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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 ''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 <math>G = (X, G) = (G^{(1)}, G^{(2)})\!</math> is a dyadic relation <math>H = (Y, H) = (H^{(1)}, H^{(2)})\!</math> that has all of its points and pairs in <math>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 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 <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>G_Y.\!</math>.  Using the symbol <math>\cap</math> to indicate the intersection of a pair of sets, the construction of <math>G_Y = \operatorname{ISOS} (G, Y)</math> can be defined as follows:

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