$\kappa$-Saturated Boolean Algebras A Boolean algebra $B$ is said to be $\kappa$-saturated if 


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*Given a Boolean algebra $A$, such that $|A|\leq \kappa$, there is a monomorphism $f:A\to B$;

*Given a Boolean algebra $A$, $|A|<\kappa$, and two monomorphisms $f,g:A\to B$, there is a isomorphism $h:B\to B$ such that $h\circ f = g$.


In The Theory of Ultrafilters by Comfort and Negrepontis it's showed that if $\kappa=\kappa^{<\kappa} = |[\kappa]^{<\kappa}|$ is infinite cardinal, then there is a unique $\kappa$-saturated Boolean up to isomorphism; to show that the authors enters in a more general theory about Jónsson classes. I'd like to know if there is a more short proof about the existence and uniqueness of $\kappa$-saturated Boolean algebras (under the assumption $\kappa=\kappa^{<\kappa}$)?
 A: I guess you are interested in $\kappa$-saturated models of size $\kappa$. Since $\kappa^{<\kappa} = \kappa$, it follows that $\kappa$ is regular. For any complete theory $T$ in a countable language, there is a standard procedure to build $\kappa$-saturated models, when $\kappa$ is an uncountable regular cardinal such that $\kappa^{<\kappa} = \kappa$. This is done by building a chain of models realising more and more types. Here by $\kappa$-saturated I am referring to the model-theoretic version of this, but one can show that a Boolean algebra which is $\kappa$-saturated in the model-theoretic sense is $\kappa$-saturated in your sense by building the required objects by increasing approximations, each of size $< \kappa$ using the model-theoretic $\kappa$-saturation. 
All of these things should be in any standard model theory book, I know that the bigger Hodges definitely does in Chapter 10. As for the complete theory to start with, pick the theory of the (unique up to isomorphism) countable atomless Boolean algebra. For the case of $\kappa =\aleph_0$, one can show that the countable atomless Boolean algebra is $\aleph_0$-saturated in your sense. 
