GO-spaces and Noetherian spectra

The Noetherian type of a space is the least κ for which the space has a κop-like base, i.e., a base in which no element has κ-many supersets. We prove some results about Noetherian types of (generalized) ordered spaces and products thereof. For example: the density of a product of not-too-many compact linear orders never exceeds its Noetherian type, with equality possible only for singular Noetherian types; we prove a similar result for products of Lindelöf GO-spaces. A countable product of compact linear orders has an -like base if and only if it is metrizable. (It is known that every metrizable space has an ωop-like base.) An infinite cardinal κ is the Noetherian type of a compact LOTS if and only if κ≠ω1 and κ is not weakly inaccessible. There is a Lindelöf LOTS with Noetherian type ω1 and there consistently is a Lindelöf LOTS with weakly inaccessible Noetherian type.