I am throwing together a wide variety of different operations into the bins labeled additive and multiplicative but it’s easy to observe a natural organization and even some relations approaching isomorphisms among and between the members of each class.

The relation between logical disjunction and the union of sets and the relation between logical conjunction and the intersection of sets ought to be clear enough for present purposes.  But the relation of set‑theoretic union to category‑theoretic co‑product and the relation of set‑theoretic intersection to syntactic concatenation deserve a closer look at this point.

The effect of a co‑product as a disjointed union, in other words, that creates an object tantamount to a disjoint union of sets in the resulting co‑product even if some of those sets intersect non‑trivially and even if some of them are identical in reality, can be achieved in several ways.

The usual conception is that of making a separate copy, for each part of the intended co‑product, of the set assigned to that part.  One imagines the set assigned to a particular part of the co‑product as being distinguished by a particular color, in other words, by the attachment of a distinct index, label, or tag, any sort of marker inherited by and passed on to every element of the set in that part.  A concrete image of the construction can be achieved by imagining each set and each element of each set is placed in an ordered pair with the sign of its color, index, label, or tag.  One describes that as the injection of each set into the corresponding part of the co‑product.

For example, given the sets and overlapping or not, one defines the indexed or marked sets and amounting to the copy of into the first part of the co‑product and the copy of into the second part of the co‑product, in the following manner.

Using the co‑product operator () for the construction, the sum, the co‑product, or the disjointed union of and in that order can be represented as the ordinary union of and

Resources

• Cactus Language • Pragmatics
• Survey of Animated Logical Graphs
• Survey of Theme One Program

cc: Academia.edu • BlueSky • Laws of Form • Ma^thstodon • Research Gate
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science