Simple, subdirectly irreducible weakly dicomplemented lattices

TL;DR

This paper explores subclasses of weakly dicomplemented lattices, focusing on their filters, ideals, and congruences, and characterizes simple and subdirectly irreducible structures using normal filters.

Contribution

It introduces new characterizations of normal filters and ideals, and establishes their role in understanding the structure and congruences of weakly dicomplemented lattices.

Findings

Normal filters form a complete lattice, not a sublattice of all filters.

Lattice of normal filters is isomorphic to that of normal ideals.

Congruences generated by normal filters are permutable.

Abstract

In this work, we exhibit several subclasses of weakly dicomplemented lattices (WDLs) based on their skeletons and dual skeletons. We investigate normal filters (resp. ideals) and show that the set of normal filters (resp. ideals) forms a complete lattice, which is not a sublattice of the lattice of all filters (ideals). The normal filter (ideal) generated by a subset and the join of two normal filters (resp. ieals) are characterized. We further prove that the lattice of normal filters is isomorphic to the lattice of normal ideals, and that the only class of filters (or ideals) that generate a congruence in WDLs is the class of normal filters. For distributive WDLs, the congruences generated by filters are characterized. Using normal filters, we characterize simple, subdirectly irreducible, and regular WDLs. Moreover, it is shown that the congruences generated by normal filters are permutable, and that regular distributive WDLs are congruence-permutable and verify the congruence extension property (CEP). Finally, we prove that, under certain conditions, the lattice of normal filters is isomorphic to the lattice of filters of the Boolean center of a distributive WDL. It is also established that the lattice of normal filters of a WDL $L$ embeds into the lattice of normal filters of the power $L^{X}$ of $L$.