# Binary relation

* [Binary Relations](#binary-relations)
* [List of properties](#list-of-properties)
* [List of relations](#list-of-relations)
* [Total](#total)
* [Trichotomous](#trichotomous)
* [Euclidean](#euclidean)
* [Serial](#serial)
* [Set-like or Local](#set-like-or-local)
* [Partial equivalence relation](#partial-equivalence-relation)
* [Reference](#reference)

## Binary Relations

* equivalence relation: symmetric, transitive, reflexive
* partial equivalence: symmetric, transitive

## List of properties

Important properties that a binary relation may have include:

* reflexivity, irreflexivity, coreflexivity
* symmetry, antisymmetry, asymmetry
* transitivity
* totality
* right Euclidean, left Euclidean, Euclidean
* trichotomy, serial properties, set-like properties

## List of relations

List of some relations by properties

* **equivalence**: reflexive, symmetric, transitive.
* **partial equivalence**: symmetric, transitive.
* **reflexive**: symmetric, transitive, serial.
* **partial order**: reflexive, antisymmetric, transitive.
* **total order** (linear order, or chain): partial order that is total.
* **well-order**: linear order where every nonempty subset has a least element.

## Total

This definition for total is different from left total. For example, `>=` is a total relation.

## Trichotomous

For example, `>` is a trichotomous relation, while the relation "divides" on natural numbers is not.

## Euclidean

A Euclidean relation is both left and right Euclidean. Equality is a Euclidean relation because if `x=y` and `x=z`, then `y=z`.

## Serial

"Is greater than" is a serial relation on the integers. But it is not a serial relation on the positive integers, because there is no `y` in the positive integers such that `1 > y`. However, "is less than" is a serial relation on the positive integers, the rational numbers and the real numbers.

Every reflexive relation is serial: for a given `x`, choose `y = x`. A serial relation can be equivalently characterized as every element having a non-empty successor neighborhood. Similarly an inverse serial relation is a relation in which every element has non-empty predecessor neighborhood.

## Set-like or Local

The usual ordering `<` on the class of ordinal numbers is set-like, while its inverse `>` is not.

## Partial equivalence relation

A partial equivalence relation (PER) `R` on a set `X` is a relation that is _symmetric_ and _transitive_.

It holds for all `a, b, c ∈ X` that: 1. if `aRb`, then `bRa` (symmetry) 2. if `aRb` and `bRc`, then `aRc` (transitivity)

If `R` is also reflexive, then `R` is an equivalence relation.

## Reference

* [Reflexive relation](http://www.wikipedia.com/en/Reflexive_relation)
* [Symmetric relation](http://www.wikipedia.com/en/Symmetric_relation)
* [Transitive relation](http://www.wikipedia.com/en/Transitive_relation)
* [Equivalence relation](http://www.wikipedia.com/en/Equivalence_relation)
* [Partial equivalence relation](http://www.wikipedia.com/en/Partial_equivalence_relation)
