\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{alternating multifunction}

\hypertarget{alternating_functions}{}\section*{{Alternating functions}}\label{alternating_functions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{constructive_aspects}{Constructive aspects}\dotfill \pageref*{constructive_aspects} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 multifunction|antisymmetric functions]]. In cases where they differ (such as in [[characteristic]] $2$), 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 \emph{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 \emph{alternation} (rather than alternatingess), although in principle one could also use \emph{alternating} as a noun.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

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

An \textbf{alternating multifunction} (or simply \emph{alternating function}) of arity $n$ from $X$ to $(Y,0)$ is a multifunction of arity $n$ from $X$ to $Y$ such that, whenever two of the function's arguments are equal, the value of the function is $0$. In arity $0$ or $1$, every multifunction is trivially alternating; in arity $2$, we can write this as the [[equational law]] $f(a,a) = 0$; in arity $3$, we have the equational laws $f(a,a,b) = 0$, $f(a,b,a) = 0$, and $f(a,b,b) = 0$; etc.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

There are many nice properties of alternating \emph{[[multilinear function|multilinear]]} functions. So suppose that $X$ and $Y$ are [[modules]] over a [[base rig]] $K$ and that $f$ is a multilinear function from $X$ to $Y$; use the usual [[zero]] element of the module $Y$ as the basepoint $0$. In the case where $Y$ is $K$ itself, we speak of an \textbf{alternating form} (a phrase which is usually taken to include multilinearity).

We will sometimes want to assume that scalar multiplication by $2$ is cancellable in $Y$ (which for example is always the case when $2$ is invertible in $K$, in particular when $K$ is a [[field]] of [[characteristic]] other than $2$), but only when stated. Since alternation requires looking at two arguments of $f$, we will often, when this leads to no loss of generality, assume that these are the first two arguments, writing $\vec{z}$ to represent all of the other arguments.

\begin{prop}
\label{alterationImpliesAntisymmetry}\hypertarget{alterationImpliesAntisymmetry}{}
An alternating [[multilinear function]] is [[antisymmetric function|antisymmetric]].

\end{prop}
\begin{proof}
\begin{displaymath}
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}) .
\end{displaymath}
Applying alternation, most of these terms vanish:

\begin{displaymath}
0 = 0 + f(x,y,\vec{z}) + f(y,x,\vec{z}) + 0 .
\end{displaymath}
Therefore,

\begin{displaymath}
f(x,y,\vec{z}) + f(y,x,\vec{z}) = 0 ,
\end{displaymath}
which is antisymmetry.

\end{proof}
\begin{prop}
\label{antisymmetryImpliesAlternation}\hypertarget{antisymmetryImpliesAlternation}{}
If multiplication by $2$ is cancellable in $Y$, then an [[antisymmetric function|antisymmetric]] function to $Y$ is alternating.

\end{prop}
\begin{proof}
By antisymmetry,

\begin{displaymath}
f(x,x,\vec{z}) + f(x,x,\vec{z}) = 0 ,
\end{displaymath}
or equivalently

\begin{displaymath}
2 f(x,x,\vec{z}) = 2 \cdot 0 .
\end{displaymath}
Cancelling $2$,

\begin{displaymath}
f(x,x,\vec{z}) = 0 ,
\end{displaymath}
which is alternation.

\end{proof}
\begin{remark}
\label{antisymmetryWarning}\hypertarget{antisymmetryWarning}{}
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 \emph{if} it is multilinear, regardless of the behaviour of $2$, but not when it is nonlinear; an antisymmetric function must be alternating \emph{if} multiplication by $2$ is cancellable in the target, regardless of linearity, but not when $2$ is noncancellable.

The simplest strongest-possible counterexamples are

\begin{displaymath}
(x,y \mapsto |y-x|)\colon \mathbb{R}^2 \to \mathbb{R} ,
\end{displaymath}
which is alternating but not antisymmetric, and

\begin{displaymath}
(x,y \mapsto x y)\colon \mathbb{F}_2^2 \to \mathbb{F}_2 ,
\end{displaymath}
which is antisymmetric but not alternating.

Of course, alternating and antisymmetric functions \emph{are} the same in the context of multilinear functions to a module in which $2$ is cancellable, in particular for multilinear functions between [[vector fields]] over the [[real numbers]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
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]] $2$ (in which multiplication by $2$ is as uncancellable as possible).

\end{remark}
\hypertarget{constructive_aspects}{}\subsection*{{Constructive aspects}}\label{constructive_aspects}

In [[constructive mathematics]], we usually assume that the arity $n$ of $f$ has [[decidable equality]], which is true if $n$ 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 $f$ there is zero. If $X$ and $Y$ are also equipped with inequalities, then $f$ is \textbf{strongly alternating} if, whenever its value is inequal to $0$ in $Y$, then arguments with inequal indices must be inequal in $X$. (In arity $2$, for example, if $f(a,b) \ne 0$, then $a \ne b$.) If the inequality on $Y$ is [[tight relation|tight]] (so that its [[negation]] is [[equality]] in $Y$), then every strongly alternating function is alternating, but the reverse requires [[excluded middle]] in general.

[[!redirects alternating multifunction]] [[!redirects alternating multifunctions]] [[!redirects alternating multimap]] [[!redirects alternating multimaps]]

[[!redirects alternating function]] [[!redirects alternating functions]] [[!redirects alternating map]] [[!redirects alternating maps]]

[[!redirects alternating multilinear function]] [[!redirects alternating multilinear functions]] [[!redirects alternating multilinear map]] [[!redirects alternating multilinear maps]]

[[!redirects alternating form]] [[!redirects alternating forms]]



\end{document}