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alternating multifunction

Alternating functions

Alternating functions

Idea

In linear algebra, alternating multilinear functions are well known, and are in many cases (over the real numbers, for example) are equivalent to antisymmetric functions?. In cases where they differ (such as in characteristic 22), it is often the alternating functions that behave better.

Actually, being alternating is not, in itself, really about linearity, and we can abstract away to a nonlinear concept of alternating function. (That said, there is one bit of very mild linear structure that is needed: a basepoint in the target set.)

The property of being alternating is called alternation (rather than alternatingess), although in principle one could also use alternating as a noun.

Definitions

Let XX be a set, and let (Y,0)(Y,0) be a pointed set (so YY is a set and 00 is one of its elements). Let nn be a natural number (or indeed any cardinal number). Recall that a multifunction of arity? nn to YY from XX is the same thing as a function to YY from the nn-fold cartesian power X nX^n.

An alternating multifunction (or simply alternating function) of arity nn from XX to (Y,0)(Y,0) is a multifunction of arity nn from XX to YY such that, whenever two of the function's arguments are equal, the value of the function is 00. In arity 00 or 11, every multifunction is trivially alternating; in arity 22, we can write this as the equational law? f(a,a)=0f(a,a) = 0; in arity 33, we have the equational laws f(a,a,b)=0f(a,a,b) = 0, f(a,b,a)=0f(a,b,a) = 0, and f(a,b,b)=0f(a,b,b) = 0; etc.

Properties

There are many nice properties of alternating multilinear functions. So suppose that XX and YY are modules over a base rig? KK and that ff is a multilinear function from XX to YY; use the usual zero element of the module YY as the basepoint 00. In the case where YY is KK itself, we speak of an alternating form (a phrase which is usually taken to include multilinearity).

We will sometimes want to assume that scalar multiplication by 22 is cancellable in YY (which for example is always the case when 22 is invertible in KK, in particular when KK is a field of characteristic other than 22), but only when stated. Since alternation requires looking at two arguments of ff, we will often, when this leads to no loss of generality, assume that these are the first two arguments, writing z\vec{z} to represent all of the other arguments.

Proposition

An alternating multilinear function is antisymmetric?.

By multilinearity,

f(x+y,x+y,z)=f(x,x,z)+f(x,y,z)+f(y,x,z)+f(y,y,z). f(x+y,x+y,\vec{z}) = f(x,x,\vec{z}) + f(x,y,\vec{z}) + f(y,x,\vec{z}) + f(y,y,\vec{z}) .

Applying alternation, most of these terms vanish:

0=0+f(x,y,z)+f(y,x,z)+0. 0 = 0 + f(x,y,\vec{z}) + f(y,x,\vec{z}) + 0 .

Therefore,

f(x,y,z)+f(y,x,z)=0, f(x,y,\vec{z}) + f(y,x,\vec{z}) = 0 ,

which is antisymmetry.

Proposition

If multiplication by 22 is cancellable in YY, then an antisymmetric? function to YY is alternating.

Proof

By antisymmetry,

f(x,x,z)+f(x,x,z)=0, f(x,x,\vec{z}) + f(x,x,\vec{z}) = 0 ,

or equivalently

2f(x,x,z)=20. 2 f(x,x,\vec{z}) = 2 \cdot 0 .

Cancelling 22,

f(x,x,z)=0, f(x,x,\vec{z}) = 0 ,

which is alternation.

Remark

It is false in both directions to state in general that alternating functions and antisymmetric functions are the same, but for different reasons. An alternating function must be antisymmetric if it is multilinear, regardless of the behaviour of 22, but not when it is nonlinear; an antisymmetric function must be alternating if multiplication by 22 is cancellable in the target, regardless of linearity, but not when 22 is noncancellable.

The simplest strongest-possible counterexamples are

(x,y|yx|): 2, (x,y \mapsto |y-x|)\colon \mathbb{R}^2 \to \mathbb{R} ,

which is alternating but not antisymmetric, and

(x,yxy):𝔽 2 2𝔽 2, (x,y \mapsto x y)\colon \mathbb{F}_2^2 \to \mathbb{F}_2 ,

which is antisymmetric but not alternating.

Of course, alternating and antisymmetric functions are the same in the context of multilinear functions to a module in which 22 is cancellable, in particular for multilinear functions between vector fields over the real numbers.

Remark

The alternating groups are really about antisymmetric functions rather than alternating functions as such. (Whereas a symmetric function is preserved by the application of any element of the symmetric group, an antisymmetric function is preserved by and only by the elements of the alternating group.) Nevertheless, this precise distinction between ‘alternating’ and ‘antisymmetric’ is well established in the theory of vector spaces over a field of characteristic 22 (in which multiplication by 22 is as uncancellable as possible).

Constructive aspects

In constructive mathematics, we usually assume that the arity nn of ff has decidable equality, which is true if nn is a natural number (which is most common) or even a (possibly infinite) extended natural number. However, as long as the arity is equipped with an inequality, then we can state the definition: whenever equal arguments have inequal indices, the value of the multifunction ff there is zero. If XX and YY are also equipped with inequalities, then ff is strongly alternating if, whenever its value is inequal to 00 in YY, then arguments with inequal indices must be inequal in XX. (In arity 22, for example, if f(a,b)0f(a,b) \ne 0, then aba \ne b.) If the inequality on YY is tight (so that its negation is equality in YY), then every strongly alternating function is alternating, but the reverse requires excluded middle in general.

Last revised on April 21, 2017 at 01:43:29. See the history of this page for a list of all contributions to it.