Notes on the lattice of fuzzy rough sets

2020

Since the theory of rough sets was introduced by Zdzislaw Pawlak, several approaches have been proposed to combine rough set theory with fuzzy set theory. In this paper, we examine one of these approaches, namely fuzzy rough sets with crisp reference sets, from a lattice-theoretic point of view. We connect the lower and upper approximations of a fuzzy relation R to the approximations of the core and support of R. We also show that the lattice of fuzzy rough sets corresponding to a fuzzy equivalence relation R and the crisp subsets of its universe is isomorphic to the lattice of rough sets for the (crisp) equivalence relation E, where E is the core of R. We establish a connection between the exact (fuzzy) sets of R and the exact (crisp) sets of the support of R.

arXiv (Cornell University), 2023

By the means of lower and upper fuzzy approximations we define quasiorders. Their properties are used to prove our main results. First, we characterize those pairs of fuzzy sets which form fuzzy rough sets w.r.t. a t-similarity relation θ on U, for certain t-norms and implicators. Then we establish conditions under which fuzzy rough sets form lattices. We show that for the min t-norm and any S-implicator defined by the max co-norm with an involutive negator, the fuzzy rough sets form a complete lattice, whenever U is finite or the range of θ and of the fuzzy sets is a fixed finite chain.

This paper builds the topological and lattice structures of L-fuzzy rough sets by introducing lower and upper sets. In particular, it is shown that when the L-relation is reflexive, the upper (resp. lower) set is equivalent to the lower (resp. upper) L-fuzzy approximation set. Then by the upper (resp. lower) set, it is indicated that an L-preorder is the equivalence condition under which the set of all the lower (resp. upper) L-fuzzy approximation sets and the Alexandrov L-topology are identical. However, associating with an L-preorder, the equivalence condition that L-interior (resp. closure) operator accords with the lower (resp. upper) L-fuzzy approximation operator is investigated. At last, it is proven that the set of all the lower (resp. upper) L-fuzzy approximation sets forms a complete lattice when the L-relation is reflexive.

IEEE Transactions on Fuzzy Systems, 2005

Rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty, it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets. The existing generalizations of fuzzy rough sets are all based on special fuzzy relations (fuzzy similarity relations,-similarity relations), it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches. In this paper, from the viewpoint of constructive approach, we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them, the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined. In axiomatic approach, we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms. The lattice and topological structures of fuzzy rough sets are also proposed. In order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets, a special lower approximation operator is applied to a fuzzy reasoning system, which coincides with the Mamdani algorithm.

2011

This paper deals with rough set approach on lattice theory. We represent the lattices for rough sets determined by an equivalence relation. Without any loss of generality, we have defined the rough set as a pair of sets (lower approximation set, upper approximation set) and then we showed that the collection of all rough sets of an approximations by an equivalence relation form a lattice by some order relation. In this paper we are able to deal with information sources in a set-theoretic manner. We also given an integrated approach to form lattices by choice function and lattice structure in rough set theory. The simple notion of this paper is to show the lattice structure in rough set theory by using indiscernible equivalence relation. Some important results are also proved. Finally, some examples are considered to illustrate the paper. c ©2011 World Academic Press, UK. All rights reserved.

Iranian Journal of Fuzzy Systems, 2022

Since Pawlak defined the notion of rough sets in 1982, many authors made wide research studying rough sets in the ordinary case and the fuzzy case. This paper introduced a new style of rough fuzzy sets based on a fuzzy ideal ℓ on a universal finite set X. New lower and new upper fuzzy sets are introduced, and consequently, fuzzy interior and fuzzy closure operators of a rough fuzzy set are discussed. These definitions, if ℓ is restricted to ℓ • = {0}, imply the fuzzification of previous definitions given in the ordinary case, and moreover in the crisp case, we get exactly these previous definitions. The new style gives us a better accuracy value of roughness than the previous styles. Rough fuzzy connectedness is introduced as a sample of applications on the recent style of roughness.

2010

Rough set model with lower and upper approximations based on lattice theory is defined and for the new model some properties are given. Lattice theory plays an important role in rough set theory and fuzzy set theory, so the new rough set model over lattice theory in this paper may be a new tool to study the relationship between fuzzy set theory and rough set theory.

Lecture Notes in Computer Science, 2004

In this paper, we discuss rough inclusions defined in Rough Mereology -a paradigm for approximate reasoning introduced by Polkowski and Skowron -as a basis for common models for rough as well as fuzzy set theories. We justify the point of view that tolerance (or, similarity) is the motif common to both theories. To this end, we demonstrate in Sect. 6 that rough inclusions (which represent a hierarchy of tolerance relations) induce rough set theoretic approximations as well as partitions and equivalence relations in the sense of fuzzy set theory. Before that, we include an account of mereological theory in Sect. 3. We also discuss granulation mechanisms based on rough inclusions with an outline of applications to rough-fuzzy-neurocomputing and computing with words in Sects. 4 and 5.

Information Sciences, 2003

This paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used. In constructive approach, a pair of lower and upper generalized approximation operators is defined. The connections between fuzzy relations and fuzzy rough approximation operators are examined. In axiomatic approach, various classes of fuzzy rough approximation operators are characterized by different sets of axioms. Axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators.

Transactions on Rough Sets, 2004

Using as example an incomplete information system with support a set of objects X, we discuss a possible algebraization of the concrete algebra of the power set of X through quasi BZ lattices. This structure enables us to define two rough approximations based on a similarity and on a preclusive relation, with the second one always better that the former. Then, we turn our attention to Pawlak rough sets and consider some of their possible algebraic structures. Finally, we will see that also Fuzzy Sets are a model of the same algebras. Particular attention is given to HW algebra which is a strong and rich structure able to characterize both rough sets and fuzzy sets.