Definitions and examples for relation composition and the two types of relation reduction that commonly arise can be found in the following articles:
Peirce’s idea of reducibility and irreducibility is the more fundamental concept, having to do with the question of whether relations can be derived from others by relational composition, and this type of operation is invoked in every variety of formal construction. Consequently, projective reducibility does nothing to defeat Peirce’s thesis about the primal nature of triadic relations.
But people sometimes confuse the two ideas of reducibility, compositional and projective, so it’s good to clarify the differences between them. Projective reducibility, when you can get it, is more of a “consolation prize” for dyadic reductionists, who tend to ignore the fact that you can’t do anything constructive without triadic relations being involved the mix. Still, it’s a useful property and good to recognize it when it occurs.