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A better understanding of derived equivalence relations (DERs) can be achieved by placing their constructions within a more general context and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation <math>L\!</math> into a dyadic relation <math>\operatorname{Der}(L),</math> with other types of operations on triadic relations.  The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations.

A better understanding of derived equivalence relations (DERs) can be achieved by placing their constructions within a more general context and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation <math>L\!</math> into a dyadic relation <math>\operatorname{Der}(L),</math> with other types of operations on triadic relations.  The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations.

To that end, let the derivation <math>\operatorname{Der}(L)</math> be expressed in the following way:

To that end, let the derivation Der(R) be expressed in the following way:

{DerR}(x, y) = Conj(o C O) (( {RSO}(x, o) , {ROS}(o, y) )).

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| <math>\upharpoonleft \operatorname{Der}(L) \upharpoonright (x, y) = \underset{o ~\in~ O}{\operatorname{Conj}} ~\underline{((}~ \upharpoonleft L_{SO} \upharpoonright (x, o) ~,~ \upharpoonleft L_{OS} \upharpoonright (o, y) ~\underline{))}~.</math>

From this abstract a form of composition, temporarily notated as "P#Q", where P c XxM and Q c MxY are otherwise arbitrary dyadic relations, and where P#Q c XxY is defined as follows:

From this abstract a form of composition, temporarily notated as "P#Q", where P c XxM and Q c MxY are otherwise arbitrary dyadic relations, and where P#Q c XxY is defined as follows: