Object Membership – Basic Structure

An abstract structure is described which provides a general model for the innermost core of object-oriented programming and modelling. The structure is called basic structure of ϵ and is introduced in the signature (O, ϵ, ϵ¯^⁽*⁾, r, .ec, .ɛɕ) with a single sort O of objects. • ϵ is the object membership relation, an indirect counterpart to set membership. • ϵ¯^⁽*⁾ is a left-infinite sequence …, ϵ¯^-1, ϵ¯⁰, ϵ¯¹ of relations, whose 0-th member is ≤ – the inheritance relation, corresponding to inclusion between sets. The 1-indexed member is the power membership relation ϵ¯ which, if .ec is total, equals the composition (.ec) ○ (≤) (in general, ϵ¯ is an abstraction of this composition). • r is the inheritance root, a distinguished object containing all objects, including r itself, as members in ϵ, thus playing the role of the universal set. • .ec is the powerclass partial map, corresponding to a relativized powerset operator. • .ɛɕ is the primary singleton partial map, corresponding to set membership between non-singleton sets and singleton sets.

As a key characteristics induced by the structure, each object is assigned a rank between 0 and a fixed limit ordinal ϖ. Objects with rank ϖ are unbounded, the remaining ones are bounded. For unbounded objects x, the images of {x} under ϵ and ϵ¯ are coincident. Therefore, object membership is formed as the union

• (ϵ) = (∊) ∪ (ϵ¯)

where ∊ is the domain-restriction of ϵ to bounded objects.

It is shown that every basic structure is a substructure of an (ϖ+1)-superstructure and thus can be embedded in the von Neumann universe of sets. The inheritance root r appears as the ϖ-th cumulation of urelement-like sets. The above equality for object membership is expressed as

• x ϵ y   iff   x ∈ y or ℙ(x) ∩ r ⊆ y.   (ℙ(x) stands for the powerset of x.)