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:

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 []. 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.

Full article:
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