Related papers: Rough sets determined by tolerances
We consider tolerances $T$ compatible with an equivalence $E$ on $U$, meaning that the relational product $E \circ T$ is included in $T$. We present the essential properties of $E$-compatible tolerances and study rough approximations…
In this paper, the ordered set of rough sets determined by a quasiorder relation $R$ is investigated. We prove that this ordered set is a complete, completely distributive lattice. We show that on this lattice can be defined three different…
We show that any regular pseudocomplemented Kleene algebra defined on an algebraic lattice is isomorphic to a rough set Kleene algebra determined by a tolerance induced by an irredundant covering.
For several types of information relations, the induced rough sets system RS does not form a lattice but only a partially ordered set. However, by studying its Dedekind-MacNeille completion DM(RS), one may reveal new important properties of…
In this paper, we consider tolerances induced by irredundant coverings. Each tolerance $R$ on $U$ determines a quasiorder $\lesssim_R$ by setting $x \lesssim_R y$ if and only if $R(x) \subseteq R(y)$. We prove that for a tolerance $R$…
The main results of the paper points out the connection between the weak ordered relations and factor lattices defined by tolerances. It is proved that for any tolerance $T$ of a lattice $L$ the Dedekind-Mac Neille completion of $L/T$ is…
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 $R$, its rough set-based Nelson…
We study uO convergence on infinitely distributive lattices, extending key properties known from Riesz spaces. We show that order continuity of uO convergence characterizes infinite distributivity. We examine O-adherence and uO adherence of…
Hamkins and Kikuchi (2016 and 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. If $\mathcal{M}$ is a model of set theory, then $\langle M,…
The congruence lattices of all algebras defined on a fixed finite set $A$ ordered by inclusion form a finite atomistic lattice $\mathcal E$. We describe the atoms and coatoms. Each meet-irreducible element of $\mathcal E$ being determined…
Covering is a common type of data structure and covering-based rough set theory is an efficient tool to process this data. Lattice is an important algebraic structure and used extensively in investigating some types of generalized rough…
We consider Kleene and Stone algebras defined on the completion DM(RS) of the ordered set of rough sets induced by a reflexive relation. We focus on cases where the completion forms a spatial and completely distributive lattice. We derive…
Given a group $G$ and a subgroup $H$, we let $\mathcal{O}_G(H)$ denote the lattice of subgroups of $G$ containing $H$. This paper provides a classification of the subgroups $H$ of $G$ such that $\mathcal{O}_{G}(H)$ is Boolean of rank at…
At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid,…
Rough Set Theory (RST), first introduced by Pawlak in 1982, is an approach for dealing with information systems where knowledge is uncertain or incomplete.\cite{Pawlak} It is of fundamental importance in many subfields of artificial…
We study the pseudo-Kleene algebras of the Dedekind-MacNeille completion of the ordered set of rough set determined by a reflexive relation. We characterize the cases when PBZ and PBZ*-lattices can be defined on these pseudo-Kleene…
Brauer and Thrall conjectured that a finite-dimensional algebra over a field of bounded representation type is actually of finite representation type and a finite-dimensional algebra (over an infinite field) of infinite representation type…
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…
We consider controlled differential equations and give new estimates for higher order Euler schemes. Our proofs are inspired by recent work of A. M. Davie who considers first and second order schemes. In order to implement the general case…
If every block of a (compatible) tolerance (relation) $T$ on a modular lattice $L$ of finite length consists of at most two elements, then we call $T$ a \emph{doubling tolerance} on $L$. We prove that, in this case, $L$ and $T$ determines a…