Changes

A number of additional definitions are relevant to sign relations whose connotative components constitute equivalence relations, if only in part.

A number of additional definitions are relevant to sign relations whose connotative components constitute equivalence relations, if only in part.

A ''dyadic relation on a single set'' (DROSS) is a non-empty set of points plus a set of ordered pairs on these points.  Until further notice, any reference to a ''dyadic relation'' is intended to be taken in this sense, in other words, as a reference to a DROSS.  In a typical notation, the dyadic relation <math>\mathfrak{G} = (X, G) = (G^{(1)}, G^{(2)})</math> is specified by giving the set of points <math>X = G^{(1)}\!</math> and the set of ordered pairs <math>G = G^{(2)} \subseteq X \times X</math> that go together to define the relation.  In contexts where the set of points is understood, it is customary to call the whole relation <math>\mathfrak{G}</math> by the name of the set <math>G.\!</math>

A ''dyadic relation on a single set'' (DROSS) is a non-empty set of points plus a set of ordered pairs on these points.  Until further notice, any reference to a ''dyadic relation'' is intended to be taken in this sense, in other words, as a reference to a DROSS.  In a typical notation, the dyadic relation <math>\underline{G} = (X, G) = (G^{(1)}, G^{(2)})</math> is specified by giving the set of points <math>X = G^{(1)}\!</math> and the set of ordered pairs <math>G = G^{(2)} \subseteq X \times X</math> that go together to define the relation.  In contexts where the set of points is understood, it is customary to call the whole relation <math>\underline{G}</math> by the name of the set <math>G.\!</math>

A ''subrelation'' of a dyadic relation <math>\mathfrak{G} = (X, G) = (G^{(1)}, G^{(2)})</math> is a dyadic relation <math>\mathfrak{H} = (Y, H) = (H^{(1)}, H^{(2)})</math> that has all of its points and pairs in <math>\mathfrak{G}</math> more precisely, that has all of its points <math>Y \subseteq X</math> and all of its pairs <math>H \subseteq G.</math>

A ''subrelation'' of a dyadic relation <math>\underline{G} = (X, G) = (G^{(1)}, G^{(2)})</math> is a dyadic relation <math>\underline{H} = (Y, H) = (H^{(1)}, H^{(2)})</math> that has all of its points and pairs in <math>\underline{G}</math> more precisely, that has all of its points <math>Y \subseteq X</math> and all of its pairs <math>H \subseteq G.</math>

The ''induced subrelation on a subset'' (ISOS), taken with respect to the dyadic relation <math>G \subseteq X \times X</math> and the subset <math>Y \subseteq X,</math> is the maximal subrelation of <math>G\!</math> whose points belong to <math>Y.\!</math>  In other words, it is the dyadic relation on <math>Y\!</math> whose extension contains all of the pairs of <math>Y \times Y</math> that appear in <math>G.\!</math>  Since the construction of an ISOS is uniquely determined by the data of <math>G\!</math> and <math>Y,\!</math> it can be represented as a function of these arguments, as in the notation <math>\operatorname{ISOS} (G, Y),</math> which can be denoted more briefly as <math>\mathfrak{G}_Y.\!</math>.  Using the symbol <math>\cap</math> to indicate the intersection of a pair of sets, the construction of <math>\mathfrak{G}_Y = \operatorname{ISOS} (G, Y)</math> can be defined as follows:

The ''induced subrelation on a subset'' (ISOS), taken with respect to the dyadic relation <math>G \subseteq X \times X</math> and the subset <math>Y \subseteq X,</math> is the maximal subrelation of <math>G\!</math> whose points belong to <math>Y.\!</math>  In other words, it is the dyadic relation on <math>Y\!</math> whose extension contains all of the pairs of <math>Y \times Y</math> that appear in <math>G.\!</math>  Since the construction of an ISOS is uniquely determined by the data of <math>G\!</math> and <math>Y,\!</math> it can be represented as a function of these arguments, as in the notation <math>\operatorname{ISOS} (G, Y),</math> which can be denoted more briefly as <math>\underline{G}_Y.\!</math>.  Using the symbol <math>\bigcap</math> to indicate the intersection of a pair of sets, the construction of <math>\underline{G}_Y = \operatorname{ISOS} (G, Y)</math> can be defined as follows:

<pre>

GY = <Y, GY> = <GY(1), GY(2)>

<math>\begin{array}{lll}

\underline{G}_Y & = & (Y, \ G_Y)

& = & (G_Y^{(1)}, \ G_Y^{(2)})

& = & (Y, \ \{ (x, y) \in Y\!\times\!Y : (x, y) \in G^{(2)} \})

\end{array}</math>

These definitions for dyadic relations can now be applied in a context where each bit of a sign relation that is being considered satisfies a special set of conditions, namely, if <math>R\!</math> is the relation bit under consideration:

 

 

These definitions for dyadic relations can now be applied in a context where each bit of a sign relation that is being considered satisfies a special set of conditions, namely, if R is the bit under consideration:

# Syntactic domain X = Sign domain S = Interpretant domain I.

# Syntactic domain X = Sign domain S = Interpretant domain I.