I will recall the very general notion of Heyting structure (as defined
by Fourman and Scott in the 70s). We will see that Iemhoff-models,
Lubarsky-models as well as Boolean- and Heyting-valued structures are
all instances of Heyting structures. Then I will introduce the notions
of loyalty and faithfulness for Heyting structures and explain how they
help us analysing the underlying logics of the previously mentioned
structures. Finally, I will explain how the work above connects to de
Jongh's theorem (resp. the de Jongh property).