918 research outputs found
A fresh perspective on canonical extensions for bounded lattices
This paper presents a novel treatment of the canonical extension of a bounded
lattice, in the spirit of thetheory of natural dualities. At the level of
objects, this can be achieved by exploiting the topological representation due
to M. Ploscica, and the canonical extension can be obtained in the same manner
as can be done in the distributive case by exploiting Priestley duality. To
encompass both objects and morphismsthe Ploscica representation is replaced by
a duality due to Allwein and Hartonas, recast in the style of Ploscica's paper.
This leads to a construction of canonical extension valid for all bounded
lattices,which is shown to be functorial, with the property that the canonical
extension functor decomposes asthe composite of two functors, each of which
acts on morphisms by composition, in the manner of hom-functors
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
Morphisms and Duality for Polarities and Lattices with Operators
Structures based on polarities have been used to provide relational semantics
for propositional logics that are modelled algebraically by non-distributive
lattices with additional operators. This article develops a first order notion
of morphism between polarity-based structures that generalises the theory of
bounded morphisms for Boolean modal logics. It defines a category of such
structures that is contravariantly dual to a given category of lattice-based
algebras whose additional operations preserve either finite joins or finite
meets. Two different versions of the Goldblatt-Thomason theorem are derived in
this setting
From non-commutative diagrams to anti-elementary classes
Anti-elementarity is a strong way of ensuring that a class of structures , in
a given first-order language, is not closed under elementary equivalence with
respect to any infinitary language of the form L . We prove
that many naturally defined classes are anti-elementary, including the
following: the class of all lattices of finitely generated convex
{\ell}-subgroups of members of any class of {\ell}-groups containing all
Archimedean {\ell}-groups; the class of all semilattices of finitely
generated {\ell}-ideals of members of any nontrivial quasivariety of
{\ell}-groups; the class of all Stone duals of spectra of
MV-algebras-this yields a negative solution for the MV-spectrum Problem;
the class of all semilattices of finitely generated two-sided ideals
of rings; the class of all semilattices of finitely generated
submodules of modules; the class of all monoids encoding the
nonstable -theory of von Neumann regular rings, respectively C*-algebras
of real rank zero; (assuming arbitrarily large Erd"os cardinals) the
class of all coordinatizable sectionally complemented modular lattices with a
large 4-frame. The main underlying principle is that under quite general
conditions, for a functor : A B, if there exists a
non-commutative diagram D of A, indexed by a common sort of poset called an
almost join-semilattice, such that D^I is a commutative
diagram for every set I, D is not isomorphic to X for
any commutative diagram X in A, then the range of is anti-elementary.Comment: 49 pages. Journal of Mathematical Logic, World Scientific Publishing,
In pres
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