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=====1.3.12.3. Digression on Derived Relations=====
=====1.3.12.3. Digression on Derived Relations=====
A better understanding of derived equivalence relations (DER's) 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 R into a dyadic relation Der(R), 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 (DER's) 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 R into a dyadic relation Der(R), 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 Der(R) 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) )).
: {DerR}(x, y) = Conj(o C O) (( {RSO}(x, o) , {ROS}(o, y) )).
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:
{P#Q}(x, y) = Conj(m C M) (( {P}(x, m) , {Q}(m, y) )).
: {P#Q}(x, y) = Conj(m C M) (( {P}(x, m) , {Q}(m, y) )).
Compare this with the usual form of composition, typically notated as "P.Q" and defined as follows:
Compare this with the usual form of composition, typically notated as "P.Q" and defined as follows:
{P.Q}(x, y) = Disj(m C M) ( {P}(x, m) . {Q}(m, y) ).
: {P.Q}(x, y) = Disj(m C M) ( {P}(x, m) . {Q}(m, y) ).
===1.4. Outlook of the Project : All Ways Lead to Inquiry===
===1.4. Outlook of the Project : All Ways Lead to Inquiry===
