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

Horváth, Eszter K.; Šešelja, Branimir; Tepavčević, Andreja

doi:10.2478/s11533-012-0103-x

Cited by 4 publications

(8 citation statements)

References 17 publications

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“…this lower bound is sharp, moreover if m; n 2; then ch .m; n/ Ä mCnC3 2˘; in addition for m; n 3 this bound is also sharp, all proved also in [15]. This result might lead us closer to the solution to the two-dimensional problem of height function with finite range, mentioned in the former section.…”

Section: Islands and Cuts Of Lattice Valued Functionssupporting

confidence: 55%

“…It is proved in [15] such that I rect .h / D I rect .h / and in h every island appears exactly in one cut.…”

Section: Islands and Cuts Of Lattice Valued Functionsmentioning

confidence: 98%

“…We denote by max .m; n/ the maximum number of different nonempty p-cuts of a standard rectangular height function on the rectangular table of size m n. It is obtained in [15] that max .m; n/ D m C n 1: If m; n 3 and the height function has maximally many rectangular islands, then the number of different nonempty cuts are strictly less than max .m; n/:…”

Section: Islands and Cuts Of Lattice Valued Functionsmentioning

confidence: 99%

“…It is an interesting fact proved in [15] that if m 3 and n 3 and a height function h W f1; 2; :::; mg f1; 2; :::; ng ! N has maximally many rectangular islands, then it has exactly two maximal islands.…”

Section: Islands and Cuts Of Lattice Valued Functionsmentioning

confidence: 99%

See 3 more Smart Citations

Islands: from coding theory to enumerative combinatorics and to lattice theory - overview and open problems

Horváth¹

2013

MMN

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Section: Islands and Cuts Of Lattice Valued Functionssupporting

confidence: 55%

“…It is proved in [15] such that I rect .h / D I rect .h / and in h every island appears exactly in one cut.…”

Section: Islands and Cuts Of Lattice Valued Functionsmentioning

confidence: 98%

Section: Islands and Cuts Of Lattice Valued Functionsmentioning

confidence: 99%

Section: Islands and Cuts Of Lattice Valued Functionsmentioning

confidence: 99%

See 2 more Smart Citations

Islands: from coding theory to enumerative combinatorics and to lattice theory - overview and open problems

Horváth¹

2013

MMN

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“…If we allow only a given finite subset of the reals as possible heights, then the problem of determining the maximum number of islands becomes considerably more difficult; see, e.g. [13,17,22]. Islands also appear naturally as cuts of lattice-valued functions [16]; furthermore, order-theoretic properties of systems of islands proved to be of interest on their own, and they have been investigated in lattices and partially ordered sets [4,6,12].…”

Section: Introductionmentioning

confidence: 99%

A general framework for island systems

Foldes¹,

Horváth²,

Radeleczki³

et al. 2015

Acta Sci. Math.

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The notion of an island defined on a rectangular board is an elementary combinatorial concept that occurred first in [3]. Results of [3] were starting points for investigations exploring several variations and various aspects of this notion. In this paper we introduce a general framework for islands that subsumes all earlier studied concepts of islands on finite boards, moreover we show that the prime implicants of a Boolean function, the formal concepts of a formal context, convex subgraphs of a simple graph, and some particular subsets of a projective plane also fit into this framework. We axiomatize those cases where islands have the property of being pairwise comparable or disjoint, or they are distant, introducing the notion of a connective island domain and of a proximity domain, respectively. In the general case the maximal systems of islands are characterised by using the concept of an admissible system. We also characterise all possible island systems in the case of connective island domains and proximity domains. 2010 Mathematics Subject Classification. 06A06.

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CD-independent subsets in meet-distributive lattices

Czédli

2013

Acta Math. Hungar.

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A subset X of a finite lattice L is CD-independent if the meet of any two incomparable elements of X equals 0. In 2009, Czédli, Hartmann and Schmidt proved that any two maximal CD-independent subsets of a finite distributive lattice have the same number of elements. In this paper, we prove that if L is a finite meet-distributive lattice, then the size of every CDindependent subset of L is at most the number of atoms of L plus the length of L. If, in addition, there is no three-element antichain of meet-irreducible elements, then we give a recursive description of maximal CD-independent subsets. Finally, to give an application of CD-independent subsets, we give a new approach to count islands on a rectangular board.Date: July 9, 2013 2000 Mathematics Subject Classification. Primary 06C10; secondary 05A05 and 05B25 .

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

Cited by 4 publications

References 17 publications

Islands: from coding theory to enumerative combinatorics and to lattice theory - overview and open problems

Islands: from coding theory to enumerative combinatorics and to lattice theory - overview and open problems

A general framework for island systems

CD-independent subsets in meet-distributive lattices

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