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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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, 16:38, 20 April 2013→6.42. Sign Processes : A Start: markup
On reflection, one observes that these processes are not completely trivial since they preserve the structure of their semantic partitions. In fact, each sign process preserves the entire topology — the family of sets closed under finite intersections and arbitrary unions — that is generated by its semantic equivalence classes. These topologies, <math>\operatorname{Top}(\text{A})\!</math> and <math>\operatorname{Top}(\text{B}),\!</math> can be viewed as partially ordered sets, <math>\operatorname{Poset}(\text{A})\!</math> and <math>\operatorname{Poset}(\text{B}),\!</math> by taking the inclusion ordering <math>(\subseteq)\!</math> as <math>(\le).\!</math> For each of the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> as things stand in their respective orderings <math>\operatorname{Poset}(\text{A})\!</math> and <math>\operatorname{Poset}(\text{B}),\!</math> the semantic equivalence classes of <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> are situated as intermediate elements that are incomparable to each other.
On reflection, one observes that these processes are not completely trivial since they preserve the structure of their semantic partitions. In fact, each sign process preserves the entire topology — the family of sets closed under finite intersections and arbitrary unions — that is generated by its semantic equivalence classes. These topologies, <math>\operatorname{Top}(\text{A})\!</math> and <math>\operatorname{Top}(\text{B}),\!</math> can be viewed as partially ordered sets, <math>\operatorname{Poset}(\text{A})\!</math> and <math>\operatorname{Poset}(\text{B}),\!</math> by taking the inclusion ordering <math>(\subseteq)\!</math> as <math>(\le).\!</math> For each of the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> as things stand in their respective orderings <math>\operatorname{Poset}(\text{A})\!</math> and <math>\operatorname{Poset}(\text{B}),\!</math> the semantic equivalence classes of <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> are situated as intermediate elements that are incomparable to each other.
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{lllll}
\operatorname{Top}(\text{A})
& = &
\operatorname{Poset}(\text{A})
& = &
\{
\varnothing,
\{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\},
\{
{}^{\backprime\backprime} \text{B} {}^{\prime\prime},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\},
S
\}.
\\[6pt]
\operatorname{Top}(\text{B})
& = &
\operatorname{Poset}(\text{B})
& = &
\{ \varnothing,
\{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\},
\{
{}^{\backprime\backprime} \text{B} {}^{\prime\prime},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\},
S
\}.
\end{array}</math>
|}
<pre>
<pre>
In anticipation of things to come, these orderings are germinal versions of the kinds of semantic hierarchies that will be used in this project to define the "ontologies", "world views", or "perspectives" corresponding to individual interpreters.
In anticipation of things to come, these orderings are germinal versions of the kinds of semantic hierarchies that will be used in this project to define the "ontologies", "world views", or "perspectives" corresponding to individual interpreters.