In mathematics, anticommutativity refers to the property of an operation being anticommutative, i. 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 e. being non commutative in a precise way. In Mathematics, commutativity is the ability to change the order of something without changing the end result Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence in physics: they are often called antisymmetric operations. 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 Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Analysis has its beginnings in the rigorous formulation of Calculus. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. 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

Definition

An n-ary operation is anticommutative if swapping the order of any two arguments negates the result. 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 For example, a binary operation * is anticommutative if for all x and y, x*y = −y*x.

More formally, a map $\scriptstyle *:A^n \longrightarrow \mathfrak{G}$ from the set of all n-tuples of elements in a set A (where n is a general integer) to a group $\scriptstyle\mathfrak{G}$ (whose operation is written in additive notation for the sake of simplicity), is anticommutative if and only if

$x_1*x_2*\dots*x_n = \sgn(\sigma) x_{\sigma(1)}*x_{\sigma(2)}*\dots* x_{\sigma(n)} \qquad \forall\boldsymbol{x} = (x_1,x_2,\dots,x_n) \in A^n$

where $\scriptstyle\sigma:(n)\longrightarrow(n)$ is an arbitrary permutation of the set (n) of first n non-zero integers and sgn(σ) is its sign. In Mathematics and related technical fields the term map or mapping is often a Synonym for function. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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 In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In several fields of Mathematics the term permutation is used with different but closely related meanings The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even This equality express the following concept

• the value of the operation is unchanged, when applied to all ordered tuples constructed by even permutation of the elements of a fixed one. Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even
• the value of the operation is the inverse of its value on a fixed tuple, when applied to all ordered tuples constructed by odd permutation to the elements of the fixed one. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even The need for the existence of this inverse element is the main reason for requiring the codomain $\scriptstyle\mathfrak{G}$ of the operation to be at least a group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, the codomain, or target, of a function f: X → Y is the set 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 In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

Note that this is an abuse of notation, since the codomain of the operation needs only to be a group: " − 1" has not a precise meaning since a multiplication is not necessarily defined on $\scriptstyle\mathfrak{G}$. 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 In Mathematics, the codomain, or target, of a function f: X → Y is the set 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 In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

Particularly important is the case n = 2. A binary operation $\scriptstyle *:A\times A\longrightarrow \mathfrak{G}$ is anticommutative if and only if

$x_1 * x_2 = -x_2 * x_1 \qquad\forall(x_1,x_2)\in A\times A$

This means that $\scriptstyle x_1 * x_2$ is the inverse of the element $\scriptstyle x_2 * x_1$ in $\scriptstyle\mathfrak{G}$. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to

Properties

If the group $\scriptstyle\mathfrak{G}$ is such that

$\mathfrak{-a} = \mathfrak{a} \iff \mathfrak{a} = \mathfrak{0}\qquad \forall \mathfrak{a} \in \mathfrak{G}$

i. e. the only element equal to its inverse is the neutral element, then for all the ordered tuples such that xj = xi for at least two different index i,j

$x_1*x_2*\dots*x_n = \mathfrak{0}$

In the case n = 2 this means

$x_1*x_1 = x_2*x_2 = \mathfrak{0}$

Examples

Anticommutative operators include: