For being able to easily retrieve the committees that are chaired or co-chaired by a club member, we add two reference properties to our Committee-ClubMember example model: the property of a club member to be the chair of a committee (ClubMember::chairedCommittee) and the property of a club member to be the co-chair of a committee (ClubMember::coChairedCommittee). We assume that any club member may chair or co-chair at most one committee (where the disjunction is non-exclusive). So, we get the following model:

Notice that there is a close correspondence between the two reference properties Committee::chair and ClubMember::chairedCommittee. They are the inverse of each other: when the club member Tom is the chair of the budget committee, expressed by the tuple ("budget committee", "Tom"), then the budget committee is the committee chaired by the club member Tom, expressed by the inverse tuple ("Tom", "budget committee"). For expressing this inverse correspondence in the diagram, we append an inverse property constraint, inverse of chair, in curly braces to the declaration of the property ClubMember::chairedCommittee, and a similar one to the property Committee::chair, as shown in the following diagram:

Using the reference path notation of OOP languages, with c referencing a Committee object, we obtain the equation:

Or, the other way around, with m referencing a ClubMember object, we obtain the equation:

Notice that when a property p₂ is the inverse of a property p₁, this implies that, the other way around, p₁ is the inverse of p₂. Therefore, when we declare the property ClubMember::chairedCommittee to be the inverse of Committee::chair, then, implicitly, Committee::chair is the inverse of ClubMember::chairedCommittee. We therefore call Committee::chair and ClubMember::chairedCommittee a pair of mutually inverse reference properties. Having such a pair in a model implies redundancy because each of the two involved reference properties can be derived from the other by inversion. This type of redundancy implies data storage overhead and update overhead, which is the price to pay for the bidirectional navigability that supports efficient object access in both directions.

In general, a bidirectional association between the classes A and B is represented by two reference properties A::bbb and B::aaa such that for any object a1 instantiating A, it holds that

For maintaining the duplicate information of a mutually inverse reference property pair, it is common to treat one of the two involved properties as the master, and the other one as the slave, and take this distinction into consideration in the code of the change methods (such as the property setters) of the affected model classes. We indicate the slave of an inverse reference property pair in a model diagram by declaring the slave property to be a derived property using the UML notation of a slash (/) as a prefix of the property name as shown in the following diagram:

The property chairedCommittee in ClubMember is now derived (as indicated by its slash prefix). Its annotation {inverse of chair} defines a derivation rule according to which it is derived by inverting the property Committee::chair.

There are two ways how to realize the derivation of a property: it may be derived on read via a read-time computation of its value, or it may be derived on update via an update-time computation performed whenever one of the variables in the derivation expression (typically, another property) changes its value. The latter case corresponds to a materialized view in a database. While a reference property that is derived on read may not guarantee efficient navigation, because the on-read computation may create unacceptable latencies, a reference property that is derived on update does provide efficient navigation.

When we designate an inverse reference property as derived by prefixing its name with a slash (/), we indicate that it is derived on update. For instance, the property /chairedCommittee in the example above is derived on update from the property chair.

In the case of a derived reference property, we have to deal with life-cycle dependencies between the affected model classes requiring special change management mechanisms based on the functionality type of the represented association (either one-to-one, many-to-one or many-to-many).

In our example of the derived inverse reference property ClubMember::chairedCommittee, which is single-valued and optional, this means that

In the case of a derived inverse reference property that is multi-valued while its inverse base property is single-valued (like Publisher::publishedBooks in Figure 12.2 below being derived from Book::publisher), the life cycle dependencies imply that

Notice that from a purely computational point of view, we are free to choose either of the two mutually inverse reference properties (like Book::authors and Author::authoredBooks) to be the master. However, in many cases, associations represent asymmetrical ontological existence dependencies that dictate which of the two mutually inverse reference properties is the master. For instance, the authorship association between the classes Book and Author represents an existential dependency of books on their authors. A book existentially depends on its author(s), while an author does not existentially depend on any of her books. Consequently, the corresponding object lifecycle dependency between Book and Author implies that their bidirectional association is maintained by maintaining Author references in Book::authors as the natural choice of master property, while Author::authoredBooks is the slave property, which is derived from Book::authors.