12,751 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
Geometric lattice structure of covering and its application to attribute reduction through matroids
The reduction of covering decision systems is an important problem in data
mining, and covering-based rough sets serve as an efficient technique to
process the problem. Geometric lattices have been widely used in many fields,
especially greedy algorithm design which plays an important role in the
reduction problems. Therefore, it is meaningful to combine coverings with
geometric lattices to solve the optimization problems. In this paper, we obtain
geometric lattices from coverings through matroids and then apply them to the
issue of attribute reduction. First, a geometric lattice structure of a
covering is constructed through transversal matroids. Then its atoms are
studied and used to describe the lattice. Second, considering that all the
closed sets of a finite matroid form a geometric lattice, we propose a
dependence space through matroids and study the attribute reduction issues of
the space, which realizes the application of geometric lattices to attribute
reduction. Furthermore, a special type of information system is taken as an
example to illustrate the application. In a word, this work points out an
interesting view, namely, geometric lattice to study the attribute reduction
issues of information systems
A conceptual approach to gene expression analysis enhanced by visual analytics
The analysis of gene expression data is a complex task for biologists wishing to understand the role of genes in the formation of diseases such as cancer. Biologists need greater support when trying to discover, and comprehend, new relationships within their data. In this paper, we describe an approach to the analysis of gene expression data where overlapping groupings are generated by Formal Concept Analysis and interactively analyzed in a tool called CUBIST. The CUBIST workflow involves querying a semantic database and converting the result into a formal context, which can be simplified to make it manageable, before it is visualized as a concept lattice and associated charts
Rough matroids based on coverings
The introduction of covering-based rough sets has made a substantial
contribution to the classical rough sets. However, many vital problems in rough
sets, including attribution reduction, are NP-hard and therefore the algorithms
for solving them are usually greedy. Matroid, as a generalization of linear
independence in vector spaces, it has a variety of applications in many fields
such as algorithm design and combinatorial optimization. An excellent
introduction to the topic of rough matroids is due to Zhu and Wang. On the
basis of their work, we study the rough matroids based on coverings in this
paper. First, we investigate some properties of the definable sets with respect
to a covering. Specifically, it is interesting that the set of all definable
sets with respect to a covering, equipped with the binary relation of inclusion
, constructs a lattice. Second, we propose the rough matroids based
on coverings, which are a generalization of the rough matroids based on
relations. Finally, some properties of rough matroids based on coverings are
explored. Moreover, an equivalent formulation of rough matroids based on
coverings is presented. These interesting and important results exhibit many
potential connections between rough sets and matroids.Comment: 15page
Generating derivative structures: Algorithm and applications
We present an algorithm for generating all derivative superstructures--for
arbitrary parent structures and for any number of atom types. This algorithm
enumerates superlattices and atomic configurations in a geometry-independent
way. The key concept is to use the quotient group associated with each
superlattice to determine all unique atomic configurations. The run time of the
algorithm scales linearly with the number of unique structures found. We show
several applications demonstrating how the algorithm can be used in materials
design problems. We predict an altogether new crystal structure in Cd-Pt and
Pd-Pt, and several new ground states in Pd-rich and Pt-rich binary systems
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