A general framework for island systems
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|Julkaisu||Acta Universitatis Szegediensis: Acta Scientiarum Mathematicarum|
|DOI - pysyväislinkit|
|Tila||Julkaistu - 2015|
The notion of an island defined on a rectangular board is an elementary combinatorial concept that occurred first in . Results of  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.