## Relations & Their Relatives • Review 1

Peirce’s notation for elementary relatives was illustrated earlier by a dyadic relation from number theory, namely ${i|j}$ for ${i}$ being a divisor of ${j}.$

Table 1 shows the first few ordered pairs of the relation on positive integers corresponding to the relative term, “divisor of”.  Thus, the ordered pair ${i\!:\!j}$ appears in the relation if and only if ${i}$ divides ${j},$ for which the usual notation is ${i|j}.$ Table 2 shows the same information in the form of a logical matrix.  This has a coefficient of ${1}$ in row ${i}$ and column ${j}$ when ${i|j},$ otherwise it has a coefficient of ${0}.$  (The zero entries have been omitted for ease of reading.) Just as matrices in linear algebra represent linear transformations, these logical arrays and matrices represent logical transformations.

The capacity of relations to generate transformations gives us a clue to the dynamics of sign relations.

The divisor of relation signified by $x|y$ is a dyadic relation on the set of positive integers $\mathbb{M}$ and thus may be understood as a subset of the cartesian product $\mathbb{M} \times \mathbb{M}.$  It is an example of a partial order, while the less than or equal to relation signified by $x \le y$ is an example of a total order relation.

The mathematics of relations can be applied most felicitously to semiotics but there we must bump the adicity or arity up to three.  We take any sign relation $L$ to be subset of a cartesian product $O \times S \times I,$ where $O$ is the set of objects under consideration in a given discussion, $S$ is the set of signs, and $I$ is the set of interpretant signs involved in the same discussion.

One thing we need to understand is the sign relation $L \subseteq O \times S \times I$ relevant to a given level of discussion may be rather more abstract than what we would call a sign process proper, that is, a structure extended through a dimension of time.  Indeed, many of the most powerful sign relations generate sign processes through iteration or recursion or similar operations.  In that event, the most penetrating analysis of the sign process or semiosis in view is achieved through grasping the generative sign relation at its core.

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