transitive relation (TransitiveRelation)

A BinaryRelation rel is transitive if (rel inst1 inst2) and (rel inst2 inst3) imply (rel inst1 inst3), for all inst1, inst2, and inst3.

Ontology

SUMO / BASE-ONTOLOGY

Superclass(es)

entity

abstract

relation

binary relation

transitive relation

Instance(s)

successor attribute closure proper part sub collection less than greater than crosses precondition sub proposition sub plan sub graph larger smaller starts finishes before during earlier superficial part interior part developmental form version sub organizations

Subclass(es)

partial ordering relation equivalence relation

Coordinate term(s)

antisymmetric relation binary predicate intransitive relation irreflexive relation reflexive relation symmetric relation trichotomizing relation unary function

Axioms (3)

if rel is an instance of transitive relation,
then for all inst1,inst2,inst3 holds: if rel(inst1,inst2) holds and rel(inst2,inst3) holds, then rel(inst1,inst3) holds

(=>
      (instance ?REL TransitiveRelation)
      (forall
            (?INST1 ?INST2 ?INST3)
            (=>
                  (and
                        (holds ?REL ?INST1 ?INST2)
                        (holds ?REL ?INST2 ?INST3))
                  (holds ?REL ?INST1 ?INST3))))

If relation is partial ordering on class, then relation is reflexive on class and relation is an instance of transitive relation and relation is an instance of antisymmetric relation.

(=>
      (partialOrderingOn ?RELATION ?CLASS)
      (and
            (reflexiveOn ?RELATION ?CLASS)
            (instance ?RELATION TransitiveRelation)
            (instance ?RELATION AntisymmetricRelation)))

If relation is equivalence relation on class, then relation is an instance of transitive relation and relation is an instance of symmetric relation and relation is reflexive on class.

(=>
      (equivalenceRelationOn ?RELATION ?CLASS)
      (and
            (instance ?RELATION TransitiveRelation)
            (instance ?RELATION SymmetricRelation)
            (reflexiveOn ?RELATION ?CLASS)))