## Sign Relations, Triadic Relations, Relations • 7

Re: Ontolog ForumJS

JS:
A binary relation is a set of ordered pairs of the elements of some other set.

That is the first definition I learned for binary relations.

Slightly more generally, a binary relation $L$ is a subset of a cartesian product $X \times Y$ of two sets, $X$ and $Y.$  In symbols, $L \subseteq X \times Y.$  Of course $X$ and $Y$ could be the same, but that’s not always the case.

I have long used the adjectives, 2-place, binary, and dyadic pretty much interchangeably in application to relations but I developed a bias toward dyadic on account of computational contexts where binary is reserved for binary numerals.

Once again, partly due to computational exigencies, I would now regard this first definition as the weak typing version.

The strong typing definition of a $k$-place relation $L \subseteq X_1 \times \ldots \times X_k$ includes the cartesian product $X_1 \times \ldots \times X_k$ as an essential part of its specification.  This serves to harmonize the definition of a $k$-place relation with the use of mathematical category theory in computer science.

When I get more time, I’ll go through the material I linked on relation theory in a slightly more leisurely manner …

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