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====1.3.10.  Recurring Themes====

====1.3.10.  Recurring Themes====

The overall purpose of the next several sections is threefold:

The overall purpose of the next several Sections is threefold:

# To continue to illustrate the salient properties of sign relations in the medium of selected examples.

# To continue to illustrate the salient properties of sign relations in the medium of selected examples.

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 G = <X, G> = <G(1), G(2)> is specified by giving the set of points X = G(1) and the set of ordered pairs G = G(2) ? X?X that go together to define the relation.  In contexts where the set of points is understood, it is customary to call the whole relation G by the name of the set G.

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>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>G\!</math> by the name of the set <math>G.\!</math>

A "subrelation" of a dyadic relation G = <X, G> = <G(1), G(2)> is a dyadic relation H = <Y, H> = <H(1), H(2)> that has all of its points and pairs in G, more precisely, that has all of its points Y ? X and all of its pairs H ? G.

A ''subrelation'' of a dyadic relation G = <X, G> = <G(1), G(2)> is a dyadic relation H = <Y, H> = <H(1), H(2)> that has all of its points and pairs in G, more precisely, that has all of its points Y ? X and all of its pairs H ? G.

The "induced subrelation on a subset" (ISOS), taken with respect to the dyadic relation G c X?X and the subset Y ? X, is the maximal subrelation of G whose points belong to Y.  In other words, it is the dyadic relation on Y whose extension contains all of the pairs of Y?Y that appear in G.  Since the construction of an ISOS is uniquely determined by the data of G and Y, it can be represented as a function of these arguments, as in the notation "ISOS (G, Y)", which can be denoted more briefly as "GY".  Using the symbol "n" to indicate the intersection of a pair of sets, the construction of GY = ISOS (G, Y) can be defined as follows:

The "induced subrelation on a subset" (ISOS), taken with respect to the dyadic relation G c X?X and the subset Y ? X, is the maximal subrelation of G whose points belong to Y.  In other words, it is the dyadic relation on Y whose extension contains all of the pairs of Y?Y that appear in G.  Since the construction of an ISOS is uniquely determined by the data of G and Y, it can be represented as a function of these arguments, as in the notation "ISOS (G, Y)", which can be denoted more briefly as "GY".  Using the symbol "n" to indicate the intersection of a pair of sets, the construction of GY = ISOS (G, Y) can be defined as follows: