On the $E$-base of Finite Lattices: Semidistributive, Modular, and Geometric Lattices

TL;DR

This paper investigates the conditions under which the $E$-base, a new implicational base from free lattice studies, accurately represents closure spaces in various classes of finite lattices, including semidistributive, modular, and geometric lattices.

Contribution

It proves that the $E$-base is valid and minimal for semidistributive lattices and characterizes modular and geometric lattices with valid $E$-bases, also showing any lattice embeds into one with a valid $E$-base.

Findings

$E$-base is valid and minimal for semidistributive lattices.

Characterization of modular and geometric lattices with valid $E$-base.

Any lattice can be embedded into a lattice with valid $E$-base.

Abstract

Implicational bases are a well-known representation of closure spaces and their closure lattices. This representation is not unique, though, and a closure space usually admits multiple bases. Among these, the canonical base, the canonical direct base as well as the $D$-base aroused significant attention due to their structural and algorithmic properties. Recently, a new base has emerged from the study of free lattices: the $E$-base. It is a refinement of the $D$-base that, unlike the aforementioned implicational bases, does not always accurately represent its associated closure space. This leads to an intriguing question: for which classes of (closure) lattices do closure spaces have valid $E$-base? Lower-bounded lattices are known to form such a class. In this paper, we prove that for semidistributive lattices, the $E$-base is both valid and minimum. We also characterize those modular and geometric lattices that have valid $E$-base. Finally, we prove that any lattice is a sublattice of a lattice with valid $E$-base.