Equivalence Relation - Connections To Other Relations

Connections To Other Relations

• A partial order is a relation that is reflexive, antisymmetric, and transitive.
• A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraic structure, and which respects the additional structure. In general, congruence relations play the role of kernels of homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many important cases congruence relations have an alternative representation as substructures of the structure on which they are defined. E.g. the congruence relations on groups correspond to the normal subgroups.
• Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric.
• A strict partial order is irreflexive, transitive, and asymmetric.
• A partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reflexive if and only if for all a∈X, there exists a b∈X such that a~b.
• A reflexive and symmetric relation is a dependency relation, if finite, and a tolerance relation if infinite.
• A preorder is reflexive and transitive.