More specifically, if we only know that a complete Boolean algebra, $\mathbf{B}$, is $\kappa$-c.c., can we give a (reasonably tight) upper bound to the size of $\mathbf{B}$ in terms of $\kappa$?


Thanks in advance.



What can we infer about the size of a complete Boolen algebra, given it is $\kappa$-c.c.?
The $\kappa$-cc condition by itself does not put any bound on the cardinality of the algebra. For example, for each $\lambda$, the Cohen algebra of regular open subsets of $2^\lambda$ is ccc, but it has cardinality $\lambda^\omega$. However, one can bound the size of $B$ using additional cardinal characteristics: for a simple bound, if $B$ is $\kappa$-cc and has a dense subset $P$ of cardinality $\lambda$, then $|B|\le\lambda^{<\kappa}$, as every element of $B$ can be written as the join of an antichain in $P$.


