and 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 (aka 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, (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 represents the identity relation, then a relation may have an inverse as follows: is called if there exists a relation called a of that satisfies if there exists a relation called a of that satisfies if it is both right-invertible and left-invertible. For an invertible homogeneous relation all right and left inverses coincide; this unique set is called its and it is denoted by In this case, 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 is the relation defined by the This is not necessarily a function: One necessary condition is that be injective, since else is multi-valued. This condition is sufficient for being a partial function, and it is clear that then is a (total) function if and only if is surjective. In that case, meaning if is bijective, may be called the inverse function of For example, the function has the inverse function However, the function has the inverse relation which is not a function, being multi-valued. Composition with relation Using composition of relations, the converse may be composed with the original relation. For example, the subset relation composed with its converse is always the universal relation: ∀A ∀B