Simple, subdirectly irreducible weakly dicomplemented lattices
Yannick Lea Tenkeu Jeufack, Leonard Kwuida
TL;DR
This work develops a structural theory for weakly dicomplemented lattices ($WDL$s) centered on normal filters and their generated congruences. It proves that the lattice of normal filters $NF(L)$ is canonically isomorphic to the lattice of normal ideals $NI(L)$, and embeds into the lattice of filters on the Boolean center under finiteness conditions; normal filters are shown to capture the essential congruences. In distributive $WDL$s, all congruences arise from normal filters and these congruences permute; the theory yields a congruence-extension property (CEP) for distributive $WDL$s and provides a clear criterion for simplicity and subdirect irreducibility via atoms in $NF(L)$. The paper also introduces several new subclasses of $WDL$s based on skeletons and dual skeletons, demonstrates a range of examples, and analyzes quotient and power constructions to connect with concept algebras and Formal Concept Analysis. These results offer a unified framework for decomposing and classifying $WDL$s with potential applications in lattice-theoretic analysis and related algebraic structures.
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 $L$.