7,375 research outputs found
Canonical extensions of locally compact frames
Canonical extension of finitary ordered structures such as lattices, posets,
proximity lattices, etc., is a certain completion which entirely describes the
topological dual of the ordered structure and it does so in a purely algebraic
and choice-free way. We adapt the general algebraic technique that constructs
them to the theory of frames.
As a result, we show that every locally compact frame embeds into a
completely distributive lattice by a construction which generalises, among
others, the canonical extensions for distributive lattices and proximity
lattices. This construction also provides a new description of a construction
by Marcel Ern\'e. Moreover, canonical extensions of frames enable us to
frame-theoretically represent monotone maps with respect to the specialisation
order
Duality and canonical extensions for stably compact spaces
We construct a canonical extension for strong proximity lattices in order to
give an algebraic, point-free description of a finitary duality for stably
compact spaces. In this setting not only morphisms, but also objects may have
distinct pi- and sigma-extensions.Comment: 29 pages, 1 figur
Representations of the Canonical group, (the semi-direct product of the Unitary and Weyl-Heisenberg groups), acting as a dynamical group on noncommuting extended phase space
The unitary irreducible representations of the covering group of the Poincare
group P define the framework for much of particle physics on the physical
Minkowski space P/L, where L is the Lorentz group. While extraordinarily
successful, it does not provide a large enough group of symmetries to encompass
observed particles with a SU(3) classification. Born proposed the reciprocity
principle that states physics must be invariant under the reciprocity transform
that is heuristically {t,e,q,p}->{t,e,p,-q} where {t,e,q,p} are the time,
energy, position, and momentum degrees of freedom. This implies that there is
reciprocally conjugate relativity principle such that the rates of change of
momentum must be bounded by b, where b is a universal constant. The appropriate
group of dynamical symmetries that embodies this is the Canonical group C(1,3)
= U(1,3) *s H(1,3) and in this theory the non-commuting space Q= C(1,3)/
SU(1,3) is the physical quantum space endowed with a metric that is the second
Casimir invariant of the Canonical group, T^2 + E^2 - Q^2/c^2-P^2/b^2 +(2h
I/bc)(Y/bc -2) where {T,E,Q,P,I,Y} are the generators of the algebra of
Os(1,3). The idea is to study the representations of the Canonical dynamical
group using Mackey's theory to determine whether the representations can
encompass the spectrum of particle states. The unitary irreducible
representations of the Canonical group contain a direct product term that is a
representation of U(1,3) that Kalman has studied as a dynamical group for
hadrons. The U(1,3) representations contain discrete series that may be
decomposed into infinite ladders where the rungs are representations of U(3)
(finite dimensional) or C(2) (with degenerate U(1)* SU(2) finite dimensional
representations) corresponding to the rest or null frames.Comment: 25 pages; V2.3, PDF (Mathematica 4.1 source removed due to technical
problems); Submitted to J.Phys.
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
Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA
Modal formulae express monadic second-order properties on Kripke frames, but
in many important cases these have first-order equivalents. Computing such
equivalents is important for both logical and computational reasons. On the
other hand, canonicity of modal formulae is important, too, because it implies
frame-completeness of logics axiomatized with canonical formulae.
Computing a first-order equivalent of a modal formula amounts to elimination
of second-order quantifiers. Two algorithms have been developed for
second-order quantifier elimination: SCAN, based on constraint resolution, and
DLS, based on a logical equivalence established by Ackermann.
In this paper we introduce a new algorithm, SQEMA, for computing first-order
equivalents (using a modal version of Ackermann's lemma) and, moreover, for
proving canonicity of modal formulae. Unlike SCAN and DLS, it works directly on
modal formulae, thus avoiding Skolemization and the subsequent problem of
unskolemization. We present the core algorithm and illustrate it with some
examples. We then prove its correctness and the canonicity of all formulae on
which the algorithm succeeds. We show that it succeeds not only on all
Sahlqvist formulae, but also on the larger class of inductive formulae,
introduced in our earlier papers. Thus, we develop a purely algorithmic
approach to proving canonical completeness in modal logic and, in particular,
establish one of the most general completeness results in modal logic so far.Comment: 26 pages, no figures, to appear in the Logical Methods in Computer
Scienc
Coherence in Modal Logic
A variety is said to be coherent if the finitely generated subalgebras of its
finitely presented members are also finitely presented. In a recent paper by
the authors it was shown that coherence forms a key ingredient of the uniform
deductive interpolation property for equational consequence in a variety, and a
general criterion was given for the failure of coherence (and hence uniform
deductive interpolation) in varieties of algebras with a term-definable
semilattice reduct. In this paper, a more general criterion is obtained and
used to prove the failure of coherence and uniform deductive interpolation for
a broad family of modal logics, including K, KT, K4, and S4
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