The Lattice of Closure Systems, Closure Operators and Implicational Systems on a Finite Set : A Survey.

We present a survey of properties of the lattice of closure systems (families of subsets of a set S containing S and closed by set intersection) on a finite set S with proofs of the more significant results. In particular, we prove that this lattice is atomistic and lower bounded and that there exists a canonical basis allowing to represent any closure system by "implicational" closure systems. The notion of closure system has many cryptomorphic versions, especially the notions of closure operator and of (full) implicational system, occuring in many fields of pure or applied mathematics and of computer science.