Saturated Boolean algebras in terms of model theory and in terms of partitions

Question

Let $\kappa$ be an infinite cardinal. A Boolean algebra $\mathbb{B}$ is said to be $\kappa$-saturated if there is no partition (i.e., collection of elements of $\mathbb{B}$ whose pairwise meet is $0$ and least upper bound is $1$) of $\mathbb{B}$, say $W$, of size $\kappa$. Is there any relationship between this and the model theoretic meaning of $\kappa$-saturated (namely that all types over sets of parameters of size $<\kappa$ are realized)?

Answer 1

As far as I know, there is no connection; it's just an unfortunate clash of terminology. It's especially unfortunate because the model-theoretic notion of saturation comes up in the theory of Boolean algebras. For example, the Boolean algebra of subsets of the natural numbers modulo finite sets is $\aleph_1$-saturated but (by a definitely non-trivial result of Hausdorff) not $\aleph_2$-saturated, in the model-theoretic sense, even if the cardinality of the continuum is large.

When (complete) Boolean algebras are used in connection with forcing, it is customary to say "$\kappa$-chain condition" instead of "$\kappa$-saturated" in the antichain sense.