Can I work only with complete atomistic boolean algebras instead of posets?

Question

I wrote a book, most of whose theorems contain something like:

$\mathfrak{A}$ is a poset with the following properties: ... Then the conclusion ... holds.

Now I am "slightly" rewriting my book with even more weak conditions. But even a slight rewrite is much work.

I wonder if I can replace $\mathfrak{A}$ with a complete atomistic boolean algebra (CABA) containing it and formulate my theorem only for complete atomistic boolean algebras only? Will this way work? Will we able to deduce theorems for posets from the special case of theorems for CABAs?

My question may look not specific enough. But it is just very general.

Please show examples (and if possible a general proof) of theorems for posets following from their special case for CABA.

Answer 1

I will answer my own question:

From https://mathoverflow.net/questions/139810/embedding-a-brouwerian-lattice-into-a-boolean-lattice: "The embedding can be made to preserve all existing meets, or all existing joins (but not both at the same time)."

Theorems freely pass from CABAs to arbitrary orders only if all meets and joins are preserved. Otherwise need to take care about every meet or join :-(

So, the answer is "no".