Preorder Relation

A Preorder Relation is a transitive reflexive binary relation.

• AKA: Preorder, Quasiorder.
• Context:
  • It can range from being a Weak Partial Order Relation (if it is also antisymmetric) to being an Equivalence Relation (if it is also symmetric).
  • It can range from being a Total Preorder Relation to being a Non-Total Preorder Relation.
• Example(s):
  • Logical Implication Relation.
  • Graph-Minor Relation.
  • …
• Counter-Example(s):
  • a Partial Order Relation.
• See: Partial Order Relation, Order Relation, Entailment Relation.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Glossary_of_order_theory#P Retrieved:2015-6-14.
  • Preorder. A preorder is a binary relation that is reflexive and transitive. Such orders may also be called quasiorders. The term preorder is also used to denote an acyclic binary relation (also called an acyclic digraph).

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Preorder
  • In mathematics, especially in order theory, preorders are binary relations that satisfy certain conditions. For example, all partial orders and equivalence relations are preorders. The name quasiorder is also common for preorders. Other names are pre-order, quasi-order, and quasi order. Many order theoretical definitions for partially ordered sets can be generalized to preorders, but the extra effort of generalization is rarely needed.

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