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| In [[mathematics]], especially in [[abstract algebra]], a '''quasigroup''' is an [[algebraic structure]] resembling a [[group (mathematics)|group]] in the sense that "[[division (mathematics)|division]]" is always possible. Quasigroups differ from groups mainly in that they need not be [[associative]].
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| A quasigroup with an identity element is called a '''loop'''.
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| {{Algebraic structures |Group}}
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| == Definitions ==
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| There are at least two equivalent formal definitions of quasigroup. One definition casts quasigroups as a set with one [[binary operation]], and the other is a version from [[universal algebra]] which describes a quasigroup by using three primitive operations. We begin with the first definition, which is easier to follow.
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| A '''quasigroup''' {{nowrap|(''Q'', *)}} is a [[Set (mathematics)|set]] ''Q'' with a binary operation * (that is, a [[magma (algebra)|magma]]), obeying the [[Latin square property]]. This states that, for each ''a'' and ''b'' in ''Q'', there exist unique elements ''x'' and ''y'' in ''Q'' such that:
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| *''a'' * ''x'' = ''b'' ;
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| *''y'' * ''a'' = ''b'' .
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| (In other words: For two elements ''a'' and ''b'', ''b'' can be found in row ''a'' and in column ''a'' of the quasigroup's multiplication table, or [[Cayley table]]. So the Cayley tables of finite quasigroups are simply [[latin square]]s.)<br>
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| The unique solutions to these equations are written {{nowrap|1=''x'' = ''a'' \ ''b''}} and {{nowrap|1=''y'' = ''b'' / ''a''}}. The operations '\' and '/' are called, respectively, '''left''' and '''right''' '''division'''.
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| The [[empty set]] equipped with the empty binary operation satisfies the definition of a quasigroup, but some authors explicitly exclude this case.<ref>{{cite book | author = Hala O. Pflugfelder | title = Quasigroups and loops: introduction | publisher = Heldermann Verlag | year = 1990 | page = 2}}</ref><ref>{{cite book | author = Richard Hubert Bruck | title = A survey of binary systems | publisher = Springer | year = 1971 | page = 1}}</ref>
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| ===Universal algebra===
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| 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 axiomatized solely by identities are called [[variety (universal algebra)|varieties]]. Many standard results in [[universal algebra]] hold only for varieties. Quasigroups are varieties if left and right division are taken as primitive.
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| A '''quasigroup''' {{nowrap|(''Q'', *, \, /)}} is a type (2,2,2) algebra satisfying the identities:
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| *''y'' = ''x'' * (''x'' \ ''y'') ;
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| *''y'' = ''x'' \ (''x'' * ''y'') ;
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| *''y'' = (''y'' / ''x'') * ''x'' ;
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| *''y'' = (''y'' * ''x'') / ''x'' .
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| Hence if {{nowrap|(''Q'', *)}} is a quasigroup according to the first definition, then {{nowrap|(''Q'', *, \, /)}} is the same quasigroup in the sense of universal algebra.
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| ===Loop===
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| A '''loop''' is a quasigroup with an [[identity element]], that is, an element ''e'' such that:
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| *''x'' * ''e'' = ''x'' and ''e'' * ''x'' = ''x'' for all ''x'' in ''Q''.
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| It follows that the identity element ''e'' is unique, and that every element of ''Q'' has a unique [[inverse element|left]] and [[inverse element|right inverse]]. Since the presence of an identity element is essential, a loop cannot be empty.
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| A '''[[Moufang loop]]''' is a loop that satisfies the Moufang identity:
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| *(''x'' * ''y'') * (''z'' * ''x'') = ''x'' * ((''y'' * ''z'') * ''x'') .
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| == Examples ==
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| * 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>}}.
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| * The [[integer]]s '''Z''' with [[subtraction]] (−) form a quasigroup.
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| * The nonzero [[rational numbers|rationals]] '''Q'''<sup>×</sup> (or the nonzero [[real number|reals]] '''R'''<sup>×</sup>) with [[division (mathematics)|division]] (÷) form a quasigroup.
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| * 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}}.
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| * 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''.<ref>{{harvnb|Colbourn|Dinitz|2007|loc=pg. 497, definition 28.12}}</ref>
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| * The set {{nowrap|1={±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).
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| * The nonzero [[octonions]] form a nonassociative loop under multiplication. The octonions are a special type of loop known as a [[Moufang loop]].
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| * An associative quasigroup is either empty or is a group, since if there is at least one element, the existence of inverses and and associativity imply the existence of an identity.
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| * 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
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| :(''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>)).
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| :Then, {{nowrap|('''F'''<sup>4</sup>, *)}} is a [[commutative]] [[Moufang loop]] that is not a group.{{Citation needed|date=March 2013}}
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| * More generally, the set of nonzero elements of any [[division algebra]] form a quasigroup.
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| == Properties ==
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| :<small>In the remainder of the article we shall denote quasigroup multiplication simply by juxtaposition.</small>
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| Quasigroups have the [[cancellation property]]: if {{nowrap|1=''ab'' = ''ac''}}, then {{nowrap|1=''b'' = ''c''}}. This follows from the uniqueness of left division of ''ab'' or ''ac'' by ''a''. Similarly, if {{nowrap|1=''ba'' = ''ca''}}, then {{nowrap|1=''b'' = ''c''}}.
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| ===Multiplication operators===
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| The definition of a quasigroup can be treated as conditions on the left and right multiplication operators {{nowrap|''L''(''x''), ''R''(''y''): ''Q'' → ''Q''}}, defined by
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| :<math>\begin{align}
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| L(x)y &= xy \\
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| R(x)y &= yx
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| \end{align}</math>
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| 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,
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| :<math>\begin{align}
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| L(x)^{-1}y &= x\backslash y \\
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| R(x)^{-1}y &= y/x
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| \end{align}</math>
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| In this notation the identities among the quasigroup's multiplication and division operations (stated in the section on [[#Universal_algebra|universal algebra]]) are
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| :<math>\begin{align}
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| L(x)L(x)^{-1} &= 1\qquad&\text{corresponding to}\qquad x(x\backslash y) &= y \\
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| L(x)^{-1}L(x) &= 1\qquad&\text{corresponding to}\qquad x\backslash(xy) &= y \\
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| R(x)R(x)^{-1} &= 1\qquad&\text{corresponding to}\qquad (y/x)x &= y \\
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| R(x)^{-1}R(x) &= 1\qquad&\text{corresponding to}\qquad (yx)/x &= y
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| \end{align}</math>
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| where 1 denotes the identity mapping on ''Q''.
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| ===Latin squares===
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| The multiplication table of a finite quasigroup is a [[Latin square]]: an {{nowrap|''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.
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| 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]].
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| ===Inverse properties===
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| Every loop element has a unique left and right inverse given by
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| :<math>x^{\lambda} = e/x \qquad x^{\lambda}x = e</math>
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| :<math>x^{\rho} = x\backslash e \qquad xx^{\rho} = e</math>
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| A loop is said to have (''two-sided'') ''inverses'' if <math>x^{\lambda} = x^{\rho}</math> for all ''x''. In this case the inverse element is usually denoted by <math>x^{-1}</math>.
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| There are some stronger notions of inverses in loops which are often useful:
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| *A loop has the ''left inverse property'' if <math>x^{\lambda}(xy) = y</math> for all <math>x</math> and <math>y</math>. Equivalently, <math>L(x)^{-1} = L(x^{\lambda})</math> or <math>x\backslash y = x^{\lambda}y</math>.
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| *A loop has the ''right inverse property'' if <math>(yx)x^{\rho} = y</math> for all <math>x</math> and <math>y</math>. Equivalently, <math>R(x)^{-1} = R(x^{\rho})</math> or <math>y/x = yx^{\rho}</math>.
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| *A loop has the ''antiautomorphic inverse property'' if <math>(xy)^{\lambda} = y^{\lambda}x^{\lambda}</math> or, equivalently, if <math>(xy)^{\rho} = y^{\rho}x^{\rho}</math>.
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| *A loop has the ''weak inverse property'' when <math>(xy)z = e</math> if and only if <math>x(yz) = e</math>. This may be stated in terms of inverses via <math>(xy)^{\lambda}x = y^{\lambda}</math> or equivalently <math>x(yx)^{\rho} = y^{\rho}</math>.
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| 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 which satisfies any two of the above four identities has the inverse property and therefore satisfies all four.
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| Any loop which satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses.
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| ==Morphisms==
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| 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).
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| ===Homotopy and isotopy===
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| {{Main|Isotopy of loops}}
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| 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
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| :<math>\alpha(x)\beta(y) = \gamma(xy)\,</math>
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| for all ''x'', ''y'' in ''Q''. A quasigroup homomorphism is just a homotopy for which the three maps are equal.
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| 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.
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| An '''autotopy''' is an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup form a group with the [[automorphism group]] as a subgroup.
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| Each 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 which is isotopic to a group need not be a group. For example, the quasigroup on '''R''' with multiplication given by {{nowrap|(''x'' + ''y'')/2}} is isotopic to the additive group {{nowrap|('''R''', +)}}, but is not itself a group. Every [[medial magma|medial]] quasigroup is isotopic to an [[abelian group]] by the [[Medial magma#Bruck–Toyoda theorem|Bruck–Toyoda theorem]].
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| ===Conjugation (parastrophe)===
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| Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation * (i.e., {{nowrap|1=''x'' * ''y'' = ''z''}}) we can form five new operations: {{nowrap|1=''x'' o ''y'' := ''y'' * ''x''}} (the '''opposite''' operation), / and \, and their opposites. That makes a total of six quasigroup operations, which are called the '''conjugates''' or '''parastrophes''' of *. Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).
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| ===Paratopy===
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| If the set ''Q'' has two quasigroup operations, * and ·, and one of them is isotopic to a conjugate of the other, the operations are said to be '''paratopic''' to each other. There are also many other names for this relation of "paratopy", e.g., '''isostrophe'''.
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| == Generalizations ==
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| === Polyadic or multiary quasigroups ===<!-- This section is linked from [[Multiary quasigroup]] -->
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| An ''n''-'''ary quasigroup''' is a set with an [[arity|''n''-ary operation]], {{nowrap|(''Q'', ''f'')}} with {{nowrap|''f'': ''Q''<sup>''n''</sup> → ''Q''}}, such that the equation {{nowrap|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''.
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| 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.
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| An example of a multiary quasigroup is an iterated group operation, {{nowrap|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.
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| 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:
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| :<math> f(x_1,\dots,x_n) = g(x_1,\dots,x_{i-1},\,h(x_i,\dots,x_j),\,x_{j+1},\dots,x_n), </math>
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| where {{nowrap|1 ≤ ''i'' < ''j'' ≤ ''n''}} and {{nowrap|(''i, j'') ≠ (1, ''n'')}}. Finite irreducible ''n''-ary quasigroups exist for all {{nowrap|''n'' > 2}}; see Akivis and Goldberg (2001) for details.
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| An ''n''-ary quasigroup with an ''n''-ary version of [[associativity]] is called an [[n-ary group]].
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| === Right- and left-quasigroups ===
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| {{expand section|date=March 2011}}
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| A '''right-quasigroup''' {{nowrap|(''Q'', *, /)}} is a type (2,2) algebra satisfying the identities:
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| *''y'' = (''y'' / ''x'') * ''x'';
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| *''y'' = (''y'' * ''x'') / ''x''.
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| Similarly, a '''left-quasigroup''' {{nowrap|(''Q'', *, \)}} is a type (2,2) algebra satisfying the identities:
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| *''y'' = ''x'' * (''x'' \ ''y'');
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| *''y'' = ''x'' \ (''x'' * ''y'').
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| == Number of small quasigroups and loops ==
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| The number of isomorphism classes of small quasigroups {{OEIS|A057991}} and loops {{OEIS|A057771}} is given here:<ref>[http://cs.anu.edu.au/~bdm/papers/ls_final.pdf McKay, Meynert, Myrvold, ''Small Latin Squares, Quasigroups and Loops'']</ref>
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| {| class="wikitable" style="text-align: center;"
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| |-
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| ![[Order (group theory)|Order]]
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| !Number of quasigroups
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| !Number of loops
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| |-
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| |0
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| |1
| |
| |0
| |
| |-
| |
| |1
| |
| |1
| |
| |1
| |
| |-
| |
| |2
| |
| |1
| |
| |1
| |
| |-
| |
| |3
| |
| |5
| |
| |1
| |
| |-
| |
| |4
| |
| |35
| |
| |2
| |
| |-
| |
| |5
| |
| |1,411
| |
| |6
| |
| |-
| |
| |6
| |
| |1,130,531
| |
| |109
| |
| |-
| |
| |7
| |
| |12,198,455,835
| |
| |23,746
| |
| |-
| |
| |8
| |
| |2,697,818,331,680,661
| |
| |106,228,849
| |
| |-
| |
| |9
| |
| |15,224,734,061,438,247,321,497
| |
| |9,365,022,303,540
| |
| |-
| |
| |10
| |
| |2,750,892,211,809,150,446,995,735,533,513
| |
| |20,890,436,195,945,769,617
| |
| |-
| |
| |11
| |
| |19,464,657,391,668,924,966,791,023,043,937,578,299,025
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| |1,478,157,455,158,044,452,849,321,016
| |
| |-
| |
| |}
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| == See also ==
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| *[[Bol loop]]
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| *[[Division ring]] – a ring in which every non-zero element has a multiplicative inverse
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| *[[Semigroup]] – an algebraic structure consisting of a set together with an associative binary operation
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| *[[Monoid]] – a semigroup with an identity element
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| *[[Planar ternary ring]] – has an additive and multiplicative loop structure
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| *[[Small Latin squares and quasigroups]]
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| *[[Problems in loop theory and quasigroup theory]]
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| *[[Mathematics of Sudoku]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * Akivis, M. A., and Vladislav V. Goldberg (2001), "Solution of Belousov's problem," ''Discussiones Mathematicae. General Algebra and Applications'' 21: 93–103.
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| * Bruck, R.H. (1958), ''A Survey of Binary Systems''. Springer-Verlag.
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| * Chein, O., H. O. Pflugfelder, and J.D.H. Smith, eds. (1990), ''Quasigroups and Loops: Theory and Applications''. Berlin: Heldermann. ISBN 3-88538-008-0.
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| * {{citation|last1=Colbourn|first1=Charles J.|last2=Dinitz|first2=Jeffrey H.|title=Handbook of Combinatorial Designs|year=2007|publisher=Chapman & Hall/ CRC|location=Boca Raton|isbn=1-58488-506-8|edition=2nd Edition}}
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| * Dudek, W.A., and Glazek, K. (2008), "Around the Hosszu-Gluskin Theorem for n-ary groups," Discrete Math. 308: 4861-4876.
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| * Pflugfelder, H.O. (1990), ''Quasigroups and Loops: Introduction''. Berlin: Heldermann. ISBN 3-88538-007-2.
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| * Smith, J.D.H. (2007), ''An Introduction to Quasigroups and their Representations''. Chapman & Hall/CRC Press. ISBN 1-58488-537-8.
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| * Smith, J.D.H. and Anna B. Romanowska (1999), ''Post-Modern Algebra''. Wiley-Interscience. ISBN 0-471-12738-8.
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| ==External links==
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| * [http://www-math.ucdenver.edu/~wcherowi/courses/m6406/csln2.html quasigroups]
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| * {{springer|title=Akivis algebra|id=p/a110450}}
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| [[Category:Non-associative algebra]]
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| [[Category:Group theory]]
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| [[Category:Latin squares]]
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