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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 reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.
A relation that is reflexive, antisymmetric, and transitive. Strict partial order A relation that is irreflexive, asymmetric, and transitive. Total order A relation that is reflexive, antisymmetric, transitive and connected. [20] Strict total order A relation that is irreflexive, asymmetric, transitive and connected. Uniqueness properties: One ...
An irreflexive, strong, [1] or strict partial order is a homogeneous relation < on a set that is transitive, irreflexive, and asymmetric; that is, it satisfies the following conditions for all ,,: Transitivity : if a < b {\displaystyle a<b} and b < c {\displaystyle b<c} then a < c {\displaystyle a<c} .
A reflexive and symmetric relation is a dependency relation (if finite), and a tolerance relation if infinite. A preorder is reflexive and transitive. A congruence relation is an equivalence relation whose domain X {\displaystyle X} is also the underlying set for an algebraic structure , and which respects the additional structure.
An antisymmetric relation on a set may be reflexive (that is, for all ), irreflexive (that is, for no ), or neither reflexive nor irreflexive. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.
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.
The incomparability relation is always symmetric and it will be reflexive if and only if < is an irreflexive relation (which is assumed by the above definition). Consequently, a strict partial order < {\displaystyle \,<\,} is a strict weak order if and only if its induced incomparability relation is an equivalence relation .
An equivalence relation is a relation that is reflexive, symmetric, and transitive, like equality expressed through the symbol "=". [74] A strict partial order is a relation that is irreflexive, anti-symmetric, and transitive, like the relation being less than expressed through the symbol "<". [75]