Eigenclass model: Difference between revisions

Revision as of 18:49, 5 November 2012

Revision as of 19:58, 5 November 2012

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then <i>x</i> is a <em>member-of</em> <i>y</i> and

<i>y</i> is a <em>container-of</em> <i>x</i>.

By axiomsaxiom (1) and (2), object membership is uniquely decomposable into the eigenclass map and the inheritance relation, which can be written as

:(&#1013;) = (<i>.ec</i>) &#9675; (&le;)

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Axiom (2) asserts antisymmetry of &le; , so that inheritance is a [[partial order]].

Furthermore, axiomsaxiom (1) and (2) assertasserts that

<span style="white-space:nowrap">(&#1013;) &#9675; (&le;) = (&#1013;)</span>, i.e.

:<i>x</i> &le; <i>y</i> &nbsp; only-if &nbsp; <i>a</i> &#1013; <i>x</i> implies <i>a</i> &#1013; <i>y</i> for every object <i>a</i> &nbsp; (every member of <i>x</i> is a member of <i>y</i>),

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<ol style="list-style-type:none; margin-left: 2ex;">

<li style="text-indent:-3ex; margin-left:3ex;">(1)

<li>(1)

Containers of an object <i>x</i> are exactly the

Every object <i>x</i> has a least container, <i>x.ec</i> (with respect to inheritance &le;).

ancestors the eigenclass of <i>x</i>,

i.e.

:(&#1013;) = (<i>.ec</i>) &#9675; (&le;) &nbsp; for a unique map <i>.ec</i>.

<li>(2)