I have an intuitive impression that a finitary relation of arity 3 or greater can always be decomposed into nested binary relations. However, I’m having trouble finding any reference for such a property, which makes me think it probably doesn’t exist. But if that’s the case, it’s not clear to me what it would actually mean for this decomposability property to not hold. So my question is “Does this property exist? If not, why not?”
I’m aware that relation composition is treated by relational calculus, but this mainly has to do with relational databases. Decomposition of relations is a common operation in this context, but I don’t know what general mathematical relevance this may have.
For context, my background is in computer science (though not specifically databases). I’m trying to work out why finitary relations of progressively greater arity (especially beyond 3) are progressively rarer in use, and thought perhaps pervasive decomposability was a possible explanation. I’m aware that the ease of writing operator expressions is surely a contributing factor historically, but I find it a bit implausible that this would be the only reason.