Eigenclass model: Difference between revisions

Revision as of 18:46, 23 April 2013

Revision as of 08:06, 1 June 2013

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Generally, the model relates to the core part of data models in which classes appear as objects. As a consequence, the model relates to [[ontology language]]s such as [[RDF Schema]] or [[Web Ontology Language|OWL]]. A good indicator of applicability is the presence of the notion of [[metaclass]].

One can distinguish two degrees of applicability. High degree is assigned to languages that allow an actual reference to an eigenclass: Ruby, Smalltalk, Objective-C and, in a lesser sense, Scala. Low degree is assigned to languages in which eigenclasses are a purely fictitious concept. In this latter case the eigenclass model can be regarded as an abstract device for joining inheritance with the instance-of relation.

== History ==

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! Inventor

! The principle

! …… in terms of the eigenclass model

|-

| style="white-space:nowrap" | c. 1980

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Let ''<u>R</u>'' denote the eigenclass chain

{''<u>r</u>'', ''<u>r</u>.ec(1)'', ''<u>r</u>.ec(2)'', &hellip;…}

of the inheritance root.

Being a distinguished subset of ''<u>H</u>'', the set ''<u>R</u>'' can be called the ''reduced helix''.

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=== {{anchor|essential_structure}} Essential structure of &#1013; ===

An ''essential structure'' of &#1013;ϵ is a structure

<span style="white-space:nowrap">''(<u>O</u>, .ec, &le;, <u>r</u>)''</span>

where

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Like with canonical structures, an essential structure is given by

<span style="white-space:nowrap">''(<u>O</u>,'' &#1013;ϵ'')''</span>

where

<span style="white-space:nowrap">(&#1013;ϵ) = (''.ec'') &#9675;○ (&le;≤)</span>.

=== The embedding ===

Any essential structure of &#1013;ϵ can be embedded into a set ''<u>V</u>'' that is formed as a cumulative hierarchy over a set of [[urelement]]s, ''<u>U</u>''.

Elements of ''<u>U</u>'' can be [[Purepure set|pure sets]]s

that are minimal both in

<span style="white-space:nowrap">''(<u>V</u>,'' ∈'')''</span> and

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:''P<sub style="margin-left:-.8ex">⋆</sub>(X)'' = (''P(X)'' ∖ {∅}) ∪ ''X''

where ''P(X)'' denotes the powerset of ''X''.

The ''ω''-th stage, called ''[[Universe (mathematics)#In ordinary mathematics| superstructure]]'' in the field of [[non-standard analysis]],{{efn|

However, we provide a slightly different construction by removing the empty set.

}}

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<!-- -->

Then

<span style="white-space:nowrap">''(<u>V</u>, .ec,'' &sube;⊆'', <u>r</u>)''</span> is an essential structure of &#1013;ϵ.

The inheritance relation, &le;≤, coincides with set inclusion, &sube;⊆, on ''<u>V</u>''.

Object membership on ''<u>V</u>'' is given by:

''x'' &#1013;ϵ ''y'' iff ''P(x)'' ∩ ''<u>r</u>'' is a subset of ''y''.

Terminal objects are the urelements.

The following are satisfied:

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| if (and only if) ''x'' ∈ ''<u>r</u>'',

|- valign=top

| ''x.ec'' &sub;⊂ ''x''

| if (and only if) ''x'' &#1013;ϵ ''x''.

|}

Any subset ''<u>O</u>'' of ''<u>V</u>'' such that ''<u>r</u>'' ∈ ''<u>O</u>'' and ''<u>O</u>.ec'' = ''<u>V</u>.ec'' ∩ ''<u>O</u>'' forms an "object system": the substructure

<span style="white-space:nowrap">''(<u>O</u>, .ec,'' &sube;⊆'', <u>r</u>)''</span> is an essential structure of object membership.

<!-- -->

Conversely, any essential structure of &#1013;ϵ can be represented by such set ''<u>O</u>''.

Moreover, such a representation exists that

for every ''x'', ''y'' from ''<u>O</u>'',

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A generalization of the canonical structure of &#1013;

allows

for a description of the core structure of [[Ontology_Ontology (information_scienceinformation science)|ontologies]] in which

classes are (among) individuals.<ref name="ome-ontology">

{{cite web | url=http://www.atalon.cz/om/object-membership/ontology/ | title=Object Membership: The ontological structure}}</ref>

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respectively.

Objects are called <em>''resources</em>''

&ndash; every object is an instance of the <code>rdfs:Resource</code> class (''<u>r</u>'').

Classes are instances of <code>rdfs:Class</code> (''<u>c</u>''),