Abstract. The notion of an island defined on a rectangular board is an elementary combinatorial concept that occurred first in [czedli]. Results of [czedli] 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.

DOI: 10.14232/actasm-013-279-7

AMS Subject Classification (1991): 06A06

Keyword(s): island system, height function, CD-independent and CDW-independent sets, admissible system, distant system, island domain, proximity domain, point-to-set proximity relation, prime implicant, formal concept, convex subgraph, connected subgraph, projective plane

Received May 8, 2013, and in revised form January 9, 2014. (Registered under 29/2013.)