Filters and congruences in weakly complemented lattices
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915535188393984 |
|---|---|
| author | Jeufack, Yannick Léa Tenkeu Kwuida, Leonard |
| author_facet | Jeufack, Yannick Léa Tenkeu Kwuida, Leonard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04960 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Filters and congruences in weakly complemented lattices Jeufack, Yannick Léa Tenkeu Kwuida, Leonard Logic Primary: 06B05, Secondary: 06B010, 06B23, 06B75 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. |
| title | Filters and congruences in weakly complemented lattices |
| topic | Logic Primary: 06B05, Secondary: 06B010, 06B23, 06B75 |
| url | https://arxiv.org/abs/2510.04960 |