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Reflexive relation. In mathematics, a binary relation on a set is reflexive if it relates every element of to itself. [1][2] An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess ...
Of particular importance are relations that satisfy certain combinations of properties. A partial order is a relation that is reflexive, antisymmetric, and transitive, [3] an equivalence relation is a relation that is reflexive, symmetric, and transitive, [4] a function is a relation that is right-unique and left-total (see below). [5] [6]
A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order. A group with a compatible total order is a totally ordered group. There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a betweenness relation.
A total order is a relation that is reflexive, antisymmetric, transitive and connected. [37] A strict total order is a relation that is irreflexive, asymmetric, transitive and connected. An equivalence relation is a relation that is reflexive, symmetric, and transitive.
Partial orders. A reflexive, weak, [1] or non-strict partial order, [2] commonly referred to simply as a partial order, is a homogeneous relation ≤ on a set that is reflexive, antisymmetric, and transitive. That is, for all it must satisfy: Reflexivity: , i.e. every element is related to itself.
The number of irreflexive relations is the same as that of reflexive relations. The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders. The number of strict weak orders is the same as that of total preorders. The total orders are the partial orders that are also total preorders.
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number is equal to itself (reflexive). If , then (symmetric).
This produces a relation that is still reflexive, antisymmetric and transitive and that strictly contains the original one. It follows that if the partial orders extending R {\displaystyle R} are themselves partially ordered by extension, then any maximal element of this extension order must be a total order.