The E-base of finite semidistributive closure lattices [LSC]
Simon Vilmin
LIS, Aix-Marseille
https://simvil.github.io/
Date(s) : 20/03/2025 iCal
11h00 - 12h30
A (finite) closure space is a pair consisting of a (finite) ground set X
and a closure operator on X. When ordered by inclusion, the fixpoints of
the closure operator—closed sets—are known to form a (closure)
lattice. Closure spaces and their associated lattices arise in several
fields of computer science, usually by means of implicit representations
such as implicational bases (IBs). An implicational base over X is a
collection of statements A –> B, aka implications, where A and B are
subsets X. While an IB represents a unique closure space, a closure
space can be represented via several different IBs. As a consequence, a
number of particular IBs have been proposed in the literature. Recently,
a new specimen coming from the study of free lattices has been
introduced in the zoo of IBs: the E-base. Unlike other commonly studied
IBs such as the canonical base or the D-base, though, the E-base does
not always faithfully represent its associated closure space. This leads
to an intriguing question: for which classes of (closure) lattices do
closure spaces have a valid E-base? Lower bounded lattices are known to
form such a class. Here, we will show that semidistributive lattices
also have valid E-base.
This result comes from a recent collaboration with Kira Adaricheva:
https://arxiv.org/abs/2502.04146.
Emplacement
Luminy - LIS, salle 04.05
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