Cut approach to islands in rectangular fuzzy relations

Cardinality of height function’s range in case of maximally many rectangular islands — computed by cuts

“…Moreover, in [10] we started to investigate rectangular islands by cuts which we continue in the present paper. This approach might help to get closer to the solution of the above mentioned open problem and also it gives some interconnection of the lattice method and tree-graph method of [1].…”

“…Hence, analyzing cuts one can get information about height functions and thus reveal combinatorial properties of rectangular islands on the board. Results from [10] for the co-domain [0 1] (real interval) remain valid; e.g, dealing with islands, we can use rectangular height functions only.…”

“…More details can be found, e.g., in [6,10,19,20]. We take the whole board to be a rectangular island (of size × ).…”

“…The height function is called a rectangular height function if for every ∈ N, every nonempty -cut of is a union of distant rectangles. There is a characterization theorem in [10] for rectangular height functions with co-domain [0 1]; the analogous theorem is valid for rectangular height functions with co-domain N (the proof is similar).…”

“…In [10] an algorithm is presented for constructing rectangular height function having the same rectangular islands as the given height function. The next statement goes one step further: Lemma 3.1.…”