## The lattices of closure systems, closure operators, and implicational systems on a finite set: A survey.(English)Zbl 1026.06008

Discrete Appl. Math. 127, No. 2, 241-269 (2003); erratum ibid. 145, No. 3, 333 (2005).
Summary: 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.

### MSC:

 06A15 Galois correspondences, closure operators (in relation to ordered sets) 68P15 Database theory
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