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PreprintCC BY 4.0arXiv

A generalization of pseudocomplementation in semilattices, lattices and posets with 0

  • Ivan Chajda
  • Helmut Länger
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Abstract

Pseudocomplemented lattices satisfy a lot of nice properties and were studied by a number of algebraists. However, even when a poset P with 0 is not pseudocomplemented, we can assign to every element a of P the set ap of all maximal elements being orthogonal to a. This assignment p is a non-trivial operator provided P satisfies the so-called Ascending Chain Condition (ACC). In particular, we investigate this operator when the poset P in question is a semilattice or a lattice. We characterize this operator and describe its properties that turn out to be similar to the corresponding properties of pseudocomplementation. Moreover, we introduce an equivalence relation Θp on P induced by p similarly as it was done by V. I. Glivenko for distributive pseudocomplemented lattices. Finally, we investigate for which lattices L with 0 satisfying the ACC this equivalence relation turns out to be a congruence and when the quotient lattice L/Θp is complemented respectively modular.

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Cite

BibTeXRIS
BibTeX
@misc{chajda2026generalization,
  title = {{A generalization of pseudocomplementation in semilattices, lattices and posets with 0}},
  author = {Ivan Chajda and Helmut Länger},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35988},
  eprint = {2610.35988},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35988},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A generalization of pseudocomplementation in semilattices, lattices and posets with 0
AU  - Ivan Chajda
AU  - Helmut Länger
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35988
UR  - https://arxiv.org/abs/2610.35988
AB  - Pseudocomplemented lattices satisfy a lot of nice properties and were studied by a number of algebraists. However, even when a poset P with 0 is not pseudocomplemented, we can assign to every element a of P the set a^p of all maximal elements being orthogonal to a. This assignment ^p is a non-trivial operator provided P satisfies the so-called Ascending Chain Condition (ACC). In particular, we investigate this operator when the poset P in question is a semilattice or a lattice. We characterize this operator and describe its properties that turn out to be similar to the corresponding properties of pseudocomplementation. Moreover, we introduce an equivalence relation Θ_p on P induced by ^p similarly as it was done by V. I. Glivenko for distributive pseudocomplemented lattices. Finally, we investigate for which lattices L with 0 satisfying the ACC this equivalence relation turns out to be a congruence and when the quotient lattice L/Θ_p is complemented respectively modular.
ER  -