Changes

=====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===