Let $I$ be an ideal and let $I^+$ denote its complement (the so-called $I$-positive sets). Now we say that $I$ is $\lambda$-saturated iff each antichain in $I^+$ has size less than $\lambda$. Further $sat(I)$ is the least cardinal $\kappa$ such that $I$ is $\kappa$-saturated.

It can be shown that if $sat(I)$ is infinite then it has to be uncountable. I think that a similar argument gives us that $sat(I)$ is a regular cardinal, provided that $I$ is a $\kappa$-complete ideal on $\kappa$.

My question now is: Do we need the $\kappa$-completeness of $I$ or is $sat(I)$ always a regular cardinal, no matter if $I$ is $\kappa$-complete or not.