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.