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\begin{document}

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\section*{involution}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{commuting_involutions}{Commuting involutions}\dotfill \pageref*{commuting_involutions} \linebreak
\noindent\hyperlink{fixed_point_free_involutions}{Fixed point free involutions}\dotfill \pageref*{fixed_point_free_involutions} \linebreak
\noindent\hyperlink{MonadOfInvolutions}{Monad of involutions}\dotfill \pageref*{MonadOfInvolutions} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

An \textbf{involution} is an [[endomorphism]] whose square is the [[identity morphism]]. Such an endomorphism must be an [[automorphism]]; indeed, it is its own [[inverse morphism|inverse]].

Where this makes sense, an \textbf{anti-involution} is an [[antihomomorphism]] instead of a [[homomorphism]] (so an antiendomorphism and necessarily an antiautomorphism).

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

\hypertarget{commuting_involutions}{}\subsubsection*{{Commuting involutions}}\label{commuting_involutions}

Two involutions $f, g : X \to X$ [[commutative diagram|commute]] if and only if their composition $f g$ is also an involution, as displayed by the following algebra:

\begin{displaymath}
f g = f (f g f g ) g = (f f) g f (g g) = g f
\end{displaymath}
\begin{displaymath}
(f g)(f g) = f (g f) g = f (f g) g = (f f)(g g)= 1
\end{displaymath}
\hypertarget{fixed_point_free_involutions}{}\subsubsection*{{Fixed point free involutions}}\label{fixed_point_free_involutions}

In [[combinatorics]], an important class of involutions are the [[fixed point]] free ones: an arbitrary involution on a [[finite set]] of cardinality $n$ may be specified by the choice of $k$ elements which are fixed together with a fixed point free involution on the remaining $(n-k)$. The number of fixed point free involutions on a set of $2n$ labelled elements is counted by the double factorial $(2n-1)!! = (2n-1)\cdot (2n-3)\cdot\dots\cdot 3\cdot 1 = \frac{(2n)!}{2^n n!}$, while arbitrary involutions on a set of $n$ labelled elements are counted by OEIS sequence \href{https://oeis.org/A000085}{A000085}, which also counts the number of [[Young tableaux]] with $n$ cells.

\hypertarget{MonadOfInvolutions}{}\subsubsection*{{Monad of involutions}}\label{MonadOfInvolutions}

An involution on a set $X$ is the same thing as an [[action]] of $\mathbb{Z}/2\mathbb{Z}$ on $X$.

More generally, let $(C,\otimes,1)$ be a [[monoidal category]] with [[distributive monoidal category|distributive]] finite [[coproducts]]. The object $2 = 1 + 1$ is equipped with an involution

\begin{displaymath}
not : 2 \to 2
\end{displaymath}
defined as the [[copairing]] $not = [inr,inl]$ of the right and left injections. Moreover, 2 can be given the structure of a [[monoid]] in $C$, with unit and multiplication

\begin{displaymath}
false : 1 \to 2 \qquad xor : 2 \otimes 2 \to 2
\end{displaymath}
defined by $false = inl$ and $xor = [id,not]$ (here we make use of the isomorphism $2 \otimes 2 \cong 2 + 2$ to define $xor$ by copairing). The mapping

\begin{displaymath}
X \mapsto 2 \otimes X \cong X + X
\end{displaymath}
thus extends to a [[monad]] on $C$, sending any object $X$ to the free object equipped with an involution over $X$. Explicitly, the unit $\eta_X : X \to 2\otimes X$ and the multiplication $\mu_X : 2\otimes 2\otimes X \to 2\otimes X$ of the monad are defined by tensoring the unit and the multiplication of the monoid with the identity on $X$, while the involution on $2 \otimes X$ is likewise defined by tensoring the involution on 2 with the identity on $X$.

We then have that involutions in $C$ are precisely the [[module over a monad|algebras]] of the monad $(2\otimes-,false\otimes-,xor\otimes-)$. In the forward direction, given an involution $f : X \to X$, we define a monad algebra structure $\alpha : 2\otimes X \to X$ on $X$ by $\alpha = [id,f]$ (again using the isomorphism $2\otimes X \cong X+X$). Conversely, given a monad algebra $\alpha : 2\otimes X \to X$, we can define an endomorphism $f : X \to X$ by $f = \alpha \circ inr$. The monad algebra laws imply that

\begin{displaymath}
\alpha \circ inr \circ \alpha \circ inr = \alpha \circ (2\otimes \alpha) \circ (2\otimes inr) \circ inr = \alpha \circ (xor\otimes id) \circ (2\otimes inr) \circ inr
\end{displaymath}
and since $xor$ is defined such that $(xor\otimes id) \circ (2\otimes inr) \circ inr = id$, we derive that $\alpha \circ inr$ is an involution.

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

\begin{itemize}%
\item [[chord diagram]]
\item [[dagger category]]
\item [[duality]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Philippe Flajolet and Robert Sedgewick, \emph{Analytic Combinatorics}, CUP, 2009. (\href{http://algo.inria.fr/flajolet/Publications/book.pdf}{author pdf})

\end{itemize}
[[!redirects involution]] [[!redirects involutions]]

[[!redirects antiinvolution]] [[!redirects antiinvolutions]] [[!redirects anti-involution]] [[!redirects anti-involutions]]



\end{document}