Changes

==Graphical representations==

==Graphical representations==

The dyadic components of sign relations have graph-theoretic representations, as <i>digraphs</i> (or <i>directed graphs</i>), which provide concise pictures of their structural and potential dynamic properties.

The dyadic components of sign relations have graph&#8209;theoretic representations, as <i>digraphs</i> (or <i>directed graphs</i>), which provide concise pictures of their structural and potential dynamic properties.

By way of terminology, a directed edge <math>(x, y)</math> is called an <i>arc</i> from point <math>x</math> to point <math>y,</math> and a self-loop <math>(x, x)</math> is called a <i>sling</i> at <math>x.</math>

By way of terminology, a directed edge <math>(x, y)</math> is called an <i>arc</i> from point <math>x</math> to point <math>y,</math> and a self-loop <math>(x, x)</math> is called a <i>sling</i> at <math>x.</math>

The denotative components <math>\mathrm{Den}(L_\mathrm{A})</math> and <math>\mathrm{Den}(L_\mathrm{B})</math> can be represented as digraphs on the six points of their common world set <math>W = O \cup S \cup I =</math> <math>\{ \mathrm{A}, \mathrm{B}, \text{“A”}, \text{“B”}, \text{“i”}, \text{“u”} \}.</math>&nbsp; The arcs are given as follows:

The denotative components <math>\mathrm{Den}(L_\mathrm{A})</math> and <math>\mathrm{Den}(L_\mathrm{B})</math> can be represented as digraphs on the six points of their common world set <math>W = O \cup S \cup I =</math> <math>\{ \mathrm{A}, \mathrm{B}, \text{“A”}, \text{“B”}, \text{“i”}, \text{“u”} \}.</math>&nbsp; The arcs are given as follows.

<ul><li><math>\mathrm{Den}(L_\mathrm{A})</math> has an arc from each point of <math>\{ \text{“A”}, \text{“i”} \}</math> to <math>\mathrm{A}</math> and an arc from each point of <math>\{ \text{“B”}, \text{“u”} \}</math> to <math>\mathrm{B}.</math></li></ul>

<p><math>\mathrm{Den}(L_\mathrm{A})</math> has an arc from each point of <math>\{ \text{“A”}, \text{“i”} \}</math> to <math>\mathrm{A}</math> and an arc from each point of <math>\{ \text{“B”}, \text{“u”} \}</math> to <math>\mathrm{B}.</math></p>

<ul><li><math>\mathrm{Den}(L_\mathrm{B})</math> has an arc from each point of <math>\{ \text{“A”}, \text{“u”} \}</math> to <math>\mathrm{A}</math> and an arc from each point of <math>\{ \text{“B”}, \text{“i”} \}</math> to <math>\mathrm{B}.</math></li></ul>

<p><math>\mathrm{Den}(L_\mathrm{B})</math> has an arc from each point of <math>\{ \text{“A”}, \text{“u”} \}</math> to <math>\mathrm{A}</math> and an arc from each point of <math>\{ \text{“B”}, \text{“i”} \}</math> to <math>\mathrm{B}.</math></p>

<math>\mathrm{Den}(L_\mathrm{A})</math> and <math>\mathrm{Den}(L_\mathrm{B})</math> can be interpreted as <i>transition digraphs</i> which chart the succession of steps or the connection of states in a computational process.&nbsp; If the graphs are read this way, the denotational arcs summarize the <i>upshots</i> of the computations involved when the interpreters <math>\mathrm{A}</math> and <math>\mathrm{B}</math> evaluate the signs in <math>S</math> according to their own frames of reference.

<math>\mathrm{Den}(L_\mathrm{A})</math> and <math>\mathrm{Den}(L_\mathrm{B})</math> can be interpreted as <i>transition digraphs</i> which chart the succession of steps or the connection of states in a computational process.&nbsp; If the graphs are read this way, the denotational arcs summarize the <i>upshots</i> of the computations involved when the interpreters <math>\mathrm{A}</math> and <math>\mathrm{B}</math> evaluate the signs in <math>S</math> according to their own frames of reference.

The connotative components <math>\mathrm{Con}(L_\mathrm{A})</math> and <math>\mathrm{Con}(L_\mathrm{B})</math> can be represented as digraphs on the four points of their common syntactic domain <math>S = I =</math> <math>\{ \text{“A”}, \text{“B”}, \text{“i”}, \text{“u”} \}.</math>&nbsp; Since <math>\mathrm{Con}(L_\mathrm{A})</math> and <math>\mathrm{Con}(L_\mathrm{B})</math> are semiotic equivalence relations, their digraphs conform to the pattern manifested by all digraphs of equivalence relations.&nbsp; In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs.&nbsp; In the present case, the arcs are given as follows:

The connotative components <math>\mathrm{Con}(L_\mathrm{A})</math> and <math>\mathrm{Con}(L_\mathrm{B})</math> can be represented as digraphs on the four points of their common syntactic domain <math>S = I =</math> <math>\{ \text{“A”}, \text{“B”}, \text{“i”}, \text{“u”} \}.</math>&nbsp; Since <math>\mathrm{Con}(L_\mathrm{A})</math> and <math>\mathrm{Con}(L_\mathrm{B})</math> are semiotic equivalence relations, their digraphs conform to the pattern manifested by all digraphs of equivalence relations.&nbsp; In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs.&nbsp; In the present case, the arcs are given as follows.

<ul><li><math>\mathrm{Con}(L_\mathrm{A})</math> has the structure of a semiotic equivalence relation on <math>S,</math> with a sling at each point of <math>S,</math> arcs in both directions between the points of <math>\{ \text{“A”}, \text{“i”} \},</math> and arcs in both directions between the points of <math>\{ \text{“B”}, \text{“u”} \}.</math></li></ul>

 

<p><math>\mathrm{Con}(L_\mathrm{A})</math> has the structure of a semiotic equivalence relation on <math>S,</math> with a sling at each point of <math>S,</math> arcs in both directions between the points of <math>\{ \text{“A”}, \text{“i”} \},</math> and arcs in both directions between the points of <math>\{ \text{“B”}, \text{“u”} \}.</math></p>

<ul><li><math>\mathrm{Con}(L_\mathrm{B})</math> has the structure of a semiotic equivalence relation on <math>S,</math> with a sling at each point of <math>S,</math> arcs in both directions between the points of <math>\{ \text{“A”}, \text{“u”} \},</math> and arcs in both directions between the points of <math>\{ \text{“B”}, \text{“i”} \}.</math></li></ul>

<p><math>\mathrm{Con}(L_\mathrm{B})</math> has the structure of a semiotic equivalence relation on <math>S,</math> with a sling at each point of <math>S,</math> arcs in both directions between the points of <math>\{ \text{“A”}, \text{“u”} \},</math> and arcs in both directions between the points of <math>\{ \text{“B”}, \text{“i”} \}.</math></p>

Taken as transition digraphs, <math>\mathrm{Con}(L_\mathrm{A})</math> and <math>\mathrm{Con}(L_\mathrm{B})</math> highlight the associations permitted between equivalent signs, as this equivalence is judged by the interpreters <math>\mathrm{A}</math> and <math>\mathrm{B},</math> respectively.

Taken as transition digraphs, <math>\mathrm{Con}(L_\mathrm{A})</math> and <math>\mathrm{Con}(L_\mathrm{B})</math> highlight the associations permitted between equivalent signs, as this equivalence is judged by the interpreters <math>\mathrm{A}</math> and <math>\mathrm{B},</math> respectively.