A Boolean algebra $B$ is defined (e.g. in Jech) to be $\kappa$-saturated if there is no partition $W$ of $B$ where $|W|=\kappa$. He seems to assume that this implies $|W|<\kappa$. But why should this be the case?

For example, say that $B$ is $\aleph_1$-saturated. Why does this imply that $B$ is $\aleph_2$-saturated? It's clearly true in the case where $B$ is complete or even $\aleph_3$-complete, but suppose we're not given that. How would one construct a partition of size $\kappa$ given a partition of size $\lambda>\kappa$ in the absence of completeness?

A Boolean algebra $B$ is defined (e.g. in Jech) to be $\kappa$-saturated if there is no partition $W$ of $B$ where $|W|=\kappa$. He seems to assume that this implies $|W|<\kappa$ for any partition $W$. But why should this be the case?

For example, say that $B$ is $\aleph_1$-saturated. Why does this imply that $B$ is $\aleph_2$-saturated? It's clearly true in the case where $B$ is complete or even $\aleph_3$-complete, but suppose we're not given that. How would one construct a partition of size $\kappa$ given a partition of size $\lambda>\kappa$ in the absence of completeness?