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\begin{document}

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\section*{binary function}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{internalisation}{Internalisation}\dotfill \pageref*{internalisation} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{binary function}, or \emph{function of two variables}, is like a [[function]] but with two [[domains]]. That is, while a function $f$ from the [[set]] $A$ to the set $C$ maps an [[element]] $x$ of $A$ to a unique element $f(x)$ of $C$, a binary function from $A$ and $B$ to $C$ maps an element $x$ of $A$ and an element $y$ of $B$ to a unique element $f(x,y)$ of $C$.

We can generalise further to \emph{multiary functions}, or \emph{functions of several variables}.

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

The possible definitions depend on [[foundations]]; for us, the simplest is probably this:

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{binary function} from $A$ and $B$ to $C$ is simply a [[function]] $f$ to $C$ from the binary [[cartesian product]] $A \times B$. We write $f(x,y)$ for $f((x,y))$.

\end{defn}
This is natural in [[structural set theory]]; it makes sense in any [[set theory]] and (suitably interpreted) in a foundational [[type theory]].

In [[material set theory]], one often declines to specify the domain of a function (since this can be recovered from it); then we can say this:

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{binary function} to $C$ is simply a [[function]] $f$ to $C$ such that every [[element]] of the [[domain]] of $f$ is an [[ordered pair]]. Again, we write $f(x,y)$ for $f((x,y))$.

\end{defn}
An equivalent (but different) definition in material set theory is this:

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{binary function} to $C$ is a [[ternary relation]] $f$ such that

\begin{itemize}%
\item given $(x,y,z) \in f$, we have $z \in C$, and
\item given $(x,y,z) \in f$ and $(x,y,z') \in f$, we have $z = z'$.

\end{itemize}
Now we write $f(x,y)$ for the unique $z$ (if any) such that $(x,y,z) \in f$.

\end{defn}
Either way, the material concept is actually more general than the structural one, because the [[domain]] of a material binary function might not be a [[cartesian product]]. The link is provided by the binary version of a [[partial function]]:

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{partial binary function} from $A$ and $B$ to $C$ is a function to $C$ from a [[subset]] of $A \times B$.

\end{defn}
It's also possible to take the concept of binary function as an undefined primitive concept, on the same level as that of [[function]]. Then we want [[axioms]] such as the following (depending on the style of [[foundations]]):

\begin{itemize}%
\item Given an [[element]] $x$ of the [[set]] $A$, an element $y$ of $B$, and a binary function from $A$ and $B$ to $C$, we have an element $f(x,y)$ of $C$.


\item If $x$ and $x'$ are [[equality|equal]] elements of $A$ and $y$ and $y'$ are equal elements of $B$, then $f(x,y) = f(x',y')$.


\item Given two such binary functions $f$ and $f'$, if $f(x,y) = f'(x,y)$ for every $x$ in $A$ and every $y$ in $B$, then $f = f'$.



\end{itemize}
\hypertarget{internalisation}{}\subsection*{{Internalisation}}\label{internalisation}

We could conceivably have the notion of binary function \emph{without} the notion of [[cartesian product]]; then a binary function could not be understood as a special case of a function. I doubt that anybody has proposed such a [[foundation of mathematics]], but there are situations where this is true in some [[internal logic]].

In particular, a binary function [[internalisation|internal to]] a [[multicategory]] is simply a [[binary morphism]] in that multicategory. This is most like a binary function between sets in the case of a [[cartesian multicategory]]. But even so, there may be no [[tensor product]] in the multicategory, and then a binary morphism cannot be understood as a special case of a [[morphism]].

\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

A binary function to the set of [[truth values]] is a [[binary relation]].

A binary function from $A$ and $A$ to $C$ may be simply called a \textbf{binary function} from $A$ to $C$.

A binary function $f$ from $A$ to $C$ is \textbf{symmetric} if $f(a,b) = f(b,a)$ always.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \textbf{binary function}, [[bilinear map]], [[multilinear map]]


\item [[binary morphism]], [[multimorphism]]


\item [[bifunctor]], [[Quillen bifunctor]]



\end{itemize}
[[!redirects binary function]] [[!redirects binary functions]]

[[!redirects function of two variables]] [[!redirects functions of two variables]] [[!redirects functions of two variables each]]

[[!redirects binary map]] [[!redirects binary maps]] [[!redirects binary mapping]] [[!redirects binary mappings]]

[[!redirects symmetric binary function]] [[!redirects symmetric binary functions]]



\end{document}