Editing Quasigroup (section)

== 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.