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, 19:04, 4 February 2009→Derived Equivalence Relations: mathematical markup
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A few of the many different expressions for this concept are recorded in Definition 11.
A few of the many different expressions for this concept are recorded in Definition 11.
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To define the ''equiference'' of signs in terms of their denotations, one says that ''<math>x\!</math> is equiferent to <math>y\!</math> under <math>L,\!</math>'' and writes <math>x ~\overset{L}{=}~ y,\!</math> to mean that <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y).</math> Taken in extension, this notion of a relation between signs induces an ''equiference relation'' on the syntactic domain.
To define the ''equiference'' of signs in terms of their denotations, one says that ''<math>x\!</math> is equiferent to <math>y\!</math> under <math>L,\!</math>'' and writes <math>x ~\overset{L}{=}~ y,\!</math> to mean that <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y).</math> Taken in extension, this notion of a relation between signs induces an ''equiference relation'' on the syntactic domain.
For each sign relation <math>L,\!</math> this yields a binary relation <math>\operatorname{Der}(L) \subseteq S \times I</math> that is defined as follows:
For each sign relation R, this yields a binary relation Der(R) c SxI that is defined as follows:
Der(R) = DerR = {<x, y> C SxI : Den(R, x) = Den(R, y)}.
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| <math>\operatorname{Der}(L) ~=~ Der^L ~=~ \{ (x, y) \in S \times I ~:~ \operatorname{Den}(L, x) = \operatorname{Den}(L, y) \}.</math>
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These definitions and notations are recorded in the following display.
These definitions and notations are recorded in the following display.
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Definition 13
Definition 13
D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)}
D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)}
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The relation Der(R) is defined and the notation "x =R y" is meaningful in every situation where Den(-,-) makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation.
The relation Der(R) is defined and the notation "x =R y" is meaningful in every situation where Den(-,-) makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation.