Extending some results of Malykhin, we prove several independence results about base properties of $\beta\omega\setminus\omega$ and its powers, especially the Noetherian type $Nt(\beta\omega\setminus\omega)$, the least $\kappa$ for which $\beta\omega\setminus\omega$ has a base that is $\kappa$-like with respect to containment. For example, $Nt(\beta\omega\setminus\omega)$ is at least 𝔰, consistently be $\omega_1$, 𝔠, 𝔠⁺, or strictly between $\omega_1$ and 𝔠. $Nt(\beta\omega\setminus\omega)$ is also consistently less than the additivity of the meager ideal. $Nt(\beta\omega\setminus\omega)$ is closely related to the existence of special kinds of splitting families.