Canonical Extensions, Esakia Spaces, and Universal Models

Canonical extensions of locally compact frames

“…To motivate our definition of canonical extension for frames we first recall the construction for distributive lattices (for details see e.g. [4,11,12,14]). Given a distributive lattice A, its canonical extension e : A → A δ is a lattice embedding of A into a complete lattice A δ uniquely described by the following two conditions Density simply means that every element of the canonical extension A δ is both a join of meets and a meet of joins of elements of A.…”

“…These definitions are adapted from the usual definitions of extensions of monotone maps for canonical extensions of distributive lattices, see e.g. [11].…”

“…For example, already with the present work we can lift operations on frames to their canonical extensions. As in the classical theory of canonical extensions, one should expect that such lifted operations will preserve certain types of equations and also that the original operations correspond to certain relations on the spectrum of the frame (in the spirit of [15,11]).…”

Free Modal Algebras Revisited: The Step-by-Step Method

“…Almost all the results can be generalized to the infinite case by defining an appropriate Stone topology on relational structures (see, e.g., [18]). We chose to stick to the finite duality to keep the arguments simple.…”

“…Dito had a number of deep observations on the step-by-step construction for free Heyting and modal algebras, and many of them were supposed to form part of this paper. Leo was interested in this method as it gives an alternative and useful perspective on Esakia spaces of free Heyting algebras (see [18] for more details on this). In fact, Esakia duality for Heyting algebras plays a prominent role in this and nearly all other approaches that apply the ideas of duality to various constructions of Heyting algebras.…”