Relations & Their Relatives : 8

At this point we can distinguish two forms of decomposability or reducibility — along with their corresponding negations, indecomposability or irreducibility – that commonly arise.

• Reducibility under relational composition $P \circ Q.$

• Reducibility under projections.  For this we need a few definitions:

Every triadic relation, say $L$ contained as a subset of the cartesian product $X \times Y \times Z,$ determines three dyadic relations, namely, the projections of $L$ on the three “planes” $X \times Y,$ $X \times Z,$ and $Y \times Z.$

In particular:

Every sign relation, say $M$ contained as a subset of the cartesian product $O \times S \times I,$ those being the sets of objects, signs, and interpretant signs respectively under discussion, determines three dyadic relations, which we may notate as follows:

• $\mathrm{proj}_{OS}{M},$ the projection of $M$ on the $O \times S$ plane;
• $\mathrm{proj}_{OI}{M},$ the projection of $M$ on the $O \times I$ plane;
• $\mathrm{proj}_{SI}{M},$ the projection of $M$ on the $S \times I$ plane.

The following Figure illustrates the situation:

Here is the critical point.  The triadic relation always determines the three dyadic projections but the three dyadic projections may or may not determine the triadic relation.  Thus we have two cases:

• If the dyadic projections determine the triadic relation, that is, there is only one triadic relation that has those three projections, then the triadic relation is said to be projectively reducible to those three dyadic relations.
• If the dyadic projections do not determine the triadic relation, that is, there is more than one triadic relation that has those same three projections, then the triadic relation is said to be projectively irreducible.

See the following article for concrete examples of both possibilities, a pair of generic triadic relations that are projectively irreducible and a pair of triadic sign relations that are projectively reducible to their dyadic projections.

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