Properties
A relation equal to its inverse is a symmetric relation (in the language of dagger categories, it is self-adjoint).
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.
However, if a relation is extendable, this need not be the case for the inverse.
The operation of taking a relation to its inverse gives the category of relations Rel the structure of a dagger category.
The set of all binary relations B(X) on a set X is a semigroup with involution with the involution being the mapping of a relation to its inverse relation.
Read more about this topic: Inverse Relation
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)