Anticommutativity

	Email		Print
	Bookmark		Link

Anticommutativity

In mathematics, anticommutativity refers to the property of an operation

Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
being anticommutative, i.e. being non-commutative
Commutativity

In mathematics, commutativity is the process to change the order of something without changing the end result. It is a fundamental property of many binary operations throughout mathematics, and many Mathematical proof depend on it....
in a precise way. Anticommutative operation

Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
s are widely used in algebra

Algebra

Algebra is a branch of mathematics concerning the study of structure , relation , and quantity. Together with geometry, mathematical analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics....
, geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, mathematical analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit , whether the limit of a sequence or the limit of a function....
and, as a consequence in physics

Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
: they are often called antisymmetric operation
Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
s.

ary operation

Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
is anticommutative if swapping the order of any two arguments negates the result. For example, a binary operation * is anticommutative if for all x and y, x*y = -y*x.

More formally, a map

Map (mathematics)

In mathematics and related technical fields, the term map or mapping is often a synonym for Function . Thus, for example, a partial map is a partial function, and a total map is a total function....
from the set of all n-tuples

Cartesian product

In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after Ren? Descartes, whose formulation of analytic geometry gave rise to this concept....
of elements in a set ' (where ' is a general integer) to a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
(whose operation

Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
is written in additive notation

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
for the sake of simplicity), is anticommutative if and only if

IFF

IFF, Iff or iff can stand for:* Identification Friend or Foe, an electronic radio-based identification system utilizing transponders...

where is an arbitrary permutation

Permutation

In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set....
of the set of first ' non-zero integers and is its sign.

Discussion

Ask a question about 'Anticommutativity'

Start a new discussion about 'Anticommutativity'

Answer questions from other users

Full Discussion Forum

Encyclopedia

In mathematics, anticommutativity refers to the property of an operation
Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
being anticommutative, i.e. being non-commutative
Commutativity

In mathematics, commutativity is the process to change the order of something without changing the end result. It is a fundamental property of many binary operations throughout mathematics, and many Mathematical proof depend on it....
in a precise way. Anticommutative operation
Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
s are widely used in algebra
Algebra

Algebra is a branch of mathematics concerning the study of structure , relation , and quantity. Together with geometry, mathematical analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics....
, geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, mathematical analysis
Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit , whether the limit of a sequence or the limit of a function....
and, as a consequence in physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
: they are often called antisymmetric operation
Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
s.

Definition
An -ary operation
Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
is anticommutative if swapping the order of any two arguments negates the result. For example, a binary operation * is anticommutative if for all x and y, x*y = -y*x.

More formally, a map
Map (mathematics)

In mathematics and related technical fields, the term map or mapping is often a synonym for Function . Thus, for example, a partial map is a partial function, and a total map is a total function....
from the set of all n-tuples
Cartesian product

In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after Ren? Descartes, whose formulation of analytic geometry gave rise to this concept....
of elements in a set ' (where ' is a general integer) to a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
(whose operation
Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
is written in additive notation
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
for the sake of simplicity), is anticommutative if and only if
IFF

IFF, Iff or iff can stand for:* Identification Friend or Foe, an electronic radio-based identification system utilizing transponders...

where is an arbitrary permutation
Permutation

In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set....
of the set of first ' non-zero integers and is its sign. This equality
Equality (mathematics)

Equality is the paradigmatic example of the more general concept of equivalence relations on a set: those binary relations which are reflexive relation, symmetric relation, and transitive relation....
expresses the following concept
the value of the operation is unchanged, when applied to all ordered tuple
Tuple

In mathematics, a tuple is a sequence of a specific number of values, called the components of the tuple. These components can be any kind of mathematical objects, where each component of a tuple is a value of a specified type....
s constructed by even permutation
Even and odd permutations

In mathematics, when X is a finite set of at least two elements, the permutations of X fall into two classes of equal size: the even permutations and the odd permutations....
of the elements of a fixed one.
the value of the operation is the inverse
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
of its value on a fixed tuple
Tuple

In mathematics, a tuple is a sequence of a specific number of values, called the components of the tuple. These components can be any kind of mathematical objects, where each component of a tuple is a value of a specified type....
, when applied to all ordered tuple
Tuple

In mathematics, a tuple is a sequence of a specific number of values, called the components of the tuple. These components can be any kind of mathematical objects, where each component of a tuple is a value of a specified type....
s constructed by odd permutation
Even and odd permutations

In mathematics, when X is a finite set of at least two elements, the permutations of X fall into two classes of equal size: the even permutations and the odd permutations....
to the elements of the fixed one. The need for the existence of this inverse element
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
is the main reason for requiring the codomain
Codomain

In mathematics, the codomain, range or target set, of a function , described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall....
of the operation
Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
to be at least a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
.

Note that this is an abuse of notation
Abuse of notation

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition ....
, since the codomain
Codomain

In mathematics, the codomain, range or target set, of a function , described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall....
of the operation
Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
needs only to be a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
: "" has not a precise meaning since a multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
is not necessarily defined on .

Particularly important is the case '. A binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
is anticommutative if and only if
IFF

IFF, Iff or iff can stand for:* Identification Friend or Foe, an electronic radio-based identification system utilizing transponders...

This means that is the inverse
Inverse element

In mathematics, the idea of inverse element generalises the concepts of additive inverse, in relation to addition, and Multiplicative inverse, in relation to multiplication....
of the element in .

Properties
If the group is such that

i.e. the only element equal to its inverse
Inverse element

In mathematics, the idea of inverse element generalises the concepts of additive inverse, in relation to addition, and Multiplicative inverse, in relation to multiplication....
is the neutral element, then for all the ordered tuple
Tuple

In mathematics, a tuple is a sequence of a specific number of values, called the components of the tuple. These components can be any kind of mathematical objects, where each component of a tuple is a value of a specified type....
s such that for at least two different index

In the case ' this means
Examples
Anticommutative operators include:
Subtraction
Subtraction

Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with....
Cross product
Cross product

In mathematics, the cross product is a binary operation on two vector s in a three-dimensional Euclidean space that results in another vector which is orthogonal to the plane containing the two input vectors....
Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
Lie ring
Lie ring

In mathematics a Lie ring is a structure related to Lie algebras that can arise as a generalisation of Lie algebras, or through the study of the lower central series of Group ....

See also

Commutativity
Commutativity

In mathematics, commutativity is the process to change the order of something without changing the end result. It is a fundamental property of many binary operations throughout mathematics, and many Mathematical proof depend on it....
Commutator
Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory....
exterior algebra
Exterior algebra

In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions....
Operation (mathematics)
Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
Symmetry in mathematics
Symmetry in mathematics

Symmetry in mathematics occurs not only in geometry, but also in other branches of mathematics. It is actually the same as Invariant : the property that something does not change under a set of Transformation s....
Particle statistics
Particle statistics

Particle statistics refers to the particular description of particles in statistical mechanics....
(for anticommutativity in physics).

External links

Weisstein, Eric W."". From MathWorld
MathWorld

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign....
--A Wolfram Web Resource.
A.T. Gainov, "", Springer-Verlag Online Encyclopaedia of Mathematics.