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, what are some common c.c.c. forcings which are this way? eg. Suslin trees.


Thanks



Is every c.c.c. non-atomic partial order of size $\omega_1$ a union of countable complete suborders?
In response to Joel's comment, here is an argument showing that it is consistent that every ccc poset of size $\omega\_1$ is the union of its nonatomic countable complete subposets.


Suppose $\text{MA}\_{\omega\_1}$ holds. It follows that there are no Suslin algebras, so any ccc poset must add a real. Furthermore, by a theorem of Pawlikowski, any poset of size $<\mathbf{add}(\mathcal{M})$ that adds a real must add a Cohen real. Therefore any ccc poset of size $\omega\_1$ adds a Cohen real. 


Now let $\mathbb{P}$ be such a poset and take $p\in\mathbb{P}$. Let $p\in A\subseteq\mathbb{P}$ be a (countable) maximal antichain. By the fact above, the cone $\mathbb{P}\upharpoonright p$ adds a Cohen real, so the usual forcing theory gives us a complete subposet $\mathbb{Q}\subseteq\mathbb{P}\upharpoonright p$ which is forcing equivalent to $\mathrm{Add}(\omega,1)$. Now, this equivalence and the fact that $\mathbb{Q}$ is ccc yield that there is a countable dense $\mathbb{Q}'\subseteq\mathbb{Q}$. Taking all of this together, $\mathbb{Q}'\cup\{p\}\cup A$ is a countable complete subposet of $\mathbb{P}$ containing $p$.



> 
> *Janusz Pawlikowski*,
>  MR 1825187 [**Cohen reals from small
>  forcings**](http://dx.doi.org/10.2307/2694924), *J. Symbolic Logic*
> **66** (2001), no. 1, 318--324.
> 
> 
> 


