Inverse relation

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For inverse relationships in statistics, see negative relationship.

In mathematics, the inverse relation of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'. In formal terms, if X and Y are sets and LX × Y is a relation from X to Y, then L-1 is the relation defined so that y L-1 x if and only if x L y. In set-builder notation, L-1 = {(y, x) ∈ Y × X | (x, y) ∈ L}.

The notation is analogous with that for an inverse function. Although many functions do not have an inverse, every relation does have a unique inverse. Despite the notation and terminology, the inverse relation is not an inverse in the sense of group inverse. However, the unary operation that maps a relation to the inverse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, more generally, induces a dagger category on the category of relations as detailed below. As a unary operation, taking the inverse (sometimes called inversion) commutes with the order-related operations of relation algebra, i.e., it commutes with union, intersection, complement etc.

The inverse relation is also called the converse relation or transpose relation— the latter in view of its similarity with the transpose of a matrix.[1] It has also been called the opposite or dual of the original relation.[2] Other notations for the inverse relation include LC, LT, L~ or ${\displaystyle {\breve {L}}}$ or L° or L.

Examples

For usual (maybe strict or partial) order relations, the converse is the naively expected "opposite" order, e.g. ${\displaystyle \leq ^{-1}=\ \geq ,~<^{-1}=\ >}$, etc.

Inverse relation of a function

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

The inverse relation of a function ${\displaystyle f:X\to Y}$ is the relation ${\displaystyle f^{-1}:Y\to X}$ defined by ${\displaystyle \operatorname {graph} \,f^{-1}=\{(y,x)\mid y=f(x)\}}$.

This is not necessarily a function: One necessary condition is that 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 f is surjective. In that case, i.e. if f is bijective, ${\displaystyle f^{-1}}$ may be called the inverse function of f.

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

Properties

In the monoid of binary endorelations on a set (with the binary operation on relations being the composition of relations), the inverse relation does not satisfy the definition of an inverse from group theory, i.e. if L is an arbitrary relation on X, then ${\displaystyle L\circ L^{-1}}$ does not equal the identity relation on X in general. The inverse relation does satisfy the (weaker) axioms of a semigroup with involution: ${\displaystyle (L^{-1})^{-1}=L}$ and ${\displaystyle (L\circ R)^{-1}=R^{-1}\circ L^{-1}}$.[3]

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 inverse relation conforms to the axioms of a dagger category (aka category with involution).[3] A relation equal to its inverse 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 heterogenous relations, Rel is also an ordered category.[3]

In relation algebra (which is an abstraction of the properties of the algebra of endorelations on a set), inversion (the operation of taking the inverse relation) commutes with other binary operations of union and intersection. Inversion also commutes with unary operation of complementation as well as with taking suprema and infima. Inversion is also compatible with the ordering of relations by inclusion.[1]

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