More Than Just Mahlo
A cardinal κ is α-Mahlo for some ordinal α if and only if κ is Mahlo and for every ordinal β<α, the set of β-Mahlo cardinals below κ is stationary in κ. We can define "hyper-Mahlo", "α-hyper-Mahlo", "weakly α-Mahlo", "weakly hyper-Mahlo", "weakly α-hyper-Mahlo", etc. by analogy with the definitions for inaccessibles.
A cardinal κ is greatly Mahlo or κ+-Mahlo if and only if it is inaccessible and there is a normal (i.e. nontrivial and closed under diagonal intersections) κ-complete filter on the power set of κ that is closed under the Mahlo operation, which maps the set of ordinals S to {αS: α has uncountable cofinality and S∩α is stationary in α}
The properties of being inaccessible, Mahlo, weakly Mahlo, α-Mahlo, greatly Mahlo, etc. are preserved if we replace the universe by an inner model.
Read more about this topic: Mahlo Cardinal