A general framework for island systems

TL;DR

This paper introduces a comprehensive framework for island systems on finite boards, unifying various concepts like Boolean function implicants, formal concepts, and convex subgraphs, with axioms for their properties.

Contribution

It generalizes existing island concepts, providing axioms and characterizations for maximal systems and different domain types, unifying multiple mathematical structures.

Findings

Unified framework for island systems on finite boards

Characterization of maximal island systems via admissible systems

Axiomatization of properties like disjointness and proximity

Abstract

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.