## Relations & Their Relatives : 4

The “divisor of” relation signified by $x|y$ is a dyadic relation on the set of positive integers $\mathbb{M},$ so it can be understood as a subset of the cartesian product $\mathbb{M} \times \mathbb{M}.$  It is an example of a partial order, whereas 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 here 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 here is that the sign relation $L \subseteq O \times S \times I$ relevant to a given level of discussion can 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 are those that generate sign processes through iteration or recursion or other operations of that sort.  When this happens, the most penetrating analysis of the sign process or semiosis in view will come through grasping the core sign relation that generates it.

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