  The notion of an island defined on a rectangular board is an elementary
combinatorial concept that occurred first in [G. Cz\'edli, The number of
rectangular islands by means of distributive lattices, European J. Combin. 30
(2009), 208-215]. Results of this paper 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 comparable or disjoint property, 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 island domains and proximity domains.
