Macneille Completions of Heyting Algebras
نویسندگان
چکیده
In this note we provide a topological description of the MacNeille completion of a Heyting algebra similar to the description of the MacNeille completion of a Boolean algebra in terms of regular open sets of its Stone space. We also show that the only varieties of Heyting algebras that are closed under MacNeille completions are the trivial variety, the variety of all Boolean algebras, and the variety of all Heyting algebras.
منابع مشابه
Comparison of MacNeille, Canonical, and Profinite Completions
Using duality theory, we give necessary and sufficient conditions for the MacNeille, canonical, and profinite completions of distributive lattices, Heyting algebras, and Boolean algebras to be isomorphic.
متن کاملMacNeille Completions of FL-algebras
We show that a large number of equations are preserved by DedekindMacNeille completions when applied to subdirectly irreducible FL-algebras/residuated lattices. These equations are identified in a systematic way, based on proof-theoretic ideas and techniques in substructural logics. It follows that a large class of varieties of Heyting algebras and FL-algebras admits completions.
متن کاملMacneille Completions of Modal Algebras
For a modal algebra (B, f), there are two natural ways to extend f to an operation on the MacNeille completion of B. The resulting structures are called the lower and upper MacNeille completions of (B, f). In this paper we consider lower and upper MacNeille completions for various varieties of modal algebras. In particular, we characterize the varieties of closure algebras and diagonalizable al...
متن کاملMacneille Transferability and Stable Classes of Heyting Algebras
A lattice P is transferable for a class of lattices K if whenever P can be embedded into the ideal lattice IK of some K ∈ K, then P can be embedded into K. There is a rich theory of transferability for lattices. Here we introduce the analogous notion of MacNeille transferability, replacing the ideal lattice IK with the MacNeille completion K. Basic properties of MacNeille transferability are de...
متن کاملTopo-canonical completions of closure algebras and Heyting algebras
We introduce and investigate topo-canonical completions of closure algebras and Heyting algebras. We develop a duality theory that is an alternative to Esakia’s duality, describe duals of topo-canonical completions in terms of the Salbany and Banaschewski compactifications, and characterize topo-canonical varieties of closure algebras and Heyting algebras. Consequently, we show that ideal compl...
متن کامل