1,175 research outputs found
On the isomorphism problem of concept algebras
Weakly dicomplemented lattices are bounded lattices equipped with two unary
operations to encode a negation on {\it concepts}. They have been introduced to
capture the equational theory of concept algebras \cite{Wi00}. They generalize
Boolean algebras. Concept algebras are concept lattices, thus complete
lattices, with a weak negation and a weak opposition. A special case of the
representation problem for weakly dicomplemented lattices, posed in
\cite{Kw04}, is whether complete {\wdl}s are isomorphic to concept algebras. In
this contribution we give a negative answer to this question (Theorem
\ref{T:main}). We also provide a new proof of a well known result due to M.H.
Stone \cite{St36}, saying that {\em each Boolean algebra is a field of sets}
(Corollary \ref{C:Stone}). Before these, we prove that the boundedness
condition on the initial definition of {\wdl}s (Definition \ref{D:wdl}) is
superfluous (Theorem \ref{T:wcl}, see also \cite{Kw09}).Comment: 15 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
Complete Boolean algebras are Bousfield lattices
Given a complete Heyting algebra we construct an algebraic tensor
triangulated category whose Bousfield lattice is the Booleanization of the
given Heyting algebra. As a consequence we deduce that any complete Boolean
algebra is the Bousfield lattice of some tensor triangulated category. Using
the same ideas we then give two further examples illustrating some interesting
behaviour of the Bousfield lattice.Comment: 10 pages, update to clarify the products occurring in the main
constructio
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