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.
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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.
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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.
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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.
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