10 research outputs found
Profinite completions and canonical extensions of Heyting algebras
We show that the profinite completions and canonical extensions of bounded distributive lattices and of Boolean algebras coincide. We characterize dual spaces of canonical extensions of bounded distributive lattices and of Heyting algebras in terms of Nachbin order-compactifications. We give the dual description of the profinite completion H of a Heyting algebra H, and characterize the dual space of H. We also give a necessary and sufficient condition for the profinite completion of H to coincide with its canonical extension, and provide a new criterion for a variety V of Heyting algebras to be finitely generated by showing that V is finitely generated if and only if the profinite completion of every member of V coincides with its canonical extension. From this we obtain a new proof of a well-known theorem that every finitely generated variety of Heyting algebras is canonical
MacNeille completion and profinite completion can coincide on finitely generated modal algebras
Following Bezhanishvili & Vosmaer, we confirm a conjecture of Yde Venema by
piecing together results from various authors. Specifically, we show that if
is a residually finite, finitely generated modal algebra such that
has equationally definable principal
congruences, then the profinite completion of is isomorphic to its
MacNeille completion, and is smooth. Specific examples of such modal
algebras are the free -algebra and the free
-algebra.Comment: 5 page
Canonical extensions and ultraproducts of polarities
J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra
with operators has evolved into an extensive theory of canonical extensions of
lattice-based algebras. After reviewing this evolution we make two
contributions. First it is shown that the failure of a variety of algebras to
be closed under canonical extensions is witnessed by a particular one of its
free algebras. The size of the set of generators of this algebra can be made a
function of a collection of varieties and is a kind of Hanf number for
canonical closure. Secondly we study the complete lattice of stable subsets of
a polarity structure, and show that if a class of polarities is closed under
ultraproducts, then its stable set lattices generate a variety that is closed
under canonical extensions. This generalises an earlier result of the author
about generation of canonically closed varieties of Boolean algebras with
operators, which was in turn an abstraction of the result that a first-order
definable class of Kripke frames determines a modal logic that is valid in its
so-called canonical frames
Profinite Heyting algebras
Abstract For a Heyting algebra A, we show that the following conditions are equivalent: (i) A is profinite; (ii) A is finitely approximable, complete, and completely joinprime generated; (iii) A is isomorphic to the Heyting algebra Up(X) of upsets of an image-finite poset X. We also show that A is isomorphic to its profinite completion iff A is finitely approximable, complete, and the kernel of every finite homomorphic image of A is a principal filter of A
Codimension and pseudometric in co-Heyting algebras
In this paper we introduce a notion of dimension and codimension for every
element of a distributive bounded lattice . These notions prove to have a
good behavior when is a co-Heyting algebra. In this case the codimension
gives rise to a pseudometric on which satisfies the ultrametric triangle
inequality. We prove that the Hausdorff completion of with respect to this
pseudometric is precisely the projective limit of all its finite dimensional
quotients. This completion has some familiar metric properties, such as the
convergence of every monotonic sequence in a compact subset. It coincides with
the profinite completion of if and only if it is compact or equivalently if
every finite dimensional quotient of is finite. In this case we say that
is precompact. If is precompact and Hausdorff, it inherits many of the
remarkable properties of its completion, specially those regarding the
join/meet irreducible elements. Since every finitely presented co-Heyting
algebra is precompact Hausdorff, all the results we prove on the algebraic
structure of the latter apply in particular to the former. As an application,
we obtain the existence for every positive integers of a term
such that in every co-Heyting algebra generated by an -tuple ,
is precisely the maximal element of codimension .Comment: 34 page