142 research outputs found
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
Direct products of bounded fuzzy lattices realized by triangular norm operators without zero divisors
In this note we continue the work of Chon, as well as Mezzomo, Bedregal, and
Santiago, by studying direct products of bounded fuzzy lattices arising from
fuzzy partially ordered sets. Chon proved that fuzzy lattices are closed under
taking direct products defined using the minimum triangular norm operator.
Mezzomo, Bedregal, and Santiago extended Chon's result to the case of bounded
fuzzy lattices under the same minimum triangular norm product construction. The
primary contribution of this study is to strengthen their result by showing
that bounded fuzzy lattices are closed under a much more general construction
of direct products; namely direct products that are defined using triangular
norm operators without zero divisors. Immediate consequences of this result are
then investigated within distributive and modular fuzzy lattices
A study of universal algebras in fuzzy set theory
This thesis attempts a synthesis of two important and fast developing branches of mathematics, namely universal algebra and fuzzy set theory. Given an abstract algebra [X,F] where X is a non-empty set and F is a set of finitary operations on X, a fuzzy algebra [I×,F] is constructed by extending operations on X to that on I×, the set of fuzzy subsets of X (I denotes the unit interval), using Zadeh's extension principle. Homomorphisms between fuzzy algebras are defined and discussed. Fuzzy subalgebras of an algebra are defined to be elements of a fuzzy algebra which respect the extended algebra operations under inclusion of fuzzy subsets. The family of fuzzy subalgebras of an algebra is an algebraic closure system in I×. Thus the set of fuzzy subalgebras is a complete lattice. A fuzzy equivalence relation on a set is defined and a partition of such a relation into a class of fuzzy subsets is derived. Using these ideas, fuzzy functions between sets, fuzzy congruence relations, and fuzzy homomorphisms are defined. The kernels of fuzzy homomorphisms are proved to be fuzzy congruence relations, paving the way for the fuzzy isomorphism theorem. Finally, we sketch some ideas on free fuzzy subalgebras and polynomial algebras. In a nutshell, we can say that this thesis treats the central ideas of universal algebras, namely subalgebras, homomorphisms, equivalence and congruence relations, isomorphism theorems and free algebra in the fuzzy set theory settin
Fuzzy Mathematics
This book provides a timely overview of topics in fuzzy mathematics. It lays the foundation for further research and applications in a broad range of areas. It contains break-through analysis on how results from the many variations and extensions of fuzzy set theory can be obtained from known results of traditional fuzzy set theory. The book contains not only theoretical results, but a wide range of applications in areas such as decision analysis, optimal allocation in possibilistics and mixed models, pattern classification, credibility measures, algorithms for modeling uncertain data, and numerical methods for solving fuzzy linear systems. The book offers an excellent reference for advanced undergraduate and graduate students in applied and theoretical fuzzy mathematics. Researchers and referees in fuzzy set theory will find the book to be of extreme value
Algebraic structures from quantum and fuzzy logics
This thesis concerns the wide research area of logic. In particular, the first
part is devoted to analyze different kinds of relational systems (orthogonal
and residuated), by investigating the properties of the algebras associated
to them. The second part is focused on algebras of logic, in particular, the
relationship between prominent quantum and fuzzy structures with certain
semirings is proved. The last chapter concerns an application of group theory
to some well known mathematical puzzles
Tameness in generalized metric structures
We broaden the framework of metric abstract elementary classes (mAECs) in
several essential ways, chiefly by allowing the metric to take values in a
well-behaved quantale. As a proof of concept we show that the result of Boney
and Zambrano on (metric) tameness under a large cardinal assumption holds in
this more general context. We briefly consider a further generalization to
partial metric spaces, and hint at connections to classes of fuzzy structures,
and structures on sheaves
The Reticulation of a Universal Algebra
The reticulation of an algebra is a bounded distributive lattice whose prime spectrum of filters or ideals is homeomorphic to the prime
spectrum of congruences of , endowed with the Stone topologies. We have
obtained a construction for the reticulation of any algebra from a
semi-degenerate congruence-modular variety in the case when the
commutator of , applied to compact congruences of , produces compact
congruences, in particular when has principal commutators;
furthermore, it turns out that weaker conditions than the fact that belongs
to a congruence-modular variety are sufficient for to have a reticulation.
This construction generalizes the reticulation of a commutative unitary ring,
as well as that of a residuated lattice, which in turn generalizes the
reticulation of a BL-algebra and that of an MV-algebra. The purpose of
constructing the reticulation for the algebras from is that of
transferring algebraic and topological properties between the variety of
bounded distributive lattices and , and a reticulation functor is
particularily useful for this transfer. We have defined and studied a
reticulation functor for our construction of the reticulation in this context
of universal algebra.Comment: 29 page
The Legendre-Fenchel transform from a category theoretic perspective
The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this paper we show how it arises in the context of category theory using categories enriched over the extended real numbers R¯¯¯¯:=[−∞,+∞]. A key ingredient is Pavlovic's 'nucleus of a profunctor' construction. The pairing between a vector space and its dual can be viewed as an R¯¯¯¯-profunctor; the construction of the nucleus of this profunctor is the construction of a lot of the theory of the Legendre-Fenchel transform. For a relation between sets viewed as a {true,false}-valued profunctor, the construction of the nucleus is the construction of the Galois connection associated to the relation.
One insight given by this approach is that the relevant structure on the function spaces involved in the Legendre-Fenchel transform is something like a metric but is asymmetric and can take negative values. This 'R¯¯¯¯-structure' is a considerable refinement of the usual partial order on real-valued function space and it allows a natural interpretation of Toland-Singer duality and of the two tropical module structures on the set of convex functions
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