23,632 research outputs found
The groupoid-based logic for lattice effect algebras
The aim of the paper is to establish a certain logic corresponding to lattice
effect algebras. First, we answer a natural question whether a lattice effect
algebra can be represented by means of a groupoid-like structure. We establish
a one-to-one correspondence between lattice effect algebras and certain
groupoids with an antitone involution. Using these groupoids, we are able to
introduce a suitable logic for lattice effect algebras.Comment: 7 page
Information completeness in Nelson algebras of rough sets induced by quasiorders
In this paper, we give an algebraic completeness theorem for constructive
logic with strong negation in terms of finite rough set-based Nelson algebras
determined by quasiorders. We show how for a quasiorder , its rough
set-based Nelson algebra can be obtained by applying the well-known
construction by Sendlewski. We prove that if the set of all -closed
elements, which may be viewed as the set of completely defined objects, is
cofinal, then the rough set-based Nelson algebra determined by a quasiorder
forms an effective lattice, that is, an algebraic model of the logic ,
which is characterised by a modal operator grasping the notion of "to be
classically valid". We present a necessary and sufficient condition under which
a Nelson algebra is isomorphic to a rough set-based effective lattice
determined by a quasiorder.Comment: 15 page
Finite closed coverings of compact quantum spaces
We show that a projective space P^\infty(Z/2) endowed with the Alexandrov
topology is a classifying space for finite closed coverings of compact quantum
spaces in the sense that any such a covering is functorially equivalent to a
sheaf over this projective space. In technical terms, we prove that the
category of finitely supported flabby sheaves of algebras is equivalent to the
category of algebras with a finite set of ideals that intersect to zero and
generate a distributive lattice. In particular, the Gelfand transform allows us
to view finite closed coverings of compact Hausdorff spaces as flabby sheaves
of commutative C*-algebras over P^\infty(Z/2).Comment: 26 pages, the Teoplitz quantum projective space removed to another
paper. This is the third version which differs from the second one by fine
tuning and removal of typo
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
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