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A Discrete Representation for Dicomplemented Lattices

Düntsch I., Kwuida L., Orłowska E.

Fundamenta Informaticae

2017

Vol. 156, nr 3/4

281--295

Dicomplemented lattices were introduced as an abstraction of Wille’s concept algebras which provided negations to a concept lattice. We prove a discrete representation theorem for the class of dicomplemented lattices. The theorem is based on a topology free version of Urquhart’s representation of general lattices.

Translative covering of a square by a sequence of arbitrary-oriented squares: Part II

Januszewski J.

Demonstratio Mathematica

2008

Vol. 41, nr 2

441-454

Given a collection of squares of arbitrary side lengths whose total area is at least 3. Then the unit square can be covered by translates of these squares.

Translative packing two squares in the unit square

Januszewski J.

Demonstratio Mathematica

2003

Vol. 36, nr 3

757--762

Any two squares with total area not greater than 0.4 can be translatively packed in the unit square. The translative packing two squares lying in parallel positions is also considered.

From binary to grey scale convex hulls

Soille P.

Fundamenta Informaticae

2000

Vol. 41, Nr 1,2

131-146

The convex hull of a set is the smallest convex set containing this set. In IR2, this definition is equivalent to the intersection of all half-planes containing the set. We show that this latter definition is nothing but an algebraic closing that can be applied to 2-D grey scale images. The resulting grey scale image is convex, in the sense that all its cross-sections are convex. An efficient translation-invariant implementation, leading to a decreasing family of convex sets that converges to an exact discrete convex hull, is proposed.