Jon Awbrey
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to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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
S = ×_{i=1}^k X_i, where the X_i are dimensions (something that can
vary), typically cast as sets, so that × here is Cartesian product.
</QUOTE>
Relational systems are just the context we need. It is usual to
begin at a moderate level of generality by considering a space X
of the following form.
X = ×_{i=1}^k X_i = X_1 × X_2 × ... × X_{k-1} × X_k.
(I’ll use X instead of S here because I want to save the letter “S” for
sign domains when we come to the special case of sign relational systems.)
We can now define a “relation” L as a subset of a cartesian product.
L ⊆ X_1 × X_2 × ... × X_{k-1} × X_k.
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 L is just a set which happens to be
a subset of the cartesian product X_1 × X_2 × ... × X_{k-1} × X_k 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
X_1 × X_2 × ... × X_{k-1} × X_k in the type-casting
L ⊆ X_1 × X_2 × ... × X_{k-1} × X_k is an essential part of the
definition of L. Employing a conventional mathematical idiom, a
k-adic relation over the nonempty sets X_1, X_2, ..., X_{k-1}, X_k
is defined as a (k+1)-tuple (L, X_1, X_2, ..., X_{k-1}, X_k) where
L is a subset of X_1 × X_2 × ... × X_{k-1} × X_k.
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 & by which virtue it enjoys more
immediate 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 ...
Regards,
Jon
Resources
=========
• Relation Theory ( https://oeis.org/wiki/Relation_theory )
• Triadic Relations ( https://oeis.org/wiki/Triadic_relation )
• Sign Relations ( https://oeis.org/wiki/Sign_relation )
> overall “variety” or size of the system, so ∏_{i=1}^k n_i,