Converse relation

Properties

In the monoid of binary endorelations on a set (with the binary operation on relations being the composition of relations), the converse relation does not satisfy the definition of an inverse from group theory, that is, if ${\displaystyle L}$ is an arbitrary relation on ${\displaystyle X,}$ then ${\displaystyle L\circ L^{\mathrm{T}}}$ does not equal the identity relation on ${\displaystyle X}$ in general. The converse relation does satisfy the (weaker) axioms of a semigroup with involution: ${\displaystyle {\left(L^{\mathrm{T}}\right)}^{\mathrm{T}}=L}$ and ${\displaystyle (L\circ R{)}^{\mathrm{T}}=R^{\mathrm{T}}\circ L^{\mathrm{T}}.}$

Since one may generally consider relations between different sets (which form a category rather than a monoid, namely the category of relations Rel), in this context the converse relation conforms to the axioms of a dagger category (a.k.a. category with involution). A relation equal to its converse is a symmetric relation; in the language of dagger categories, it is self-adjoint.

Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relations as sets), and actually an involutive quantale. Similarly, the category of heterogeneous relations, Rel is also an ordered category.

In the calculus of relations, conversion (the unary operation of taking the converse relation) commutes with other binary operations of union and intersection. Conversion also commutes with unary operation of complementation as well as with taking suprema and infima. Conversion is also compatible with the ordering of relations by inclusion.

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, connected, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its converse is too.

Inverses

If ${\displaystyle I}$ represents the identity relation, then a relation ${\displaystyle R}$ may have an inverse as follows: ${\displaystyle R}$ is called

right-invertible

if there exists a relation ${\displaystyle X,}$ called a right inverse of ${\displaystyle R,}$ that satisfies ${\displaystyle R\circ X=I.}$

left-invertible

if there exists a relation ${\displaystyle Y,}$ called a left inverse of ${\displaystyle R,}$ that satisfies ${\displaystyle Y\circ R=I.}$

invertible

if it is both right-invertible and left-invertible.

For an invertible homogeneous relation ${\displaystyle R,}$ all right and left inverses coincide; this unique set is called its inverse and it is denoted by ${\displaystyle R^{-1}.}$ In this case, ${\displaystyle R^{-1}=R^{\mathrm{T}}}$ holds.

Converse relation of a function

A function is invertible if and only if its converse relation is a function, in which case the converse relation is the inverse function.

The converse relation of a function ${\displaystyle f:X\to Y}$ is the relation ${\displaystyle f^{-1}\subseteq Y\times X}$ defined by the ${\displaystyle \mathrm{graph}{f}^{-1}=\{(y,x)\in Y\times X:y=f(x)\}.}$

This is not necessarily a function: One necessary condition is that ${\displaystyle f}$ be injective, since else ${\displaystyle f^{-1}}$ is multi-valued. This condition is sufficient for ${\displaystyle f^{-1}}$ being a partial function, and it is clear that ${\displaystyle f^{-1}}$ then is a (total) function if and only if ${\displaystyle f}$ is surjective. In that case, meaning if ${\displaystyle f}$ is bijective, ${\displaystyle f^{-1}}$ may be called the inverse function of ${\displaystyle f.}$

For example, the function ${\displaystyle f(x)=2x+2}$ has the inverse function ${\displaystyle f^{-1}(x)=\frac{x}{2}-1.}$

However, the function ${\displaystyle g(x)=x^2}$ has the inverse relation ${\displaystyle g^{-1}(x)=\pm \sqrt{x},}$ which is not a function, being multi-valued.

Composition with relation

Using composition of relations, a relation can be composed with its converse.

For the subset relation ${\displaystyle \subseteq }$ on the power set ${\displaystyle \mathcal{P}(U)}$ of a universe ${\displaystyle U}$, both compositions with its converse are the universal relation on ${\displaystyle \mathcal{P}(U)}$:

${\displaystyle (\subseteq )\circ (\supseteq )=\mathcal{P}(U)\times \mathcal{P}(U)\text{and}(\supseteq )\circ (\subseteq )=\mathcal{P}(U)\times \mathcal{P}(U).}$

Indeed, for any ${\displaystyle A,C\subseteq U}$,

${\displaystyle A((\subseteq )\circ (\supseteq ))C⟺\mathrm{\exists }B\subseteq U:A\subseteq B\wedge C\subseteq B}$

which holds by taking ${\displaystyle B=A\cup C}$; similarly,

${\displaystyle A((\supseteq )\circ (\subseteq ))C⟺\mathrm{\exists }B\subseteq U:B\subseteq A\wedge B\subseteq C,}$

which holds by taking ${\displaystyle B=A\cap C}$.

Now consider the set membership relation ${\displaystyle \in \subseteq U\times \mathcal{P}(U)}$ and its converse ${\displaystyle \ni \subseteq \mathcal{P}(U)\times U}$. For sets ${\displaystyle A,B\subseteq U}$,

${\displaystyle A(\ni \circ \in )B⟺\mathrm{\exists }z\in U:z\in A\wedge z\in B⟺A\cap B\ne \mathrm{\varnothing },}$

so ${\displaystyle \ni \circ \in }$ is the "nonempty intersection" relation on ${\displaystyle \mathcal{P}(U)}$. Conversely, for elements ${\displaystyle x,y\in U}$,

${\displaystyle x(\in \circ \ni )y⟺\mathrm{\exists }A\subseteq U:x\in A\wedge y\in A,}$

which always holds (e.g. for ${\displaystyle A=\{x,y\}}$); hence ${\displaystyle \in \circ \ni <!--\; -->=U\times U}$ is the universal relation on ${\displaystyle U}$.

The compositions are used to classify relations according to type: for a relation Q, when the identity relation on the range of Q contains Q^TQ, then Q is called univalent. When the identity relation on the domain of Q is contained in QQ^T, then Q is called total. When Q is both univalent and total then it is a function. When Q^T is univalent, then Q is termed injective. When Q^T is total, Q is termed surjective.

If Q is univalent, then QQ^T is an equivalence relation on the domain of Q, see Transitive relation#Related properties.