Just by way of clarifying and emphasizing a few points —
I use the word relation to mean a special type of mathematical object, namely, a designated subset included within a cartesian product of sets.
Whatever else this definition of a relation may have going for or against it, it does serve to pick out a class of formal structures that work in good stead as intermediary objects between the world of phenomena and our human capacity for coping with whatever reality may emanate in them. So that is mainly how I aim to use it here.
When I’m being careful, then, I’ll try to use words that maintain a distinction between objects, formal or otherwise, and the symbolic modifications of media we use to reference those objects.
For instance, I took some care with this statement from my last post:
The mathematical examples are typical of many in linguistic, logical, and mathematical contexts where we start out with compact, ready-made axioms, definitions, equations, expressions, formulas, predicates, or terms that denote the relations of interest.
For example, we might be discussing dyadic relative terms like “parent of —” or “square of —” and triadic relative terms like “giver of — to —” or “sum of — and —”.
I used “axioms, definitions, equations, expressions, formulas, predicates, terms” along with “dyadic relative terms” and “triadic relative terms” for various sorts of symbolic entities that serve to denote or describe formal objects of thought and discussion, while I tried to reserve “relations” for the objects themselves.