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In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped arguments. The notion '' inverse'' refers to a group structure on the operation's
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
, possibly with another operation.
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 ...
is an anticommutative operation because commuting the operands of gives for example, Another prominent example of an anticommutative operation is the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
. In
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developmen ...
, where
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
is of central importance, these operations are mostly called antisymmetric operations, and are extended in an
associative 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 ...
setting to cover more than two
arguments An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
.


Definition

If A, B are two
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 comm ...
s, a
bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, ...
f\colon A^2 \to B is anticommutative if for all x, y \in A we have :f(x, y) = - f(y, x). More generally, a
multilinear map In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W ar ...
g : A^n \to B is anticommutative if for all x_1, \dots x_n \in A we have :g(x_1,x_2, \dots x_n) = \text(\sigma) g(x_,x_,\dots x_) where \text(\sigma) is the
sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
of the
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...
\sigma.


Properties

If the abelian group B has no 2-
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
, implying that if x = -x then x = 0, then any anticommutative bilinear map f\colon A^2 \to B satisfies :f(x, x) = 0. More generally, by transposing two elements, any anticommutative multilinear map g\colon A^n \to B satisfies :g(x_1, x_2, \dots x_n) = 0 if any of the x_i are equal; such a map is said to be alternating. Conversely, using multilinearity, any alternating map is anticommutative. In the binary case this works as follows: if f\colon A^2 \to B is alternating then by bilinearity we have :f(x+y, x+y) = f(x, x) + f(x, y) + f(y, x) + f(y, y) = f(x, y) + f(y, x) = 0 and the proof in the multilinear case is the same but in only two of the inputs.


Examples

Examples of anticommutative binary operations include: *
Cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
* Lie bracket of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
* Lie bracket of a
Lie ring In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
*
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 ...


See also

*
Commutativity 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 ...
*
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. Group theory The commutator of two elements, ...
*
Exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
*
Graded-commutative ring In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy :xy = (-1)^ yx, where , ''x'' , and , ''y'' , d ...
*
Operation (mathematics) In mathematics, an operation is a function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most commonly studied operat ...
*
Symmetry in mathematics Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. Given a structured obje ...
*
Particle statistics Particle statistics is a particular description of multiple particles in statistical mechanics. A key prerequisite concept is that of a statistical ensemble (an idealization comprising the state space of possible states of a system, each labeled w ...
(for anticommutativity in physics).


References

*.


External links

*. Which references th
Original Russian work
*{{MathWorld , title=Anticommutative , urlname=Anticommutative Properties of binary operations