# Reflexivity in RO

We first introduce a pattern used in RO for combination with property chains, then we discuss the general treatment of reflexivity in RO, and how relations in RO may be *locally* reflexive but are rarely *globally* reflexive.

## Defining Property Chains involving Reflexivity

When defining property chains over *R* and *R2* we typically name this according to the composition and include an axiom:

```
ObjectProperty: {R}_{R2}
SubObjectPropertyOf: R o R2
```

we may also want to make this a reflexive property chain:

```
ObjectProperty: R
SubObjectPropertyOf: {R}_{R2}
```

When this ODP is used, the property chain axiom should be tagged with an is a defining property chain axiom where second argument is reflexive axiom.

In this case, the parent relation is a grouping relation for query purposes. Consider:

- http://purl.obolibrary.org/obo/RO_0002177 'attached to part of'
- http://purl.obolibrary.org/obo/RO_0002371 'attached to'

Consider a ligament or tendon attached to some projection P that is part of a bone B, that is part of the limb skeleton. We may be interested in all attachmed structures for any part of the body, and this named property chain provides a convenient way of querying for it.

In all cases of this ODP, the parent property can be safely 'unfolded' away with no loss of information. i.e. expanded to an actual chain with two property expressions or assertions.

The placement of the base form as a child of the parent is justified via the local reflexivity of the 2nd argument.

## Reflexivity

In many formal treatments of relations, the Reflexivity characteristic is frequently used. For example, in mereology, 'part of' is frequently characterized as reflexive. This can seem confusing when compared with everyday usage (no one typically considers an object to be part of itself). However, it is convenient from the point of view of mathematical formulation.

For example, it is common to have a property chain axiom:

`overlaps <- has-part o part-of`

If has-part and part-of are *not* reflexive then this inference will be incomplete.
We can get around this by adding individual subPropertyOf axioms:

`has-part -> overlaps`

`part-of`

-> overlaps`

However, the rules are easier to maintain for the reflexive case.

Another case is declaring two structures to have no parts in common:

`(part-of some X) disjointWith (part-of some Y)`

This is also incomplete if reflexivity is not declared.

## Proper-X relations

Some ontologies introduce a relation `proper_R`

to mark the non-reflexive case of R.

E.g. proper-part-of.

In RO, we typically do not include a 'proper' form of the relation. This is because it inflates the ontology, and complicates reasoning. Also, it does not actually add information or provide real utility or intuition.

## Local Reflexivity

We rarely if ever declare a relation to be reflexive. Reflexivity is incompatible with domain and range assertions. Consider:

```
ObjectProperty: P
Characteristics: Reflexive
Domain: C
Class: C
DisjointWith: D
```

reflexivity of P literally means that every individual in the domain stands in a P relation to itself. This would include any instances of D. However, it an instance of D cannot be the subject of a P relation, hence D is unsatisfiable.

This is rarely a concern in formal treatments, where a restricted domain is assumed; however, RO covers all domains so full reflexivity is almost always inappropriate.

Instead, here we would make use of a 'local reflexivity' axiom. E.g.

```
ObjectProperty: P
Domain: C
Class: C
DisjointWith: D
SubClassOf: P Self
```

In practice we may not have asserted all cases of local reflexivity (TODO: tracker link).