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1

Hausdorff limits of one parameter families of definable sets in o-minimal structures

Kocel-Cynk B.

Czasopismo Techniczne. Nauki Podstawowe

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2016

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Y. 113, iss. 1-NP

73--80

EN

We give an elementary proof of the following theorem on definability of Hausdorff limits of one parameter families of definable sets: let A [subset of] R×Rn be a bounded definable subset in o-minimal structure on (R,+,∙) such that for any y∈(0,c), c>0, the fibre Ay:={x∈Rn:(y,x)∈A} is a Lipschitz cell with constant L independent of y. Then the Hausdorff limit lim[y→0] Āy exists and is definable.

PL

W prezentowanej pracy przedstawiamy elementarny dowód następującego twierdzenia o definiowalności granicy Hausdorffa jednoparametrowej rodziny zbiorów definiowalnych: niech A [pozdbiór] R×Rn będzie ograniczonym zbiorem definiowalnym w strukturze o-minimalnej typu (R,+,∙) takim, że dla dowolnego y∈(0,c), c>0, wókno Ay:={x∈Rn:(y,x)∈A} jest komórką Lipschitza ze staą L niezależną od y. Wtedy granica Hausdorffa lim[y→0] Āy istnieje i jest definiowalna.

2

Algebras of Definable and Rough Sets in Quasi Order-based Approximation Spaces

Kumar A., Banerjee M.

Fundamenta Informaticae

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2015

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Vol. 141, nr 1

37--55

EN

A pair of approximation operators, based on the notion of granules in generalized approximation spaces, was studied in an earlier work by the authors. In this article, we investigate algebraic structures formed by the definable sets and also by the rough sets determined by this pair of approximation operators. The definable sets are open sets of an Alexandrov topological space, and form a completely distributive lattice in which the set of completely join irreducible elements is join dense. The collection of rough sets also forms a similar structure. Representation results for such classes of completely distributive lattices as well as Heyting algebras in terms of definable and rough sets are obtained. Further, two unary operators on rough sets are considered, making the latter constitute a structure that is named a ‘rough lattice’. Representation results for rough lattices are proved.

3

A Note on 3-valued Rough Logic Accepting Decision Rules

Polkowski L.

Fundamenta Informaticae

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2004

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Vol. 61, nr 1

37--45

EN

Rough sets carry, intuitively, a 3-valued logical structure related to the three regions into which any rough set x divides the universe., viz., the lower definable set i(x), the upper definable set c(x), and the boundary region c(x)\i(x) witnessing the vagueness of associated knowledge. In spite of this intuition, the currently known way of relating rough sets and 3-valued logics is only via 3-valued ukasiewicz algebras (Pagliani) that endow spaces of disjoint representations of rough sets with its structure. Here, we point to a 3-valued rough logic RL of unary predicates in which values of logical formulas are given as intensions over possible worlds that are definable sets in a model of rough set theory (RZF). This logic is closely related to the ukasiewicz 3-valued logic, i.e., its theorems are theorems of the ukasiewicz 3-valued logic and theorems of the ukasiewicz 3-valued logic are in one-to-one correspondence with acceptable formulas of rough logic. The formulas of rough logic have denotations and are evaluated in any universe U in which a structure of RZF has been established. RZF is introduced in this note as a variant of set theory in which elementship is defined via containment, i.e., it acquires a mereological character (for this, see the cited exposition of Lesniewski's ideas). As an application of rough logic RL, decision rules and dependencies in information systems are characterized as acceptable formulas of this logic whereas functional dependencies turn out to be theorems of rough logic RL.

4

The Whitney property of a fiber of a definable mapping

Kocel-Cynk B.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 2

141-147

EN

We prove the following theorem Let S be a polynomially bounded o-minimal structure on (R,+,.) and let f : A --> [R^n] be a continuous, definable function on a compact definable set [A is subset of R^m]. Then there exist a positive real number [alpha belongs to R+] and a definable function C : f(A) --> R+ such that for any x [belongs to] f(A) and any two points p and q in the same connected component of [f^-1](x) there exists a piece-wise analytic curve gamma joinning p and q in [f^-1](x) with length [gamma is less than or equal to C(x)][...]. As a consequence we obtain the regular separation with parameter for definable sets.