Filters and congruences in weakly complemented lattices

Abstract

In this paper, we show that given a weakly dicomplemented lattice (WDL) $\mathcal{L}=(L; \vee, \wedge, ^{\Delta}, ^{\nabla}, 0, 1)$, $^{\Delta}$ 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) $(L,\wedge, \vee, ^{\Delta}, 0, 1)$.\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.