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# binary relation (BinaryRelation)

BinaryRelations are relations that are true only of pairs of things. BinaryRelations are represented as slots in frame systems.

## Ontology

SUMO / BASE-ONTOLOGY

## Class(es)

 class

inheritable relation

binary relation

## Superclass(es)

 entity

abstract

relation

binary relation

distributes

## Subclass(es)

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

## Coordinate term(s)

binary function  binary predicate  case role  function  intentional relation  list  object attitude  partial valued relation  predicate  probability relation  propositional attitude  quaternary function  quaternary predicate  quaternary relation  quintary predicate  quintary relation  relation extended to quantities  single valued relation  spatial relation  temporal relation  ternary function  ternary predicate  ternary relation  total valued relation  unary function  variable arity relation

## Constrains relations

equivalence relation on  inverse  irreflexive on  partial ordering on  reflexive on  total ordering on  trichotomizing on

## Axioms (4)

relation is disjointly decomposed into binary relation,ternary relation,quaternary relation,quintary relation,variable arity relation.
`(disjointDecomposition Relation BinaryRelation TernaryRelation QuaternaryRelation QuintaryRelation VariableArityRelation)`

If rel is an instance of binary relation, then there don't exist item1,item2,item3, so that rel(item1,item2,item3,) holds.
```(=>
(instance ?REL BinaryRelation)
(not
(exists
(?ITEM1 ?ITEM2 ?ITEM3 @ROW)
(holds ?REL ?ITEM1 ?ITEM2 ?ITEM3 @ROW))))```

If
, then rel is an instance of asymmetric relation.
```(=>
(and
(instance ?REL BinaryRelation)
(or
(domain ?REL 1 ?CLASS1)
(domainSubclass ?REL 1 ?CLASS1))
(or
(domain ?REL 2 ?CLASS2)
(domainSubclass ?REL 2 ?CLASS2)
(range ?REL ?CLASS2)
(rangeSubclass ?REL ?CLASS2))
(disjoint ?CLASS1 ?CLASS2))
(instance ?REL AsymmetricRelation))```

```(=>
(and
(instance ?REL RelationExtendedToQuantities)
(instance ?REL BinaryRelation)
(instance ?NUMBER1 RealNumber)
(instance ?NUMBER2 RealNumber)
(holds ?REL ?NUMBER1 ?NUMBER2))
(forall
(?UNIT)
(=>
(instance ?UNIT UnitOfMeasure)
(holds
?REL
(MeasureFn ?NUMBER1 ?UNIT)
(MeasureFn ?NUMBER2 ?UNIT)))))```