Filters and congruences in weakly complemented lattices

TL;DR

The paper addresses the structure of filters and congruences in weakly dicomplemented lattices by constructing a dual weak complementation on the filter lattice F(L) and examining the skeleton and dual skeleton. It proves that dense filters form a nearlattice and that principal filters form a dual weakly complemented lattice dually isomorphic to the original lattice, with S-filters providing a bridge between F(L) and F(S(L)). It establishes a bijection between prime filters of the dual skeleton S(L) and S-primary filters of L, and characterizes congruences generated by S-filters in distributive WCLs, including when these congruences commute. The results yield a robust framework for understanding spectral/topological representations of WDLs and distributive WCLs via filters and S-filters, with implications for simple and subdirectly irreducible structures.

Abstract

In this paper, we show that given a weakly dicomplemented lattice (WDL) $\mathcal{L}=(L; \vee, \wedge, ^Δ, ^{\nabla}, 0, 1)$, $^Δ$ induces a structure of a dual weakly complemented lattice in the lattice $(F(L), \subseteq)$ of filters of $\mathcal{L}$. We prove that the set of dense elements of $F(L)$ forms a nearlattice, and the set of principal filters of $\mathcal{L}$ forms a dual weakly complemented lattice that is dually isomorphic to the weakly complemented lattice (WCL) $(L,\wedge, \vee, ^Δ, 0, 1)$.\par Each filter of the dual skeleton $\overline{S}(L)$ of $L$ constitutes a base of some filter in $L$, called an S-filter, and it is proved that S-filters form a complete lattice isomorphic to the complete lattice of filters of $\overline{S}(L)$. S-primary filters are introduced and investigated, and it is shown that there exists a bijection between the set of prime filters of $\overline{S}(L)$ and the set of S-primary filters of $\mathcal{L}$. Furthermore, each maximal filter of a WDL $\mathcal{L}$ is a primary filter, though there exist primary filters of $\mathcal{L}$ that are not maximal.The congruences generated by filters in a distributive weakly complemented lattice are characterized. Finally, simple and subdirectly irreducible distributive weakly complemented lattices are also characterized using S-filters.

Theorems & Definitions (74)

• Definition 2.1
• Proposition 2.2
• Remark 2.3
• Proposition 2.4
• Definition 2.5
• Definition 2.6
• Remark 2.7
• Proposition 2.8
• proof
• Remark 2.9