The lattices of closure systems, closure operators, and implicational systems on a finite set: a survey

Abstract

Closure systems (i.e. families of subsets of a set S containing S and closed by set intersection) or, equivalently, closure operators and full implicational systems appear in many fields in pure or applied mathematics and computer science. We present a survey of properties of the lattice of closure systems on a finite set S with proofs of the more significant results. In particular we show that this lattice is atomistic and lower bounded and that there exists a canonical basis for the representation of any closure system by “implicational” closure systems. Since the lattices of closure operators and of full implicational systems are anti-isomorphic with the lattice of closure systems they have the dual properties.