Decomposition of composition and a partial proof of a conjecture

Composition of binary relations can be decomposed into two operations: \otimes and \mathrm{dom}:
g \otimes f = \left\{ ( ( x ; z) ; y) \, | \, x f y \wedge y g z \right\}.

Composition of binary relations is decomposed as: g \circ f = \mathrm{dom} (g\otimes f).

I introduce similar decomposition of reloids, and using this try to prove that composition with a principal reloid is distributive over join of reloids.

The proof is partial, there are some white spots in it. The idea is very elegant, but I have failed to make a complete proof. Please email me or comment on this blog if you find a complete proof.

See this note about the proof (PDF).

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