# Projections between complete boolean algebras

Let $$P$$ and $$Q$$ be complete boolean algebras. Suppose that $$\dot H$$ is a $$P$$-name such that $$1_P\Vdash\dot H$$ is $$Q$$-generic. For each $$p\in P$$, let $$A_p$$ be the set of $$q\in Q$$ such that $$p\Vdash q\in\dot H$$. Then the map $$\sigma:P\rightarrow Q$$, given by $$\sigma(p)=\prod A_p$$, is a projection, in the sense that $$\sigma$$ is order-preserving and for all $$p,q$$ with $$q\leq\sigma(p)$$, there is $$p'\leq p$$ such that $$\sigma(p')\leq q$$. I wonder if this projection has any connection with any other arbitrary projection (between $$P$$ and $$Q$$). More specifically, suppose $$\pi:P\rightarrow Q$$ is a projection and let $$p_1,p_2\in P$$ (possibly $$p_1\nleq p_2$$). If $$\sigma(p_1)\leq\sigma(p_2)$$, is it true that $$\pi(p_1)\leq\pi(p_2)$$?

• Perhaps you should refine the question by asking what if $\sigma$ is defined using the $P$-name $\pi(\dot{G})$. Apr 24 at 18:25

The answer is no. Let $$P$$ arise from product forcing $$Q\times Q$$. So forcing with $$P$$ adds two mutually generic filters for $$Q$$, one on each factor. Let $$\sigma$$ be the projection onto the first coordinate, and $$\pi$$ be the projection onto the second coordinate. Let $$p_1=(q,1)$$ and $$p_2=(1,q)$$ for some nontrivial $$q\in Q$$. Note that $$\sigma(p_1)=q\leq 1=\sigma(p_2)$$, but $$\pi(p_1)=1\not\leq q=\pi(p_2)$$.