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# 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)

```(=>
(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)))```