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Quasiorders, Tolerance Relations and Corresponding “Partitions”

100%

Nowak M.

Bulletin of the Section of Logic

2016

tom 45

nr 2

The paper deals with a generalization of the notion of partition for wider classes of binary relations than equivalences: for quasiorders and tolerance relations. The counterpart of partition for the quasiorders is based on a generalization of the notion of equivalence class while it is shown that such a generalization does not work in case of tolerances. Some results from [5] are proved in a much more simple way. The third kind of “partition” corresponding to tolerances, not occurring in [5], is introduced.

Rough Approximations Based on Valued Tolerance Relations

88%

Qin K. , Luo J. , Pei Z.

Fundamenta Informaticae

2015

tom Vol. 142, nr 1-4

183--194

Rough set approach for knowledge discovery in incomplete information systems has been extensively studied. This paper conduct a further study of valued tolerance relation based rough approximations. We make an analysis of the existing rough approximabilities and propose a new approach for lower (upper) approximability, which is a generalization of Pawlak approximation operators for complete information system. The approach has also been generalized to fuzzy cases. Some basic properties of the approximation operators are examined.

Tolerances Induced by Irredundant Coverings

75%

Järvinen J. , Radeleczki S.

Fundamenta Informaticae

2015

tom Vol. 137, nr 3

341--353

In this paper, we consider tolerances induced by irredundant coverings. Each tolerance R on U determines a quasiorder .≤R by setting x .≤R y if and only if R(x) ⊆ R(y). We prove that for a tolerance R induced by a covering H of U, the covering H is irredundant if and only if the quasiordered set (U,.≤R ) is bounded by minimal elements and the tolerance R coincides with the product .≤R ◦ .≤R . We also show that in such a case H = {↑m | m is minimal in (U,.≤R )}, and for each minimal m, we have R(m) = ↑m. Additionally, this irredundant covering H inducing R consists of some blocks of the tolerance R. We give necessary and sufficient conditions under which H and the set of R-blocks coincide. These results are established by applying the notion of Helly numbers of quasiordered sets.

Distributive lattices with a given skeleton

75%

Grygiel J.

Discussiones Mathematicae - General Algebra and Applications

2004

tom 24

nr 1

75-94

We present a construction of finite distributive lattices with a given skeleton. In the case of an H-irreducible skeleton K the construction provides all finite distributive lattices based on K, in particular the minimal one.

Kierunkowe zbiory podobieństwa a problem niekompletności danych

63%

Adamus E.

Metody Informatyki Stosowanej

2008

tom nr 1 (Tom 13)

7-15

W pracy przedstawiona została metoda warunkowego uzupełniania niekompletnych danych dopełnieniami klas podobieństwa.

The problem of the incomplete data is quite common especially in the case of the actual measurement samples. In this connection, it has been vastly commented in the literaturt, especially in the rough set theory. The rough set theory was meant as a tool for imprecise and inconsistent information systems. The aim of this work is to supplement the incomplete data relying on the relations designed to this problem, (similarity and tolerance relation). Basing on the opposite information to the incomplete object we know the area of permitted values for this object. The method proposed in the article works on the assumption that we possess with the opposite information to the supplemented sample in our information system.

The Structure of Multigranular Rough Sets

63%

Järvinen J. , Radeleczki S.

Fundamenta Informaticae

2020

tom Vol. 176, nr 1

17--41

We study multigranulation spaces of two equivalences. The lattice-theoretical properties of so-called “optimistic” and “pessimistic” multigranular approximation systems are given. We also consider the ordered sets of rough sets determined by these approximation pairs.