is preserved by ρ, τ, and σ,

decidability

is preserved by ρ, τ, and σ,

finite model property

is preserved by ρ and σ,

tabularity

is preserved by ρ and τ,

Kripke completeness

definability on Kripke frames is preserved by ρ and τ.

first-order

The value of modal companions and the Blok–Esakia theorem as a tool for investigation of intermediate logics comes from the fact that many interesting properties of logics are preserved by some or all of the mappings ρ, σ, and τ. For example,

Alexander Chagrov and Michael Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, 1997.

Vladimir V. Rybakov, Admissibility of Logical Inference Rules, vol. 136 of Studies in Logic and the Foundations of Mathematics, Elsevier, 1997.