma (algebra)|magma]], indicating that a quasigroup has to satisfy the closure property), obeying the '''Latin square property'''. This states that, for each {{mvar|a}} and {{mvar|b}} in {{mvar|Q}}, there exist unique elements {{mvar|x}} and {{mvar|y}} in {{mvar|Q}} such that both
<math display=block>a \ast x = b</math>
<math display=block>y \ast a = b</math> 
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, or [[Cayley table]]. This property ensures that the Cayley table of a finite quasigroup, and, in particular, a finite group, is a [[Latin square]].)  The requirement that {{mvar|x}} and {{mvar|y}} be unique can be replaced by the requirement that the magma be [[Cancellation property|cancellative]].{{sfn|ps=|Rubin|Rubin|1985|p=[https://archive.org/details/equivalentsofaxi0000rubi/page/109 109]}}{{efn|For clarity, cancellativity alone is insufficient: the requirement for existence of a solution must be retained.}}

The unique solutions to these equations are written {{math|''x'' {{=}} ''a'' \ ''b''}} and {{math|''y'' {{=}} ''b'' / ''a''}}.  The operations '{{math|\}}' and '{{math|/}}' are called, respectively, [[left division]] and [[right division]].  With regard to the Cayley table, the first equation (left division) means that the {{mvar|b}} entry in the {{mvar|a}} row is in the {{mvar|x}} column while the second equation (right division) means that the {{mvar|b}} entry in the {{mvar|a}} column is in the {{mvar|y}} row.

The [[empty set]] equipped with the [[Function_(mathematics)#Standard_functions|empty binary operation]] satisfies this definition of a quasigroup.  Some authors accept the empty quasigroup but others explicitly exclude it.{{sfn|ps=|Pflugfelder|1990|p=2}}{{sfn|ps=|Bruck|1971|p=1}}

=== Universal algebra ===
Given some [[algebraic structure]], an [[mathematical identity|identity]] is an equation in which all variables are tacitly [[universal quantifier|universally quantified]], and in which all [[Operation (mathematics)|operations]] are among the primitive operations proper to the structure. Algebraic structures that satisfy axioms that are given solely by identities are called a [[variety (universal algebra)|variety]]. Many standard results in [[universal algebra]] hold only for varieties. Quasigroups form a variety if left and right division are taken as primitive.

A '''right-quasigroup''' {{math|(''Q'', ∗, /)}} is a type {{nowrap|(2, 2)}} algebra that satisfy both identities:
<math display=block>y = (y / x) \ast x</math>
<math display=block>y = (y \ast x) / x</math>

A '''left-quasigroup''' {{math|(''Q'', ∗, \)}} is a type {{nowrap|(2, 2)}} algebra that satisfy both identities:
<math display=block>y = x \ast (x \backslash y)</math>
<math display=block>y = x \backslash (x \ast y)</math>

A '''quasigroup''' {{math|(''Q'', ∗, \, /)}} is a type {{nowrap|(2, 2, 2)}} algebra (i.e., equipped with three binary operations) that satisfy the identities:{{efn|1=There are six identities that these operations satisfy, namely:{{sfn|ps=|Shcherbacov|Pushkashu|Shcherbacov|2021|p=1}}
<math display=block>y = (y / x) \ast x</math>
<math display=block>y = x \backslash (x \ast y)</math>
<math display=block>y = x / (y \backslash x)</math>
<math display=block>y = (y \ast x) / x</math>
<math display=block>y = x \ast (x \backslash y)</math>
<math display=block>y = (x / y) \backslash x</math>
Of these, the first three imply the last three, and vice versa, leading to either set of three identities being sufficient to equationally specify a quasigroup.{{sfn|ps=|Shcherbacov|Pushkashu|Shcherbacov|2021|p=3|loc=Thm.&nbsp;1,&nbsp;2}}
}}
<math display=block>y = (y / x) \ast x</math>
<math display=block>y = (y \ast x) / x</math>
<math display=block>y = x \ast (x \backslash y)</math>
<math display=block>y = x \backslash (x \ast y)</math>

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

== Loops ==
A '''loop''' is a quasigroup with an [[identity element]]; that is, an element, ''e'', such that
: ''x'' ∗ ''e'' = ''x'' and ''e'' ∗ ''x'' = ''x'' for all ''x'' in ''Q''.

It follows that the identity element, ''e'', is unique, and that every element of ''Q'' has unique [[inverse element|left]] and [[inverse element|right inverse]]s (which need not be the same).

A quasigroup with an [[idempotent element]] is called a '''pique''' ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an [[abelian group]], {{nowrap|(''A'', +)}}, taking its subtraction operation as quasigroup multiplication yields a pique {{nowrap|(''A'', −)}} with the group identity (zero) turned into a "pointed idempotent".  (That is, there is a [[Quasigroup#Homotopy and isotopy|principal isotopy]] {{nowrap|(''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, a '''[[Bol loop]]''' is a loop that satisfies  either:
: ''x'' ∗ (''y'' ∗ (''x'' ∗ ''z'')) = (''x'' ∗ (''y'' ∗ ''x'')) ∗ ''z''{{quad}} for each ''x'', ''y'' and ''z'' in ''Q'' (a ''left Bol loop''),
or else
: ((''z'' ∗ ''x'') ∗ ''y'') ∗ ''x'' = ''z'' ∗ ((''x'' ∗ ''y'') ∗ ''x''){{quad}} for each ''x'', ''y'' and ''z'' in ''Q'' (a ''right Bol loop'').

A loop that is both a left and right Bol loop is a '''[[Moufang loop]]'''.  This is equivalent to any one of the following single Moufang identities holding for all ''x'', ''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 the [[Chicago Loop]], as their originators were studying quasigroups in Chicago at the time.<ref>{{cite web |last1=Smith |first1=Jonathan D. H. |title=Codes, Errors, and Loops |url=https://www.youtube.com/watch?v=NvHJ_dOG5Qc&t=2094 |website=Recording of the Codes & Expansions Seminar |date=2 April 2024 |access-date=2 April 2024}}</ref>

== Symmetries ==
{{harvtxt|Smith|2007}} names the following important properties and subclasses:

=== Semisymmetry ===
A quasigroup is '''semisymmetric''' if any of the following equivalent identities hold for all ''x'', ''y'':{{efn|The first two equations are equivalent to the last two by direct application of the cancellation property of quasigroups.  The last pair are shown to be equivalent by setting {{math|1=''x'' = ((''x'' ∗ ''y'') ∗ ''x'') ∗ (''x'' ∗ ''y'') = ''y'' ∗ (''x'' ∗ ''y'')}}.}}
: {{math|1=''x'' ∗ ''y'' = ''y'' / ''x''}}
: {{math|1=''y'' ∗ ''x'' = ''x'' \ ''y''}}
: {{math|1=''x'' = (''y'' ∗ ''x'') ∗ ''y''}}
: {{math|1=''x'' = ''y'' ∗ (''x'' ∗ ''y'').}}

Although this class may seem special, every quasigroup ''Q'' induces a semisymmetric quasigroup ''Q''Δ on the direct product cube ''Q''<sup>3</sup> via the following operation:
: {{math|1=(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) ⋅ (''y''<sub>1</sub>, ''y''<sub>2</sub>, ''y''<sub>3</sub>) = (''y''<sub>3</sub> / ''x''<sub>2</sub>, ''y''<sub>1</sub> \ ''x''<sub>3</sub>, ''x''<sub>1</sub> ∗ ''y''<sub>2</sub>) = (''x''<sub>2</sub> // ''y''<sub>3</sub>, ''x''<sub>3</sub> \\ ''y''<sub>1</sub>, ''x''<sub>1</sub> ∗ ''y''<sub>2</sub>),}}
where "{{math|1=//}}" and "{{math|1=\\}}" are the [[#Conjugation (parastrophe)|conjugate division operations]] given by {{math|1=''y'' // ''x'' = ''x'' / ''y''}} and {{math|1=''y'' \\ ''x'' = ''x'' \ ''y''}}.

=== Triality ===
{{expand section|date=February 2015}}
A quasigroup may exhibit semisymmetric [[triality]].<ref>{{cite book |last=Smith |first=Jonathan D. H. |url=https://jdhsmith.math.iastate.edu/math/GTaH.pdf |title=Groups, Triality, and Hyperquasigroups |publisher=Iowa State University}}</ref>

=== Total symmetry ===
A narrower class is a '''totally symmetric quasigroup''' (sometimes abbreviated '''TS-quasigroup''') in which all [[#Conjugation_(parastrophe)|conjugates]] coincide as one operation: {{math|1=''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. {{math|1=''x'' ∗ ''y'' = ''y'' ∗ ''x''}}.

Idempotent total symmetric quasigroups are precisely (i.e. in a bijection with) [[Steiner system|Steiner triples]], so such a quasigroup is also called a '''Steiner quasigroup''', and sometimes the latter is even abbreviated as '''squag'''.  The term '''sloop''' refers to an analogue for loops, namely, totally symmetric loops that satisfy {{math|1=''x'' ∗ ''x'' = 1}} instead of {{math|1=''x'' ∗ ''x'' = ''x''}}. Without idempotency, total symmetric quasigroups correspond to the geometric notion of [[extended Steiner triple]], also called Generalized Elliptic Cubic Curve (GECC).

=== Total antisymmetry ===
A quasigroup {{nowrap|(''Q'', ∗)}} is called '''weakly totally anti-symmetric'''  if for all {{nowrap|''c'', ''x'', ''y'' ∈ ''Q''}}, the following implication holds.{{sfn|ps=|Damm|2007}}
: (''c'' ∗ ''x'') ∗ ''y'' = (''c'' ∗ ''y'') ∗ ''x'' implies that ''x'' = ''y''.

A quasigroup {{nowrap|(''Q'', ∗)}} is called '''totally anti-symmetric''' if, in addition, for all {{nowrap|''x'', ''y'' ∈ ''Q''}}, the following implication holds:{{sfn|ps=|Damm|2007}}
: ''x'' ∗ ''y'' = ''y'' ∗ ''x'' implies that ''x'' = ''y''.

This property is required, for example, in the [[Damm algorithm]].

== Examples ==
* Every [[group (mathematics)|group]] is a loop, because {{nowrap|1=''a'' ∗ ''x'' = ''b''}} [[if and only if]] {{nowrap|1=''x'' = ''a''<sup>−1</sup> ∗ ''b''}}, and {{nowrap|1=''y'' ∗ ''a'' = ''b''}} if and only if {{nowrap|1=''y'' = ''b'' ∗ ''a''<sup>−1</sup>}}.
* The [[integer]]s '''Z''' (or the [[rational numbers|rationals]] '''Q''' or the [[real number|reals]] '''R''') with [[subtraction]] (−) form a quasigroup. These quasigroups are not loops because there is no identity element (0 is a right identity because {{nowrap|1=''a'' − 0 = ''a''}}, but not a left identity because, in general, {{nowrap|1=0 − ''a'' ≠ ''a''}}).
* The nonzero rationals '''Q'''<sup>×</sup> (or the nonzero reals '''R'''<sup>×</sup>) with [[division (mathematics)|division]] (÷) form a quasigroup.
* Any [[vector space]] over a [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] not equal to 2 forms an [[idempotent]], [[commutative]] quasigroup under the operation {{nowrap|1=''x'' ∗ ''y'' = (''x'' + ''y'') / 2}}.
* Every [[Steiner system|Steiner triple system]] defines an [[idempotent]], [[commutative]] quasigroup: {{nowrap|1=''a'' ∗ ''b''}} is the third element of the triple containing ''a'' and ''b''. These quasigroups also satisfy {{nowrap|1=(''x'' ∗ ''y'') ∗ ''y'' = ''x''}} for all ''x'' and ''y'' in the quasigroup. These quasigroups are known as ''Steiner quasigroups''.{{sfn|ps=|Colbourn|Dinitz|2007|p=497|loc=definition&nbsp;28.12}}
* The set {{nowrap|1={{mset|±1, ±i, ±j, ±k}}}} where {{nowrap|1=ii = jj = kk = +1}} and with all other products as in the [[quaternion group]] forms a nonassociative loop of order 8. See [[hyperbolic quaternion#Historical review|hyperbolic quaternions]] for its application. (The hyperbolic quaternions themselves do ''not'' form a loop or quasigroup.)
* The nonzero [[octonions]] form a nonassociative loop under multiplication. The octonions are a special type of loop known as a [[Moufang loop]].
* An associative quasigroup is either empty or is a group, since if there is at least one element, the [[Quasigroup#Inverse_properties|invertibility]] 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 to [[Hans Zassenhaus]]. On the underlying set of the four-dimensional [[vector space]] '''F'''<sup>4</sup> over the 3-element [[Galois field]] {{nowrap|1='''F''' = '''Z'''/3'''Z'''}} define
*: (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ''x''<sub>4</sub>) ∗ (''y''<sub>1</sub>, ''y''<sub>2</sub>, ''y''<sub>3</sub>, ''y''<sub>4</sub>) = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ''x''<sub>4</sub>) + (''y''<sub>1</sub>, ''y''<sub>2</sub>, ''y''<sub>3</sub>, ''y''<sub>4</sub>) + (0, 0, 0, (''x''<sub>3</sub> − ''y''<sub>3</sub>)(''x''<sub>1</sub>''y''<sub>2</sub> − ''x''<sub>2</sub>''y''<sub>1</sub>)).
: Then, {{nowrap|('''F'''<sup>4</sup>, ∗)}} is a [[commutative]] [[Moufang loop]] that is not a group.{{sfn|ps=|Romanowska|Smith|1999|p=93}}
* More generally, the nonzero elements of any [[division algebra]] form a quasigroup with the operation of multiplication in the algebra.

== Properties ==
: <small>In the remainder of the article we shall denote quasigroup [[multiplication by juxtaposition|multiplication simply by juxtaposition]].</small>

Quasigroups have the [[cancellation property]]: if {{math|1=''ab'' = ''ac''}}, then {{math|1=''b'' = ''c''}}. This follows from the uniqueness of left division of ''ab'' or ''ac'' by ''a''. Similarly, if {{math|1=''ba'' = ''ca''}}, then {{math|1=''b'' = ''c''}}.

The Latin square property of quasigroups implies that, given any two of the three variables in {{math|1=''xy'' = ''z''}}, the third variable is uniquely determined.

=== Multiplication operators ===
The definition of a quasigroup can be treated as conditions on the left and right [[multiplication operator]]s {{math|''L''{{sub|x}}, ''R''{{sub|x}} : ''Q'' → ''Q''}}, defined by 
: ''L''<sub>''x''</sub>(''y'') = ''xy''
: ''R''<sub>''x''</sub>(''y'') = ''yx''

The definition says that both mappings are [[bijection]]s from ''Q'' to itself. A magma ''Q'' is a quasigroup precisely when all these operators, for every ''x'' in ''Q'', are bijective. The inverse mappings are left and right division, that is, 
: {{math|1=''L''{{su|lh=.9|b=''x''|p=−1}}(''y'') = ''x'' \ ''y''}}
: {{math|1=''R''{{su|lh=.9|b=''x''|p=−1}}(''y'') = ''y'' / ''x''}}

In this notation the identities among the quasigroup's multiplication and division operations (stated in the section on [[#Universal_algebra|universal algebra]]) are
: {{math|1=''L''<sub>''x''</sub>''L''{{su|lh=.9|b=''x''|p=−1}} = {{math|id}} {{space|10}} corresponding to {{space|10}} ''x''(''x'' \ ''y'') = ''y''}}
: {{math|1=''L''{{su|lh=.9|b=''x''|p=−1}}''L''<sub>''x''</sub> = {{math|id}} {{space|10}} corresponding to {{space|10}} ''x'' \ (''xy'') = ''y''}}
: {{math|1=''R''<sub>''x''</sub>''R''{{su|lh=.9|b=''x''|p=−1}} = {{math|id}} {{space|10}} corresponding to {{space|10}} (''y'' / ''x'')''x'' = ''y''}}
: {{math|1=''R''{{su|lh=.9|b=''x''|p=−1}}''R''<sub>''x''</sub> = {{math|id}} {{space|10}} corresponding to {{space|10}} (''yx'') / ''x'' = ''y''}}
where {{math|id}} denotes the identity mapping on ''Q''.

=== Latin squares ===
<div style="float:right;">
{| class="wikitable" style="text-align:center;font-weight:bold;width:275px;height:275px;outline:2px solid;"
|+ A Latin square, the unbordered multiplication table for a quasigroup whose 10 elements are the digits 0–9.
|0||4||8||2||3||9||6||7||1||5
|-
|3||6||2||8||7||1||9||5||0||4
|-
|8||9||3||1||0||6||4||2||5||7
|-
|1||7||6||5||4||8||0||3||2||9
|-
|2||1||9||0||6||7||5||8||4||3
|-
|5||2||7||4||9||3||1||0||8||6
|-
|4||3||0||6||1||5||2||9||7||8
|-
|9||8||5||7||2||0||3||4||6||1
|-
|7||0||1||9||5||4||8||6||3||2
|-
|6||5||4||3||8||2||7||1||9||0
|}
</div>
{{main|Latin square}}

The multiplication table of a finite quasigroup is a [[Latin square]]: an {{math|''n'' × ''n''}} table filled with ''n'' 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. See ''[[Small Latin squares and quasigroups]]''.

==== Infinite quasigroups ====
For a [[countably infinite]] quasigroup ''Q'', it is possible to imagine an infinite array in which every row and every column corresponds to some element ''q'' of ''Q'', and where the element {{math|''a'' ∗ ''b''}} is in the row corresponding to ''a'' and the column responding to ''b''. 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 an [[uncountably infinite]] quasigroup, such as the group of non-zero [[real number]]s 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 a [[sequence]]. (This is somewhat misleading however, as the reals can be written in a sequence of length {{tmath|1= \mathfrak{c} }}, assuming the [[well-ordering theorem]].){{clear}}

=== Inverse properties ===
The binary operation of a quasigroup is '''invertible''' in the sense that both ''L''<sub>''x''</sub> and ''R''<sub>''x''</sub>, the [[Quasigroup#Multiplication_operators|left and right multiplication operators]], are bijective, and hence [[invertible function|invertible]].

Every loop element has a unique left and right inverse given by
: {{math|1=''x''<sup>''λ''</sup> = ''e'' / ''x'' {{space|10}} ''x''<sup>''λ''</sup>''x'' = ''e''}}
: {{math|1=''x''<sup>''ρ''</sup> = ''x'' \ ''e'' {{space|10}} ''xx''<sup>''ρ''</sup> = ''e''}}

A loop is said to have (''two-sided'') ''inverses'' if {{math|1=''x''<sup>''λ''</sup> = ''x''<sup>''ρ''</sup>}} for all ''x''. In this case the inverse element is usually denoted by {{itco|''x''}}<sup>−1</sup>.

There are some stronger notions of inverses in loops that are often useful:
* A loop has the ''left inverse property'' if {{math|1=''x''<sup>''λ''</sup>(''xy'') = ''y''}} for all ''x'' and ''y''. Equivalently, {{math|1=''L''{{su|lh=.9|b=''x''|p=−1}} = ''L''<sub>''x''<sup>''λ''</sup></sub>}} or {{math|1=''x'' \ ''y'' = ''x''<sup>''λ''</sup>''y''}}.
* A loop has the ''right inverse property'' if {{math|1=(''yx'')''x''<sup>''ρ''</sup> = ''y''}} for all ''x'' and&nbsp;''y''. Equivalently, {{math|1=''R''{{su|lh=.9|b=''x''|p=−1}} = ''R''<sub>''x''<sup>''ρ''</sup></sub>}} or {{math|1=''y'' / ''x'' = ''yx''<sup>''ρ''</sup>}}.
* A loop has the ''antiautomorphic inverse property'' if {{math|1=(''xy'')<sup>''λ''</sup> = ''y''<sup>''λ''</sup>''x''<sup>''λ''</sup>}} or, equivalently, if {{math|1=(''xy'')<sup>''ρ''</sup> = ''y''<sup>''ρ''</sup>''x''<sup>''ρ''</sup>}}.
* A loop has the ''weak inverse property'' when {{math|1=(''xy'')''z'' = ''e''}} if and only if {{math|1=''x''(''yz'') = ''e''}}. This may be stated in terms of inverses via {{math|1=(''xy'')<sup>''λ''</sup>''x'' = ''y''<sup>''λ''</sup>}} or equivalently {{math|1=''x''(''yx'')<sup>''ρ''</sup> = ''y''<sup>''ρ''</sup>}}.

A loop has the ''inverse 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.

== Morphisms ==
A quasigroup or loop [[homomorphism]] is a [[map (mathematics)|map]] {{nowrap|''f'' : ''Q'' → ''P''}} between two quasigroups such that {{nowrap|1=''f''(''xy'') = ''f''(''x'')''f''(''y'')}}. Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).

=== Homotopy and isotopy ===
{{Main|Isotopy of loops}}
Let ''Q'' and ''P'' be quasigroups. A '''quasigroup homotopy''' from ''Q'' to ''P'' is a triple {{nowrap|(''α'', ''β'', ''γ'')}} of maps from ''Q'' to ''P'' such that
: ''α''(''x'')''β''(''y'') = ''γ''(''xy'')
for all ''x'', ''y'' in ''Q''. A quasigroup homomorphism is just a homotopy for which the three maps are equal.

An '''isotopy''' is a homotopy for which each of the three maps {{nowrap|(''α'', ''β'', ''γ'')}} is a [[bijection]]. Two quasigroups are '''isotopic''' if there is an isotopy between them. In terms of Latin squares, an isotopy {{nowrap|(''α'', ''β'', ''γ'')}} is given by a permutation of rows ''α'', a permutation of columns ''β'', and a permutation on the underlying element set ''γ''.

An '''autotopy''' is an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup forms a group with the [[automorphism 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 on '''R''' with multiplication given by {{nowrap|(''x'', ''y'') ↦ (''x'' + ''y'')/2}} is isotopic to the additive group {{nowrap|('''R''', +)}}, but is not itself a group as it has no identity element. Every [[medial magma|medial]] quasigroup is isotopic to an [[abelian group]] by the [[Medial magma#Bruck–Murdoch–Toyoda_theorem|Bruck–Toyoda theorem]].

=== Conjugation (parastrophe) ===
Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation&nbsp;∗ (i.e., {{math|1=''x'' ∗ ''y'' = ''z''}}) we can form five new operations: {{math|1=''x'' o ''y'' := ''y'' ∗ ''x''}} (the '''opposite''' operation), {{math|1=/}} and {{math|1=\}}, and their opposites.  That makes a total of six quasigroup operations, which are called the '''conjugates''' or '''parastrophes''' of&nbsp;∗.   Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).

=== Isostrophe (paratopy) ===
If the set ''Q'' has two quasigroup operations, ∗ and&nbsp;·, and one of them is isotopic to a conjugate of the other, the operations are said to be '''isostrophic''' to each other.  There are also many other names for this relation of "isostrophe", e.g., '''paratopy'''.

== Generalizations ==
=== Polyadic or multiary quasigroups ===
<!-- This section is linked from [[Multiary quasigroup]] -->
An ''n''-'''ary quasigroup''' is a set with an [[arity|''n''-ary operation]], {{math|(''Q'', ''f'')}} with {{math|''f'' : ''Q''<sup>''n''</sup> → ''Q''}}, such that the equation {{math|1=''f''(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'') = ''y''}} has a unique solution for any one variable if all the other ''n'' variables are specified arbitrarily.  '''Polyadic''' or '''multiary''' means ''n''-ary for some nonnegative integer ''n''.

A 0-ary, or '''nullary''', quasigroup is just a constant element of ''Q''.  A 1-ary, or '''unary''', quasigroup is a bijection of ''Q'' to itself.  A '''binary''', or 2-ary, quasigroup is an ordinary quasigroup.

An example of a multiary quasigroup is an iterated group operation, {{math|1=''y'' = ''x''<sub>1</sub> · ''x''<sub>2</sub> · ··· · ''x''<sub>''n''</sub>}}; 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.  An ''n''-ary quasigroup is '''irreducible''' if its operation cannot be factored into the composition of two operations in the following way:
: ''f''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) = ''g''(''x''<sub>1</sub>, ..., ''x''<sub>''i''−1</sub>, ''h''(''x''<sub>''i''</sub>, ..., ''x''<sub>''j''</sub>), ''x''<sub>''j''+1</sub>, ..., ''x''<sub>''n''</sub>),
where {{math|1 ≤ ''i'' &lt; ''j'' ≤ ''n''}} and {{math|(''i'', ''j'') ≠ (1, ''n'')}}.  Finite irreducible ''n''-ary quasigroups exist for all {{math|''n'' > 2}}; see {{harvtxt|Akivis|Goldberg|2001}} for details.

An ''n''-ary quasigroup with an ''n''-ary version of [[associativity]] is called an [[n-ary group|''n''-ary group]].

== Number of small quasigroups and loops ==
{{main|Small Latin squares and quasigroups}}
The number of isomorphism classes of small quasigroups {{OEIS|A057991}} and loops {{OEIS|A057771}} is given here:{{sfn|ps=|McKay|Meynert|Myrvold|2007}}

{| class="wikitable" style="text-align: right;"   
|-
  ! [[Order (group theory)|Order]]
  ! Number of quasigroups
  ! Number of loops
|-
  | 0
  | 1
  | 0
|-
  | 1
  | 1
  | 1
|-
  | 2
  | 1
  | 1
|-
  | 3
  | 5
  | 1
|-
  | 4
  | 35
  | 2
|-
  | 5
  | {{val|1,411}}
  | 6
|-
  | 6
  | {{val|1,130,531}}
  | 109
|-
  | 7
  | {{val|12,198,455,835}}
  | {{val|23,746}}
|-
  | 8
  | {{val|2,697,818,331,680,661}}
  | {{val|106,228,849}}
|-
  | 9
  | {{val|15,224,734,061,438,247,321,497}}
  | {{val|9,365,022,303,540}}
|-
  | 10
  | {{val|2,750,892,211,809,150,446,995,735,533,513}}
  | {{val|20,890,436,195,945,769,617}}
|-
  | 11
  | {{val|19,464,657,391,668,924,966,791,023,043,937,578,299,025}}
  | {{val|1,478,157,455,158,044,452,849,321,016}}
|-
|}

== See also ==
* [[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]]

== Notes ==
{{notelist}}

== References ==
=== Citations ===
{{reflist|30em}}

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{{refend}}

== External links ==
* [http://www-math.ucdenver.edu/~wcherowi/courses/m6406/csln2.html quasigroups]
* {{eom|title=Quasi-group}}

[[Category:Non-associative algebra]]
[[Category:Group theory]]
[[Category:Latin squares]]