Object Membership: The Core Structure of Object Technology

An abstract structure is described that is believed to be central to object-oriented programming and modelling. In its main form, the structure is built from three relations between objects, ϵ, ≤ and.ec. The most fundamental relation, ϵ, is called object membership and is a refinement of the instance-of relation. The ≤ relation is the inheritance between objects. Finally,.ec is a partial map that forms a distinguished subrelation of ϵ. The structure arises as a generalization of the innermost core of the object model of the Ruby programming language. In Ruby, the.ec map is total – every object has an eigenclass. The following equalities hold: (≥) = (.ec) ○ (϶) and (ϵ) = (.ec) ○ (≤). The second equality says that object membership is the composition of infinite regress of eigenclasses with inheritance. As an essential feature, the structure supports circular objects, i.e. objects x such that x ϵ x. This is also the main indicator of applicability. It is shown how the structure applies to class-based programming languages (Ruby, Python, Java, Scala, Smalltalk, Objective-C, CLOS, Perl and Dylan), prototypal languages (JavaScript), and ontology languages (RDF Schema, OWL Full). A general mathematical structure of object membership is described in an affiliate document [45]. A gradual set-theoretic representation is provided with the exact correspondence of ϵ, ≤,.ec and derived constituents of the structure to fundamental notions of set theory. As a result, a uniform and mathematically precise view of an essential part of object technology is provided