> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/220-relation-theory/relation-types.md).

# Types of relations

Relations: division based on general properties

* Finiteness of sets
  * finitary relation
  * infinitary relation
* Geneity of sets
  * homogeneous relation: between entities of the same type
  * heterogeneous relation: between different type of entities
* Arity of relation
  * nullary: makes no sense as a relation
  * unary  : is more of a property then relation
  * binary : usual relation between 2 sets (even if they are the same set)
  * ternary: relation between 3 sets (rare)
  * arity > 2 are generalizations, they are rels on/between indexed sets
* Cardinality of relation
  * null (empty) relation
  * full (total) relation

Relations: division based on axioms

* Reflexivity
  * reflexive relation
  * irreflexive relation
  * coreflexive relation
  * quasi-reflexive relation
  * *negated* (¬)
    * non-reflexive relation
    * non-irreflexive relation
* Symmetry
  * symmetric relation
  * asymmetric relation
  * antisymmetric relation
  * *negated* (¬)
    * non-symmetric relation
    * non-asymmetric relation
    * non-antisymmetric relation
* Transitivity
  * transitive relation
  * *negated* (¬)
    * intransitive  relation
    * non-transitive
* reflexive: `Sy+Tr+Serial`
* equivalence, `EQ` = `Re+Sy+Tr`
* partial equivalence, `pEQ`: `Sy+Tr`
* partial order: `pOrd` = `Re+vS+Tr`
* linear (total) order: partial order that is total, `Re+vS+Tr+`
* linear (total) order: partial order that is total, `Re+vS+Tr+Total`
* well-order: linear order where every non-empty subset has a least element.
* The relationship of one set being a subset of another is called inclusion or sometimes containment.