Coarsenings In Deutsch Et Al's Constructor Theory

Disclaimer: I posted a questions on constructor theory here a few days ago but received two closing votes, I guess because it consisted of several subquestions, so I deleted it and now try to focus on a single issue.

I'm reading Constructor Theory Of Probability and became a bit confused about the notion of coarsening: This seems very natural to me, $$X'$$ is a coarsening of $$X$$ iff there exists a partition $$C$$ on $$X$$ such that $$X'=\{\cup c:c\in C\}$$. The second part I understand in the sense that arbitrary maps $$f:S\to X$$ on the state space are associated with the variables $$X_f:=\{f^{-1}(x):x\in X\}$$, and if $$f_1$$, $$f_2$$ map into any ring $$R$$, we have $$X_{f_1+f_2}$$ and $$X_{f_1f_2}$$ being coarsenings of $$X_{(f_1,f_2)}$$ (where $$(f_1,f_2)$$ is interpreted as map into $$R^2$$).

But in the following part, they seem to use a different notion: If $$\forall x\in X:x\cap y=\emptyset$$ holds, as indicated here, which is equivalent to $$x\cap(\cup X)=\emptyset$$, how can some coarsening of $$X$$ be sharp in $$y$$, as $$\cup X=\cup X'$$ for any coarsening $$X'$$? (The definition of a $$X$$ being sharp in $$a$$ is $$\exists x\in X:a\subseteq x$$.)

As far as their example with the photon number goes, they earlier claim 'The $$z$$-component of the spin is a variable, represented as the set of two intrinsic attributes: that of the $$z$$-component of the spin being $$1/2$$ and $$−1/2$$', so here I understand $$X$$ to equal $$\{\{|n\rangle\}:n\in\omega\}$$ (the eigenspaces of the projector). Then it is true that $$(1/\sqrt{2})(|0\rangle+|1\rangle)$$ lies in the eigenspace (what do they mean by $$+1$$-eigenspace?) of $$|0\rangle\langle 0|+|1\rangle\langle1|$$ but if we interpret projectors as variables this way, we obtain a different meaning of addition, especially $$X_{P_1+P_2}$$ does not equal $$X_{P_1}+X_{P_2}$$ in the way I understand it!

Would it be appropriate to define a coarsening $$X'$$ of $$X$$ by $$\forall x\in X:\exists x'\in X':x\subseteq x'$$, or equivalently, whenever $$X$$ is sharp in some attribute $$a$$, so must be $$X'$$? Because that's mostly compatible with those two paragraphs.