In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, a quasigroup is an
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
resembling a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
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 need not be
associative and need not have an
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
.
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, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
, and the other, from
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, ...
, defines a quasigroup as having three primitive operations. The
homomorphic image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''Q'' with a binary operation ∗ (that is, a
magma
Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural sa ...
, indicating that a quasigroup 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 mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplicat ...
. 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 Latin sq ...
.) The requirement that ''x'' and ''y'' be unique can be replaced by the requirement that the magma be
cancellative
In mathematics, the notion of cancellative is a generalization of the notion of invertible.
An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that .
A ...
.
The unique solutions to these equations are written and . The operations '\' and '/' are called, respectively,
left division
Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication.
At an elementary level the division of two natural numb ...
and
right division. With regard to the Cayley table, the first equation (left division) means that the ''b'' entry in the ''a'' row marks the ''x'' column while the second equation (right division) means that the ''b'' entry in the ''a'' column marks the ''y'' row.
The
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
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, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
, an
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
is an equation in which all variables are tacitly
universally quantified
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...
, and in which all
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
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
. Many standard results in
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, ...
hold only for varieties. Quasigroups form a variety 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, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
; 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
* "Left", a song by Nickelback from the album ''Curb'', 1996
Direction
* Left (direction), the relative direction opposite of right
* L ...
and
right inverses (which need not be the same).
A quasigroup with an
idempotent element
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
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, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
, , 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. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in .
A loop, ''L'', is said to be a left Bol loop if it sat ...
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 Moufang is the family name of the following people:
* Christoph Moufang (1817–1890), a Roman Catholic cleric
* Ruth Moufang (1905–1977), a German mathematician, after whom several concepts in mathematics are named:
** Moufang–Lie algebra
** ...
. 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
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:
:
where "//" and "\\" are the
conjugate division operations given by
and
.
Triality
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 refers to an analogue for loops, namely, totally symmetric loops that satisfy instead of . 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 In error detection, the Damm algorithm is a check digit algorithm that detects all single-digit errors and all adjacent transposition errors. It was presented by H. Michael Damm in 2004.
Strengths and weaknesses Strengths
The Damm algorithm is ...
.
Examples
* Every
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
is a loop, because
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
, and if and only if .
* The
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s Z (or the
rationals
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
Q or the
reals R) with
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
(−) 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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
characteristic not equal to 2 forms an
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
,
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
quasigroup under the operation .
* Every
Steiner triple system
250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line.
In combinatorial mathematics, a Steiner system (named after Jakob Steiner) ...
defines an
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
,
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
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
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
\ of the quaternions under multiplication. It is given by the group presentation
:\mathrm_8 ...
forms a nonassociative loop of order 8. See
hyperbolic quaternions for its application. (The hyperbolic quaternions themselves do ''not'' form a loop or quasigroup.)
* The nonzero
octonions
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...
form a nonassociative loop under multiplication. The octonions are a special type of loop known as a
Moufang loop Moufang is the family name of the following people:
* Christoph Moufang (1817–1890), a Roman Catholic cleric
* Ruth Moufang (1905–1977), a German mathematician, after whom several concepts in mathematics are named:
** Moufang–Lie algebra
** ...
.
* An associative quasigroup is either empty or is a group, since if there is at least one element, the
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
F
4 over the 3-element
Galois field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtra ...
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
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
Moufang loop Moufang is the family name of the following people:
* Christoph Moufang (1817–1890), a Roman Catholic cleric
* Ruth Moufang (1905–1977), a German mathematician, after whom several concepts in mathematics are named:
** Moufang–Lie algebra
** ...
that is not a group.
* More generally, the nonzero elements of any
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
form a quasigroup.
Properties
:
In the remainder of the article we shall denote quasigroup multiplication simply by juxtaposition.
Quasigroups have the
cancellation property
In mathematics, the notion of cancellative is a generalization of the notion of invertible.
An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that .
An ...
: 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 operator
In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is,
T_f\varphi(x) = f(x) \varphi (x) \quad
for all in th ...
s , defined by
:
The definition says that both mappings are
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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,
:
In this notation the identities among the quasigroup's multiplication and division operations (stated in the section on
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, ...
) are
:
where 1 denotes the identity mapping on ''Q''.
Latin squares
![10 x10 lateinisches quadrat](https://upload.wikimedia.org/wikipedia/commons/7/75/10_x10_lateinisches_quadrat.svg)
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 Latin sq ...
: 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
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
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 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
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
quasigroup, such as the group of non-zero
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
. (This is somewhat misleading however, as the reals can be written in a sequence of length
, assuming the
well-ordering theorem
In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set ''X'' is ''well-ordered'' by a strict total order if every non-empty subset of ''X'' has a least element under the orde ...
.)
Inverse properties
The binary operation of a quasigroup is invertible in the sense that both
and
, the
left and right multiplication operators, are bijective, and hence
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
.
Every loop element has a unique left and right inverse given by
:
:
A loop is said to have (''two-sided'') ''inverses'' if
for all ''x''. In this case the inverse element is usually denoted by
.
There are some stronger notions of inverses in loops which are often useful:
*A loop has the ''left inverse property'' if
for all
and
. Equivalently,
or
.
*A loop has the ''right inverse property'' if
for all
and
. Equivalently,
or
.
*A loop has the ''antiautomorphic inverse property'' if
or, equivalently, if
.
*A loop has the ''weak inverse property'' when
if and only if
. This may be stated in terms of inverses via
or equivalently
.
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
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
is a
map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
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
:
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
. 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
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
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 quasigroup is isotopic to an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
by the
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
Polyadic or multiary quasigroups
An ''n''-ary quasigroup is a set with an
''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:
:
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
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
is called an
''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:
See also
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Division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
– a ring in which every non-zero element has a multiplicative inverse
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Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
– an algebraic structure consisting of a set together with an associative binary operation
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Monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
– a semigroup with an identity element
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Planar ternary ring
In mathematics, an algebraic structure (R,T) consisting of a non-empty set R and a ternary mapping T \colon R^3 \to R \, may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marsh ...
– has an additive and multiplicative loop structure
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Problems in loop theory and quasigroup theory
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Mathematics of Sudoku
The mathematics of Sudoku refers to the use of mathematics to study Sudoku puzzles to answer questions such as ''"How many filled Sudoku grids are there?"'', "''What is the minimal number of clues in a valid puzzle?''" and ''"In what ways can S ...
Notes
References
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External links
quasigroups* {{eom, title=Quasi-group
Non-associative algebra
Group theory
Latin squares