# Normal quasi-uniformity elegantly defined

Error in the proof of below theorem!! http://www.mathematics21.org/binaries/addons.pdf

My earlier attempt to describe normal reloids (where the normality is taken from “Each regular paratopological group is completely regular” article, not from the customary normality of the induced topological space) failed.

After a little more thought, I have however proved (see addons.pdf) a cute algebraic description of normal quasi-uniformities:

Theorem The following are pairwise equivalent:

1. Endoreloid $f$ is normal.
2. $(\mathsf{FCD}) f^{- 1} \circ (\mathsf{FCD}) f \sqsubseteq (\mathsf{FCD}) f \circ (\mathsf{FCD}) f$.
3. $(\mathsf{FCD}) (f^{- 1} \circ f) \sqsubseteq (\mathsf{FCD}) (f \circ f)$.

I continue to investigate the article by Taras Banakh, Alex Ravsky, how can it be generalized with reloids instead of uniform spaces and funcoids instead of topological spaces.