Representations for the Largest Extension of a Closure System

Abstract

An extension of a closure system on a finite set S is a closure system on the same set S containing the first one as a sublattice. A closure system can be represented in different ways, e.g. by an implicational base or by the set of its meet-irreducible elements. When a closure system is described by an implicational base, we provide a characterization of the implicational base for the largest extension. We also show that the largest extension can be handled by a small modification of the implicational base of the input closure system. This answers a question asked in Yoshikawa et al. (J. Math. Psychol. 77, 82–93, 2017). Second, we are interested in computing the largest extension when the closure system is given by the set of all its meet-irreducible elements. We give an incremental polynomial time algorithm to compute the largest extension of a closure system, and left open whether the number of meet-irreducible elements grows exponentially.