I’m still collecting my wits from the mind-numbing events of the past two weeks so I’ll copy your last remarks here and work through them step by step.
- I think what you have is sound, and can be described in a number of ways. In years past in seeking ways to both qualify and quantify variety in systems I characterized this distinction as between “dimensional variety” and “cardinal variety”. Thankfully, this seems straightforward from a mathematical perspective, namely in a standard relational system where the are dimensions (something that can vary), typically cast as sets, so that here is Cartesian product.
Relational systems are just the context we need. It is usual to begin at a moderate level of generality by considering a space of the following form.
(I’ll use instead of here because I want to save the letter for sign domains when we come to the special case of sign relational systems.)
We can now define a relation as a subset of a cartesian product.
There are two common ways of understanding the subset symbol in this context. Using language from computer science I’ll call them the weak typing and strong typing interpretations.
- Under weak typing conventions is just a set which happens to be a subset of the cartesian product but which could just as easily be cast as a subset of any other qualified superset. The mention of a particular cartesian product is accessory but not necessary to the definition of the relation itself.
- Under strong typing conventions the cartesian product in the type-casting is an essential part of the definition of Employing a conventional mathematical idiom, a -adic relation over the nonempty sets is defined as a -tuple where is a subset of
We have at this point opened two fronts of interest in cybernetics, namely, the generation of variety and the recognition of constraint. There’s more detail on this brand of relation theory in the resource article linked below. I’ll be taking the strong typing approach to relations from this point on, largely because it comports more naturally with category theory and thus enjoys ready applications to systems and their transformations.
But my eye-brain system is going fuzzy on me now, so I’ll break here and continue later …