On Nontrivial Weak Dicomplementations and the Lattice Congruences that Preserve Them

TL;DR

This paper investigates the structure and number of lattice congruences in weakly complemented lattices, identifying the maximum counts for finite cases and characterizing their structures, with implications for weakly dicomplemented lattices.

Contribution

It determines the maximum number of congruences in finite (dual) weakly complemented lattices and characterizes their structures, extending results to weakly dicomplemented lattices.

Findings

Largest congruence counts for n-element lattices are 2^{n-2}+1, 2^{n-3}+1, 5*2^{n-6}+1, 2^{n-4}+1 for n≥7.

Structures of lattices with these maximum congruences are characterized.

Results extend to weakly dicomplemented lattices.

Abstract

We study the existence of nontrivial and of representable (dual) weak complementations, along with the lattice congruences that preserve them, in different constructions of bounded lattices, then use this study to determine the finite (dual) weakly complemented lattices with the largest numbers of congruences, along with the structures of their congruence lattices. It turns out that, if $n\geq 7$ is a natural number, then the four largest numbers of congruences of the $n$--element (dual) weakly complemented lattices are: $2^{n-2}+1$, $2^{n-3}+1$, $5\cdot 2^{n-6}+1$ and $2^{n-4}+1$. For smaller numbers of elements, several intermediate numbers of congruences appear between the elements of this sequence. After determining these numbers, along with the structures of the (dual) weakly complemented lattices having these numbers of congruences, we derive a similar result for weakly dicomplemented lattices.