Quasigroup

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, especially in
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, a quasigroup is an
algebraic structure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
resembling a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
in the sense that "
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
" is always possible. Quasigroups differ from groups mainly in that they are not necessarily
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. A quasigroup with an identity element is called a loop.

# Definitions

There are at least two structurally equivalent formal definitions of quasigroup. One defines a quasigroup as a set with one
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, and the other, from
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spa ...
, defines a quasigroup as having three primitive operations. The homomorphic
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of a quasigroup defined with a single binary operation, however, need not be a quasigroup. We begin with the first definition.

## Algebra

A quasigroup is a non-empty set ''Q'' with a binary operation ∗ (that is, a
magma Magma () is the molten or semi-molten natural material from which all igneous rock Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main The three types of rocks, rock types, the others ...
, indicating that a quasigoup has to satisfy closure property), 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 both :''a'' ∗ ''x'' = ''b'', :''y'' ∗ ''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, or
Cayley table Named after the 19th century British British may refer to: Peoples, culture, and language * British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies. ** Britishness, the British identity a ...
. This property ensures that the Cayley table of a finite quasigroup, and, in particular, finite group, is a
Latin square In combinatorics and in experimental design, a Latin square is an ''n'' × ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 La ...
.) The uniqueness requirement can be replaced by the requirement that the magma be
cancellative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The unique solutions to these equations are written and . The operations '\' and '/' are called, respectively, left division and right division. The
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

equipped with the empty binary operation satisfies this definition of a quasigroup. Some authors accept the empty quasigroup but others explicitly exclude it.

## Universal algebra

Given some
algebraic structure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, an
identity Identity may refer to: Social sciences * Identity (social science) Identity is the qualities, beliefs, personality, looks and/or expressions that make a person (self-identity One's self-concept (also called self-construction, se ...
is an equation in which all variables are tacitly
universally quantified In mathematical logic, a universal quantification is a type of Quantification (logic), quantifier, a logical constant which is interpretation (logic), interpreted as "given any" or "for all". It expresses that a predicate (mathematical logic), pre ...
, and in which all
operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical or dental specialty that ...
are among the primitive operations proper to the structure. Algebraic structures axiomatized solely by identities are called
varieties Variety may refer to: Science and technology Mathematics * Algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, ...
. Many standard results in
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spa ...
hold only for varieties. Quasigroups are varieties if left and right division are taken as primitive. A quasigroup is a type (2,2,2) algebra (i.e., equipped with three binary operations) satisfying the identities: :''y'' = ''x'' ∗ (''x'' \ ''y''), :''y'' = ''x'' \ (''x'' ∗ ''y''), :''y'' = (''y'' / ''x'') ∗ ''x'', :''y'' = (''y'' ∗ ''x'') / ''x''. 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 is a quasigroup according to the first definition, then is the same quasigroup in the sense of universal algebra. And vice versa: if is a quasigroup according to the sense of universal algebra, then is a quasigroup according to the first definition.

# Loops

A loop is a quasigroup with an
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
; 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
left Left may refer to: Music * ''Left'' (Hope of the States album), 2006 * ''Left'' (Monkey House album), 2016 Direction * Left (direction) Body relative directions (also known as egocentric coordinates) are orientation (geometry), geometri ...
and right inverses (which need not be the same). A quasigroup with an
idempotent element Idempotence (, ) is the property of certain operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, , taking its subtraction operation as quasigroup multiplication yields a pique with the group identity (zero) turned into a "pointed idempotent". (That is, there is a principal isotopy .) A loop that is associative is a group. A group can have a non-associative pique isotope, but it cannot have a nonassociative loop isotope. There are weaker associativity properties that have been given special names. For instance, a
Bol loop In mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of Group (mathematics), group. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in . A loop (algebra), loop, ''L'', is ...
is a loop that satisfies either: :''x'' ∗ (''y'' ∗ (''x'' ∗ ''z'')) = (''x'' ∗ (''y'' ∗ ''x'')) ∗ ''z'' for each ''x'', ''y'' and ''z'' in ''Q'' (a ''left Bol loop''), or else :((''z'' ∗ ''x'') ∗ ''y'') ∗ ''x'' = ''z'' ∗ ((''x'' ∗ ''y'') ∗ ''x'') 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. 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''), or :(''x'' ∗ ''y'') ∗ (''z'' ∗ ''x'') = (''x'' ∗ (''y'' ∗ ''z'')) ∗ ''x''.

# Symmetries

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

## Semisymmetry

A quasigroup is semisymmetric if the following equivalent identities hold: :''x'' ∗ ''y'' = ''y'' / ''x'', :''y'' ∗ ''x'' = ''x'' \ ''y'', :''x'' = (''y'' ∗ ''x'') ∗ ''y'', :''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''3 via the following operation: :$\left(x_1, x_2, x_3\right) \cdot \left(y_1, y_2, y_3\right) = \left(y_3/x_2, y_1 \backslash x_3 , x_1 * y_2\right) = \left(x_2//y_3, x_3 \backslash \backslash y_1, x_1 * y_2\right) ,$ where "//" and "\\" are the conjugate division operations given by $y // x = x / y$ and $y \backslash\backslash x = x \backslash y$.

## Total symmetry

A narrower class is a totally symmetric quasigroup (sometimes abbreviated TS-quasigroup) in which all conjugates coincide as one operation: . Another way to define (the same notion of) totally symmetric quasigroup is as a semisymmetric quasigroup which also is commutative, i.e. . Idempotent total symmetric quasigroups are precisely (i.e. in a bijection with) Steiner triples, so such a quasigroup is also called a Steiner quasigroup, and sometimes the latter is even abbreviated as squag; the term sloop is defined similarly for a Steiner quasigroup that is also a loop. 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 is called totally anti-symmetric if for all , both of the following implications hold: # (''c'' ∗ ''x'') ∗ ''y'' = (''c'' ∗ ''y'') ∗ ''x'' implies that ''x'' = ''y'' # ''x'' ∗ ''y'' = ''y'' ∗ ''x'' implies that ''x'' = ''y''. It is called weakly totally anti-symmetric if only the first implication holds. This property is required, for example, in the Damm algorithm.

# Examples

* Every
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
is a loop, because if and only if , and if and only if . * The integers Z (or the rational numbers, rationals Q or the real number, reals R) with subtraction (−) form a quasigroup. These quasiqroups are not loops because there is no identity element (0 is a right identity because , but not a left identity because, in general, ). * The nonzero rationals Q× (or the nonzero reals R×) with
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
(÷) 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 . * Every Steiner system, Steiner triple system defines an idempotent, commutative quasigroup: is the third element of the triple containing ''a'' and ''b''. These quasigroups also satisfy for all ''x'' and ''y'' in the quasigroup. These quasigroups are known as ''Steiner quasigroups''. * The set where 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. * 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 F4 over the 3-element Galois field define :(''x''1, ''x''2, ''x''3, ''x''4) ∗ (''y''1, ''y''2, ''y''3, ''y''4) = (''x''1, ''x''2, ''x''3, ''x''4) + (''y''1, ''y''2, ''y''3, ''y''4) + (0, 0, 0, (''x''3 − ''y''3)(''x''1''y''2 − ''x''2''y''1)). :Then, is a commutative
Moufang loop In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
that is not a group. * More generally, the nonzero elements of any division algebra form a quasigroup.

# Properties

:In the remainder of the article we shall denote quasigroup multiplication by juxtaposition, multiplication simply by juxtaposition. Quasigroups have the cancellation property: if , then . This follows from the uniqueness of left division of ''ab'' or ''ac'' by ''a''. Similarly, if , then . The Latin square property of quasigroups implies that, given any two of the three variables in , the third variable is uniquely determined.

## Multiplication operators

The definition of a quasigroup can be treated as conditions on the left and right multiplication operators , defined by :$\begin L_x\left(y\right) &= xy \\ R_x\left(y\right) &= yx \\ \end$ The definition says that both mappings are bijections 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, :$\begin L_x^\left(y\right) &= x\backslash y \\ R_x^\left(y\right) &= y/x \end$ In this notation the identities among the quasigroup's multiplication and division operations (stated in the section on #Universal_algebra, universal algebra) are :$\begin L_xL_x^ &= 1\qquad&\text\qquad x\left(x\backslash y\right) &= y \\ L_x^L_x &= 1\qquad&\text\qquad x\backslash\left(xy\right) &= y \\ R_xR_x^ &= 1\qquad&\text\qquad \left(y/x\right)x &= y \\ R_x^R_x &= 1\qquad&\text\qquad \left(yx\right)/x &= y \end$ where 1 denotes the identity mapping on ''Q''.

## Latin squares

The multiplication table of a finite quasigroup is a
Latin square In combinatorics and in experimental design, a Latin square is an ''n'' × ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 La ...
: an 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 ''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 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 a sequence. (This is somewhat misleading however, as the reals can be written in a sequence of length $\mathfrak$, assuming the Well-Ordering Theorem.)

## Inverse properties

The binary operation of a quasigroup is invertible in the sense that both $L_x$ and $R_x$, 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 :$x^ = e/x \qquad x^x = e$ :$x^ = x\backslash e \qquad xx^ = e$ A loop is said to have (''two-sided'') ''inverses'' if $x^ = x^$ for all ''x''. In this case the inverse element is usually denoted by $x^$. There are some stronger notions of inverses in loops which are often useful: *A loop has the ''left inverse property'' if $x^\left(xy\right) = y$ for all $x$ and $y$. Equivalently, $L_x^ = L_$ or $x\backslash y = x^y$. *A loop has the ''right inverse property'' if $\left(yx\right)x^ = y$ for all $x$ and $y$. Equivalently, $R_x^ = R_$ or $y/x = yx^$. *A loop has the ''antiautomorphic inverse property'' if $\left(xy\right)^ = y^x^$ or, equivalently, if $\left(xy\right)^ = y^x^$. *A loop has the ''weak inverse property'' when $\left(xy\right)z = e$ if and only if $x\left(yz\right) = e$. This may be stated in terms of inverses via $\left(xy\right)^x = y^$ or equivalently $x\left(yx\right)^ = y^$. 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. Any loop which satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses.

# Morphisms

A quasigroup or loop homomorphism is a map (mathematics), map between two quasigroups such that . Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).

## Homotopy and isotopy

Let ''Q'' and ''P'' be quasigroups. A quasigroup homotopy from ''Q'' to ''P'' is a triple of maps from ''Q'' to ''P'' such that :$\alpha\left(x\right)\beta\left(y\right) = \gamma\left(xy\right)\,$ 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 is a bijection. Two quasigroups are isotopic if there is an isotopy between them. In terms of Latin squares, an isotopy 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 form 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 which is isotopic to a group need not be a group. For example, the quasigroup on R with multiplication given by is isotopic to the additive group , but is not itself a group. Every medial magma, medial quasigroup is isotopic to an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
by the Medial magma#Bruck–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 ∗ (i.e., ) we can form five new operations: (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).

## Isostrophe (paratopy)

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 isostrophic to each other. There are also many other names for this relation of "isostrophe", e.g., paratopy.

# Generalizations

An ''n''-ary quasigroup is a set with an arity, ''n''-ary operation, with , such that the equation 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, ; 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: :$f\left(x_1,\dots,x_n\right) = g\left(x_1,\dots,x_,\,h\left(x_i,\dots,x_j\right),\,x_,\dots,x_n\right),$ where and . Finite irreducible ''n''-ary quasigroups exist for all ; see Akivis and Goldberg (2001) for details. An ''n''-ary quasigroup with an ''n''-ary version of associativity is called an n-ary group, ''n''-ary group.

## Right- and left-quasigroups

A right-quasigroup is a type (2,2) algebra satisfying both identities: ''y'' = (''y'' / ''x'') ∗ ''x''; ''y'' = (''y'' ∗ ''x'') / ''x''. Similarly, a left-quasigroup is a type (2,2) algebra satisfying both identities: ''y'' = ''x'' ∗ (''x'' \ ''y''); ''y'' = ''x'' \ (''x'' ∗ ''y'').

# Number of small quasigroups and loops

The number of isomorphism classes of small quasigroups and loops is given here:

*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

# References

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