Is every c.c.c. non-atomic partial order of size $\omega_1$ a union of countable complete suborders? We say that $\mathbb{P}$ is a complete suborder of $\mathbb{Q}$, if it is a suborder, and maximal antichains in $\mathbb{P}$ remain maximal antichains in $\mathbb{Q}$ As the title says, is every c.c.c. non-atomic partial order of size $\omega_1$ a union of countable complete suborders? If the answer is no, are there some common c.c.c. forcings which are this way? eg. Suslin trees. Thanks

Question was answered at mathoverflow.net/questions/248291