The number of rectangular islands by means of distributive lattices
European Journal of Combinatorics
The minimum cardinality of maximal systems of rectangular islands
European Journal of Combinatorics
Graph Theory
Cut approach to islands in rectangular fuzzy relations
Fuzzy Sets and Systems
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Islands are combinatorial objects that can be intuitively defined on a board consisting of a finite number of cells. It is a fundamental property that two islands are either containing or disjoint. Czedli determined the maximum number of rectangular islands. Pluhar solved the same problem for bricks, and Horvath, Nemeth and Pluhar for triangular islands. Here, we give a much shorter proof for these results, and also for new, analogous results on toroidal and some other boards.